research topics in mathematics for phd

251+ Math Research Topics [2024 Updated]

Mathematics, often dubbed as the language of the universe, holds immense significance in shaping our understanding of the world around us. It’s not just about crunching numbers or solving equations; it’s about unraveling mysteries, making predictions, and creating innovative solutions to complex problems. In this blog, we embark on a journey into the realm of math research topics, exploring various branches of mathematics and their real-world applications.

How Do You Write A Math Research Topic?

Writing a math research topic involves several steps to ensure clarity, relevance, and feasibility. Here’s a guide to help you craft a compelling math research topic:

• Identify Your Interests: Start by exploring areas of mathematics that interest you. Whether it’s pure mathematics, applied mathematics, or interdisciplinary topics, choose a field that aligns with your passion and expertise.
• Narrow Down Your Focus: Mathematics is a broad field, so it’s essential to narrow down your focus to a specific area or problem. Consider the scope of your research and choose a topic that is manageable within your resources and time frame.
• Review Existing Literature: Conduct a thorough literature review to understand the current state of research in your chosen area. Identify gaps, controversies, or unanswered questions that could form the basis of your research topic.
• Formulate a Research Question: Based on your exploration and literature review, formulate a clear and concise research question. Your research question should be specific, measurable, achievable, relevant, and time-bound (SMART).
• Consider Feasibility: Assess the feasibility of your research topic in terms of available resources, data availability, and research methodologies. Ensure that your topic is realistic and achievable within the constraints of your project.
• Consult with Experts: Seek feedback from mentors, advisors, or experts in the field to validate your research topic and refine your ideas. Their insights can help you identify potential challenges and opportunities for improvement.
• Refine and Iterate: Refine your research topic based on feedback and further reflection. Iterate on your ideas to ensure clarity, coherence, and relevance to the broader context of mathematics research.
• Craft a Title: Once you have finalized your research topic, craft a compelling title that succinctly summarizes the essence of your research. Your title should be descriptive, engaging, and reflective of the key themes of your study.
• Write a Research Proposal: Develop a comprehensive research proposal outlining the background, objectives, methodology, and expected outcomes of your research. Your research proposal should provide a clear roadmap for your study and justify the significance of your research topic.

By following these steps, you can effectively write a math research topic that is well-defined, relevant, and poised to make a meaningful contribution to the field of mathematics.

251+ Math Research Topics: Beginners To Advanced

• Prime Number Distribution in Arithmetic Progressions
• Diophantine Equations and their Solutions
• Applications of Modular Arithmetic in Cryptography
• The Riemann Hypothesis and its Implications
• Graph Theory: Exploring Connectivity and Coloring Problems
• Knot Theory: Unraveling the Mathematics of Knots and Links
• Fractal Geometry: Understanding Self-Similarity and Dimensionality
• Differential Equations: Modeling Physical Phenomena and Dynamical Systems
• Chaos Theory: Investigating Deterministic Chaos and Strange Attractors
• Combinatorial Optimization: Algorithms for Solving Optimization Problems
• Computational Complexity: Analyzing the Complexity of Algorithms
• Game Theory: Mathematical Models of Strategic Interactions
• Number Theory: Exploring Properties of Integers and Primes
• Algebraic Topology: Studying Topological Invariants and Homotopy Theory
• Analytic Number Theory: Investigating Properties of Prime Numbers
• Algebraic Geometry: Geometry Arising from Algebraic Equations
• Galois Theory: Understanding Field Extensions and Solvability of Equations
• Representation Theory: Studying Symmetry in Linear Spaces
• Harmonic Analysis: Analyzing Functions on Groups and Manifolds
• Mathematical Logic: Foundations of Mathematics and Formal Systems
• Set Theory: Exploring Infinite Sets and Cardinal Numbers
• Real Analysis: Rigorous Study of Real Numbers and Functions
• Complex Analysis: Analytic Functions and Complex Integration
• Measure Theory: Foundations of Lebesgue Integration and Probability
• Topological Groups: Investigating Topological Structures on Groups
• Lie Groups and Lie Algebras: Geometry of Continuous Symmetry
• Differential Geometry: Curvature and Topology of Smooth Manifolds
• Algebraic Combinatorics: Enumerative and Algebraic Aspects of Combinatorics
• Ramsey Theory: Investigating Structure in Large Discrete Structures
• Analytic Geometry: Studying Geometry Using Analytic Methods
• Hyperbolic Geometry: Non-Euclidean Geometry of Curved Spaces
• Nonlinear Dynamics: Chaos, Bifurcations, and Strange Attractors
• Homological Algebra: Studying Homology and Cohomology of Algebraic Structures
• Topological Vector Spaces: Vector Spaces with Topological Structure
• Representation Theory of Finite Groups: Decomposition of Group Representations
• Category Theory: Abstract Structures and Universal Properties
• Operator Theory: Spectral Theory and Functional Analysis of Operators
• Algebraic Number Theory: Study of Algebraic Structures in Number Fields
• Cryptanalysis: Breaking Cryptographic Systems Using Mathematical Methods
• Discrete Mathematics: Combinatorics, Graph Theory, and Number Theory
• Mathematical Biology: Modeling Biological Systems Using Mathematical Tools
• Population Dynamics: Mathematical Models of Population Growth and Interaction
• Epidemiology: Mathematical Modeling of Disease Spread and Control
• Mathematical Ecology: Dynamics of Ecological Systems and Food Webs
• Evolutionary Game Theory: Evolutionary Dynamics and Strategic Behavior
• Mathematical Neuroscience: Modeling Brain Dynamics and Neural Networks
• Mathematical Physics: Mathematical Models in Physical Sciences
• Quantum Mechanics: Foundations and Applications of Quantum Theory
• Statistical Mechanics: Statistical Methods in Physics and Thermodynamics
• Fluid Dynamics: Modeling Flow of Fluids Using Partial Differential Equations
• Mathematical Finance: Stochastic Models in Finance and Risk Management
• Option Pricing Models: Black-Scholes Model and Beyond
• Portfolio Optimization: Maximizing Returns and Minimizing Risk
• Stochastic Calculus: Calculus of Stochastic Processes and Itô Calculus
• Financial Time Series Analysis: Modeling and Forecasting Financial Data
• Operations Research: Optimization of Decision-Making Processes
• Linear Programming: Optimization Problems with Linear Constraints
• Integer Programming: Optimization Problems with Integer Solutions
• Network Flow Optimization: Modeling and Solving Flow Network Problems
• Combinatorial Game Theory: Analysis of Games with Perfect Information
• Algorithmic Game Theory: Computational Aspects of Game-Theoretic Problems
• Fair Division: Methods for Fairly Allocating Resources Among Parties
• Auction Theory: Modeling Auction Mechanisms and Bidding Strategies
• Voting Theory: Mathematical Models of Voting Systems and Social Choice
• Social Network Analysis: Mathematical Analysis of Social Networks
• Algorithm Analysis: Complexity Analysis of Algorithms and Data Structures
• Machine Learning: Statistical Learning Algorithms and Data Mining
• Deep Learning: Neural Network Models with Multiple Layers
• Reinforcement Learning: Learning by Interaction and Feedback
• Natural Language Processing: Statistical and Computational Analysis of Language
• Computer Vision: Mathematical Models for Image Analysis and Recognition
• Computational Geometry: Algorithms for Geometric Problems
• Symbolic Computation: Manipulation of Mathematical Expressions
• Numerical Analysis: Algorithms for Solving Numerical Problems
• Finite Element Method: Numerical Solution of Partial Differential Equations
• Monte Carlo Methods: Statistical Simulation Techniques
• High-Performance Computing: Parallel and Distributed Computing Techniques
• Quantum Computing: Quantum Algorithms and Quantum Information Theory
• Quantum Information Theory: Study of Quantum Communication and Computation
• Quantum Error Correction: Methods for Protecting Quantum Information from Errors
• Topological Quantum Computing: Using Topological Properties for Quantum Computation
• Quantum Algorithms: Efficient Algorithms for Quantum Computers
• Quantum Cryptography: Secure Communication Using Quantum Key Distribution
• Topological Data Analysis: Analyzing Shape and Structure of Data Sets
• Persistent Homology: Topological Invariants for Data Analysis
• Mapper Algorithm: Method for Visualization and Analysis of High-Dimensional Data
• Algebraic Statistics: Statistical Methods Based on Algebraic Geometry
• Tropical Geometry: Geometric Methods for Studying Polynomial Equations
• Model Theory: Study of Mathematical Structures and Their Interpretations
• Descriptive Set Theory: Study of Borel and Analytic Sets
• Ergodic Theory: Study of Measure-Preserving Transformations
• Combinatorial Number Theory: Intersection of Combinatorics and Number Theory
• Additive Combinatorics: Study of Additive Properties of Sets
• Arithmetic Geometry: Interplay Between Number Theory and Algebraic Geometry
• Proof Theory: Study of Formal Proofs and Logical Inference
• Reverse Mathematics: Study of Logical Strength of Mathematical Theorems
• Nonstandard Analysis: Alternative Approach to Analysis Using Infinitesimals
• Computable Analysis: Study of Computable Functions and Real Numbers
• Graph Theory: Study of Graphs and Networks
• Random Graphs: Probabilistic Models of Graphs and Connectivity
• Spectral Graph Theory: Analysis of Graphs Using Eigenvalues and Eigenvectors
• Algebraic Graph Theory: Study of Algebraic Structures in Graphs
• Metric Geometry: Study of Geometric Structures Using Metrics
• Geometric Measure Theory: Study of Measures on Geometric Spaces
• Discrete Differential Geometry: Study of Differential Geometry on Discrete Spaces
• Algebraic Coding Theory: Study of Error-Correcting Codes
• Information Theory: Study of Information and Communication
• Coding Theory: Study of Error-Correcting Codes
• Cryptography: Study of Secure Communication and Encryption
• Finite Fields: Study of Fields with Finite Number of Elements
• Elliptic Curves: Study of Curves Defined by Cubic Equations
• Hyperelliptic Curves: Study of Curves Defined by Higher-Degree Equations
• Modular Forms: Analytic Functions with Certain Transformation Properties
• L-functions: Analytic Functions Associated with Number Theory
• Zeta Functions: Analytic Functions with Special Properties
• Analytic Number Theory: Study of Number Theoretic Functions Using Analysis
• Dirichlet Series: Analytic Functions Represented by Infinite Series
• Euler Products: Product Representations of Analytic Functions
• Arithmetic Dynamics: Study of Iterative Processes on Algebraic Structures
• Dynamics of Rational Maps: Study of Dynamical Systems Defined by Rational Functions
• Julia Sets: Fractal Sets Associated with Dynamical Systems
• Mandelbrot Set: Fractal Set Associated with Iterations of Complex Quadratic Polynomials
• Arithmetic Geometry: Study of Algebraic Geometry Over Number Fields
• Diophantine Geometry: Study of Solutions of Diophantine Equations Using Geometry
• Arithmetic of Elliptic Curves: Study of Elliptic Curves Over Number Fields
• Rational Points on Curves: Study of Rational Solutions of Algebraic Equations
• Galois Representations: Study of Representations of Galois Groups
• Automorphic Forms: Analytic Functions with Certain Transformation Properties
• L-functions: Analytic Functions Associated with Automorphic Forms
• Selberg Trace Formula: Tool for Studying Spectral Theory and Automorphic Forms
• Langlands Program: Program to Unify Number Theory and Representation Theory
• Hodge Theory: Study of Harmonic Forms on Complex Manifolds
• Riemann Surfaces: One-dimensional Complex Manifolds
• Shimura Varieties: Algebraic Varieties Associated with Automorphic Forms
• Modular Curves: Algebraic Curves Associated with Modular Forms
• Hyperbolic Manifolds: Manifolds with Constant Negative Curvature
• Teichmüller Theory: Study of Moduli Spaces of Riemann Surfaces
• Mirror Symmetry: Duality Between Calabi-Yau Manifolds
• Kähler Geometry: Study of Hermitian Manifolds with Special Symmetries
• Algebraic Groups: Linear Algebraic Groups and Their Representations
• Lie Algebras: Study of Algebraic Structures Arising from Lie Groups
• Representation Theory of Lie Algebras: Study of Representations of Lie Algebras
• Quantum Groups: Deformation of Lie Groups and Lie Algebras
• Algebraic Topology: Study of Topological Spaces Using Algebraic Methods
• Homotopy Theory: Study of Continuous Deformations of Spaces
• Homology Theory: Study of Algebraic Invariants of Topological Spaces
• Cohomology Theory: Study of Dual Concepts to Homology Theory
• Singular Homology: Homology Theory Defined Using Simplicial Complexes
• Sheaf Theory: Study of Sheaves and Their Cohomology
• Differential Forms: Study of Multilinear Differential Forms
• De Rham Cohomology: Cohomology Theory Defined Using Differential Forms
• Morse Theory: Study of Critical Points of Smooth Functions
• Symplectic Geometry: Study of Symplectic Manifolds and Their Geometry
• Floer Homology: Study of Symplectic Manifolds Using Pseudoholomorphic Curves
• Gromov-Witten Invariants: Invariants of Symplectic Manifolds Associated with Pseudoholomorphic Curves
• Mirror Symmetry: Duality Between Symplectic and Complex Geometry
• Calabi-Yau Manifolds: Ricci-Flat Complex Manifolds
• Moduli Spaces: Spaces Parameterizing Geometric Objects
• Donaldson-Thomas Invariants: Invariants Counting Sheaves on Calabi-Yau Manifolds
• Algebraic K-Theory: Study of Algebraic Invariants of Rings and Modules
• Homological Algebra: Study of Homology and Cohomology of Algebraic Structures
• Derived Categories: Categories Arising from Homological Algebra
• Stable Homotopy Theory: Homotopy Theory with Stable Homotopy Groups
• Model Categories: Categories with Certain Homotopical Properties
• Higher Category Theory: Study of Higher Categories and Homotopy Theory
• Higher Topos Theory: Study of Higher Categorical Structures
• Higher Algebra: Study of Higher Categorical Structures in Algebra
• Higher Algebraic Geometry: Study of Higher Categorical Structures in Algebraic Geometry
• Higher Representation Theory: Study of Higher Categorical Structures in Representation Theory
• Higher Category Theory: Study of Higher Categorical Structures
• Homotopical Algebra: Study of Algebraic Structures in Homotopy Theory
• Homotopical Groups: Study of Groups with Homotopical Structure
• Homotopical Categories: Study of Categories with Homotopical Structure
• Homotopy Groups: Algebraic Invariants of Topological Spaces
• Homotopy Type Theory: Study of Foundations of Mathematics Using Homotopy Theory

In conclusion, the world of mathematics is vast and multifaceted, offering endless opportunities for exploration and discovery. Whether delving into the abstract realms of pure mathematics or applying mathematical principles to solve real-world problems, mathematicians play a vital role in advancing human knowledge and shaping the future of our world.

By embracing diverse math research topics and interdisciplinary collaborations, we can unlock new possibilities and harness the power of mathematics to address the challenges of today and tomorrow. So, let’s embark on this journey together as we unravel the mysteries of numbers and explore the boundless horizons of mathematical inquiry.

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• Doing a PhD in Mathematics
• Doing a PhD

What Does a PhD in Maths Involve?

Maths is a vast subject, both in breadth and in depth. As such, there’s a significant number of different areas you can research as a math student. These areas usually fall into one of three categories: pure mathematics, applied mathematics or statistics. Some examples of topics you can research are:

• Number theory
• Numerical analysis
• String theory
• Random matrix theory
• Graph theory
• Quantum mechanics
• Statistical forecasting
• Matroid theory
• Control theory

Besides this, because maths focuses on addressing interdisciplinary real-world problems, you may work and collaborate with other STEM researchers. For example, your research topic may relate to:

• Biomechanics and transport processes
• Evidence-based medicine
• Fluid dynamics
• Financial mathematics
• Machine learning
• Theoretical and Computational Optimisation

What you do day-to-day will largely depend on your specific research topic. However, you’ll likely:

• Continually read literature – This will be to help develop your knowledge and identify current gaps in the overall body of knowledge surrounding your research topic.
• Undertake research specific to your topic – This can include defining ideas, proving theorems and identifying relationships between models.
• Collect and analyse data – This could comprise developing computational models, running simulations and interpreting forecasts etc.
• Liaise with others – This could take many forms. For example, you may work shoulder-to-shoulder with individuals from different disciplines supporting your research, e.g. Computer scientists for machine learning-based projects. Alternatively, you may need frequent input from those who supplied the data for your research, e.g. Financial institutions or biological research colleagues.
• Attend a wide range of lectures, seminars and events.

Browse PhD Opportunities in Mathematics

Application of artificial intelligence to multiphysics problems in materials design, study of the human-vehicle interactions by a high-end dynamic driving simulator, physical layer algorithm design in 6g non-terrestrial communications, machine learning for autonomous robot exploration, detecting subtle but clinically significant cognitive change in an ageing population, how long does it take to get a phd in maths.

The average programme duration for a mathematics PhD in the UK is 3 to 4 years for a full-time studying. Although not all universities offer part-time maths PhD programmes, those that do have a typical programme duration of 5 to 7 years.

Again, although the exact arrangement will depend on the university, most maths doctorates will require you to first register for an MPhil . At the end of your first year, your supervisor will assess your progress to decide whether you should be registered for a PhD.

Additional Learning Modules

Some Mathematics departments will require you to enrol on to taught modules as part of your programme. These are to help improve your knowledge and understanding of broader subjects within your field, for example, Fourier Analysis, Differential Geometry and Riemann Surfaces. Even if taught modules aren’t compulsory in several universities, your supervisor will still encourage you to attend them for your development.

Most UK universities will also have access to specialised mathematical training courses. The most common of these include Pure Mathematics courses hosted by Mathematics Access Grid Conferencing ( MAGIC ) and London Taught Course Centre ( LTCC ) and Statistics courses hosted by Academy for PhD Training in Statistics ( APTS ).

What Are the Typical Entry Requirements for A PhD in Maths?

In the UK, the typical entry requirements for a Maths PhD is an upper second-class (2:1) Master’s degree (or international equivalent) in Mathematics or Statistics [1] .

However, there is some variation on this. From writing, the lowest entry requirement is an upper second-class (2:1) Bachelor’s degree in any math-related subject. The highest entry requirement is a first-class (1st) honours Master’s degree in a Mathematics or Statistics degree only.

It’s worth noting if you’re applying to a position which comes with funding provided directly by the Department, the entry requirements will usually be on the higher side because of their competitiveness.

In terms of English Language requirements, most mathematics departments require at least an overall IELTS (International English Language Testing System) score of 6.5, with no less than 6.0 in each individual subtest.

Tips to Consider when Making Your Application

When applying to any mathematics PhD, you’ll be expected to have a good understanding of both your subject field and the specific research topic you are applying to. To help show this, it’s advisable that you demonstrate recent engagement in your research topic. This could be by describing the significance of a research paper you recently read and outlining which parts interested you the most, and why. Additionally, you can discuss a recent mathematics event you attended and suggest ways in how what you learnt might apply to your research topic.

As with most STEM PhDs, most maths PhD professors prefer you to discuss your application with them directly before putting in a formal application. The benefits of this is two folds. First, you’ll get more information on what their department has to offer. Second, the supervisor can better discover your interest in the project and gauge whether you’d be a suitable candidate. Therefore, we encourage you to contact potential supervisors for positions you’re interested in before making any formal applications.

How Much Does a Maths PhD Typically Cost?

The typical tuition fee for a PhD in Maths in the UK is £4,407 per year for UK/EU students and £20,230 per year for international students. This, alongside the range in tuition fees you can expect, is summarised below:

Note: The above tuition fees are based on 12 UK Universities [1]  for 2020/21 Mathematic PhD positions. The typical fee has been taken as the median value.

In addition to the above, it’s not unheard of for research students to be charged a bench fee. In case you’re unfamiliar with a bench fee, it’s an annual fee additional to your tuition, which covers the cost of specialist equipment or resources associated with your research. This can include the upkeep of supercomputers you may use, training in specialist analysis software, or travelling to conferences. The exact fee will depend on your specific research topic; however, it should be minimal for most mathematic projects.

What Specific Funding Opportunities Are There for A PhD in Mathematics?

Alongside the usual funding opportunities available to all PhD Research students such as doctoral loans, departmental scholarships, there are a few other sources of funding available to math PhD students. Examples of these include:

You can find more information on these funding sources here: DiscoverPhDs funding guide .

What Specific Skills Do You Gain from Doing a PhD in Mathematics?

A doctorate in Mathematics not only demonstrates your commitment to continuous learning, but it also provides you with highly marketable skills. Besides subject-specific skills, you’ll also gain many transferable skills which will prove useful in almost all industries. A sample of these skills is listed below.

• Logical ability to consider and analyse complex issues,
• Commitment and persistence towards reaching research goals,
• Outstanding verbal and written skills,
• Strong attention to detail,
• The ability to liaise with others from unique disciple backgrounds and work as part of a team
• Holistic deduction and reasoning skills,
• Forming and explaining mathematical and logical solutions to a wide range of real-world problems,
• Exceptional numeracy skills.

What Jobs Can You Get with A Maths PhD?

One of the greatest benefits maths PostDocs will have is the ability to pursue a wide range of career paths. This is because all sciences are built on core principles which, to varying extents, are supported by the core principles of mathematics. As a result, it’s not uncommon to ask students what path they intend to follow after completing their degree and receive entirely different answers. Although not extensive by any means, the most common career paths Math PostDocs take are listed below:

• Academia – Many individuals teach undergraduate students at the university they studied at or ones they gained ties to during their research. This path is usually the preferred among students who want to continue focusing on mathematical theories and concepts as part of their career.
• Postdoctoral Researcher – Others continue researching with their University or with an independent organisation. This can be a popular path because of the opportunities it provides in collaborative working, supervising others, undertaking research and attending conferences etc.
• Finance – Because of their deepened analytical skills, it’s no surprise that many PostDocs choose a career in finance. This involves working for some of the most significant players in the financial district in prime locations including London, Frankfurt and Hong Kong. Specific job titles can include Actuarial, Investment Analyst or Risk Modeller.
• Computer Programming – Some students whose research involves computational mathematics launch their career as a computer programmer. Due to their background, they’ll typically work on specialised projects which require high levels of understanding on the problem at hand. For example, they may work with physicists and biomedical engineers to develop a software package that supports their more complex research.
• Data Analyst – Those who enjoy number crunching and developing complex models often go into data analytics. This can involve various niches such as forecasting or optimisation, across various fields such as marketing and weather.

What Are Some of The Typical Employers Who Hire Maths PostDocs?

As mentioned above, there’s a high demand for skilled mathematicians and statisticians across a broad range of sectors. Some typical employers are:

• Education – All UK and international universities
• Governments – STFC and Department for Transport
• Healthcare & Pharmaceuticals – NHS, GSK, Pfizer
• Finance & Banking – e.g. Barclays Capital, PwC and J. P. Morgan
• Computing – IBM, Microsoft and Facebook
• Engineering – Boeing, Shell and Dyson

The above is only a small selection of employers. In reality, mathematic PostDocs can work in almost any industry, assuming the role is numerical-based or data-driven.

How Much Can You Earn with A PhD in Maths?

As a mathematics PhD PostDoc, your earning potential will mostly depend on your chosen career path. Due to the wide range of options, it’s impossible to provide an arbitrary value for the typical salary you can expect.

However, if you pursue one of the below paths or enter their respective industry, you can roughly expect to earn [3] :

Academic Lecturer

• Approximately £30,000 – £35,000 starting salary
• Approximately £40,000 with a few years experience
• Approximately £45,000 – £55,000 with 10 years experience
• Approximately £60,000 and over with significant experience and a leadership role. Certain academic positions can earn over £80,000 depending on the management duties.

Actuary or Finance

• Approximately £35,000 starting salary
• Approximately £45,000 – £55,000 with a few years experience
• Approximately £70,000 and over with 10 years experience
• Approximately £180,000 and above with significant experience and a leadership role.

Aerospace or Mechanical Engineering

• Approximately £28,000 starting salary
• Approximately £35,000 – £40,000 with a few years experience
• Approximately £60,000 and over with 10 years experience

Data Analyst

• Approximately £45,000 – £50,000 with a few years experience
• Approximately £90,000 and above with significant experience and a leadership role.

Again, we stress that the above are indicative values only. Actual salaries will depend on the specific organisation and position and responsibilities of the individual.

Facts and Statistics About Maths PhD Holders

The below chart provides useful insight into the destination of Math PostDocs after completing their PhD. The most popular career paths from other of highest to lowest is education, information and communication, finance and scientific research, manufacturing and government.

Note: The above chart is based on ‘UK Higher Education Leavers’ data [2] between 2012/13 and 2016/17 and contains a data size of 200 PostDocs. The data was obtained from the Higher Education Statistics Agency ( HESA ).

Which Noteworthy People Hold a PhD in Maths?

Alan turing.

Alan Turing was a British Mathematician, WW2 code-breaker and arguably the father of computer science. Alongside his lengthy list of achievements, Turning achieved a PhD in Mathematics at Princeton University, New Jersey. His thesis titled ‘Systems of Logic Based on Ordinals’ focused on the concepts of ordinal logic and relative computing; you can read it online here . To this day, Turning pioneering works continues to play a fundamental role in shaping the development of artificial intelligence (AI).

Ruth Lawrence

Ruth Lawrence is a famous British–Israeli Mathematician well known within the academic community. Lawrence earned her PhD in Mathematics from Oxford University at the young age of 17! Her work focused on algebraic topology and knot theory; you can read her interesting collection of research papers here . Among her many contributions to Maths, her most notable include the representation of the braid groups, more formally known as Lawrence–Krammer representations.

Emmy Noether

Emmy Noether was a German mathematician who received her PhD from the University of Erlangen, Germany. Her research has significantly contributed to both abstract algebra and theoretical physics. Additionally, she proved a groundbreaking theorem important to Albert Einstein’s general theory of relativity. In doing so, her theorem, Noether’s theorem , is regarded as one of the most influential developments in physics.

Other Useful Resources

Institute of Mathematics and its Applications (IMA) – IMA is the UK’s professional body for mathematicians. It contains a wide range of useful information, from the benefits of further education in Maths to details on grants and upcoming events.

Maths Careers – Math Careers is a site associated with IMA that provides a wide range of advice to mathematicians of all ages. It has a section dedicated to undergraduates and graduates and contains a handful of information about progressing into research.

Resources for Graduate Students – Produced by Dr Mak Tomford, this webpage contains an extensive collection of detailed advice for Mathematic PhD students. Although the site uses US terminology in places, don’t let that put you off as this resource will prove incredibly helpful in both applying to and undertaking your PhD.

Student Interviews – Still wondering whether a PhD is for you? If so, our collection of PhD interviews would be a great place to get an insider perspective. We’ve interviewed a wide range of PhD students across the UK to find out what doing a PhD is like, how it’s helped them and what advice they have for other prospective students who may be thinking of applying to one. You can read our insightful collection of interviews here .

[1] Universities used to determine the typical (median) and range of entry requirements and tuition fees for 2020/21 Mathematics PhD positions.

• http://www.lse.ac.uk/study-at-lse/Graduate/Degree-programmes-2020/MPhilPhD-Mathematics
• https://www.ox.ac.uk/admissions/graduate/courses/dphil-mathematics?wssl=1
• https://www.graduate.study.cam.ac.uk/courses/directory/mapmpdpms
• https://www.ucl.ac.uk/prospective-students/graduate/research-degrees/mathematics-mphil-phd
• http://www.bristol.ac.uk/study/postgraduate/2020/sci/phd-mathematics/
• https://www.surrey.ac.uk/postgraduate/mathematics-phd
• https://www.maths.ed.ac.uk/school-of-mathematics/studying-here/pgr/phd-application
• https://www.lancaster.ac.uk/study/postgraduate/postgraduate-courses/mathematics-phd/
• https://www.sussex.ac.uk/study/phd/degrees/mathematics-phd
• https://www.manchester.ac.uk/study/postgraduate-research/programmes/list/05325/phd-pure-mathematics/
• https://warwick.ac.uk/study/postgraduate/research/courses-2020/mathematicsphd/
• https://www.exeter.ac.uk/pg-research/degrees/mathematics/

[2] Higher Education Leavers Statistics: UK, 2016/17 – Outcomes by subject studied – https://www.hesa.ac.uk/news/28-06-2018/sfr250-higher-education-leaver-statistics-subjects

[3] Typical salaries have been extracted from a combination of the below resources. It should be noted that although every effort has been made to keep the reported salaries as relevant to Math PostDocs as possible (i.e. filtering for positions which specify a PhD qualification as one of their requirements/preferences), small inaccuracies may exist due to data availability.

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Research Areas

Analysis & pde, applied math, combinatorics, financial math, number theory, probability, representation theory, symplectic geometry & topology.

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PhD projects

Several School members offer supervision for PhD research projects in the School of Mathematics and Statistics.

Navigate via the tabs below to view project offerings by School members in the areas of Applied Mathematics, Pure Mathematics and Statistics. (This list was updated September 2022.)

Please note that this is not an exhaustive list of all potential projects and supervisors available in the School. 

Information about PhD research offerings and potential supervisors can be found in various locations. It's worth browsing the current research students list to see what research our PhD students are currently working on, and with whom.

There is also a past research students list which provides links to the theses of former students and the names of their supervisors. 

It's also recommended to browse our Staff Directory , where our staff members' names are linked to their research profiles which provide details about their areas of research and often include the topics they are open to supervising students in.

We host PhD information sessions in the School of Mathematics and Statistics twice a year. Keep an eye on our events page for session information. 

• Applied mathematics
• Pure mathematics
• Real world problem solving using dynamical systems, stochastic modelling and queueing theory for stochastic transport and signalling in cells. 
• Real & Computational Algebraic Geometry: Possible subjects include nonnegativity of real polynomials, polynomial system solving, semialgebraic sets, and algorithmic aspects of real algebraic & convex geometry.
• Polynomial & Convex Optimization: Potential topics include convex relaxations, designing algorithms, exploiting structure (e.g. sparsity), and applications in science & engineering.
• Dynamical Systems and Ergodic Theory: Projects that combine techniques from nonlinear dynamics, ergodic theory, functional analysis, differential geometry, or machine learning and can range from pure mathematical theory through to numerical techniques and applications (including ocean/atmosphere/fluids/blood flow), depending on the student.
• Optimisation: Projects are occasionally available in optimisation, mainly using either techniques from mixed integer programming to solve applied problems (e.g. transport, medicine,…) or mathematical problems arising from dynamics.
• Modelling and analysis of ocean biogeochemical cycles including isotope dynamics, inverse modelling of hydrographic data to detect climate-driven circulation changes, and analysis of large-scale ocean transport. PhD students should be highly motivated, have a strong background in applied mathematics and/or theoretical physics, and will have the opportunity to contribute to shaping their project.
• Data-Driven Multi-stage Robust Optimization: The aim of this study is to develop mathematical principles for multi-stage robust optimization problems, which can identify true optimal solutions and can readily be validated by common computer algorithms, to design associated  data-driven numerical methods to locate these solutions and to provide an advanced optimization framework to solve a wide range of real-life optimization models of multi-stage technical decision-making under evolving uncertainty.
• Semi-algebraic Global Optimization: The goal of this study is to examine classes of semi-algebraic global optimization problems, where the constraints are defined by polynomial equations and inequalities. These problems have numerous locally best solutions that are not globally best. We develop mathematical principles and numerical methods which can identify and locate the globally best solutions.
• Detection and cloaking of surface water waves created by submerged objects
• Decomposition of ocean currents into wave-like and eddy-like components
• Theory and application of Quasi-Monte Carlo methods: for high dimensional integration, approximation, and related problems.
• Computational Mathematics: with specialised topics in radial basis functions, random fields, uncertainty quantification, partial differential equations on spheres and manifolds, stochastic partial differential equations. 
• Discrete Integrable Systems: These are birational maps with particularly ordered dynamics and their study is a nice motivation for using algebraic geometry, symmetry, ideal theory and number theory in the study of dynamical systems.
• Arithmetic Dynamics:  This field is the study of iterated rational maps over the integers or rationals or over finite fields, rather than the complex or real numbers. I am particularly interested in how the usual structures present in dynamical systems over the continuum manifest themselves over discrete spaces.
• Convex geometry: Focused on the study of the facial structure of convex sets and the relations between the geometry of convex optimisation problems and performance of numerical methods. The project can be oriented towards convex algebraic geometry, experimental mathematics or classical convex analysis.
• Algebraic and Geometric Aspects of Integrable Systems: The ubiquitous nature of integrable systems is reflected in their (apparent or disguised) presence in a wide range of areas in both mathematics and (mathematical) physics. Projects focus on the algebraic and/or geometric aspects of discrete and/or continuous integrable systems, depending on the individual student's background and preferences. 
• Analysis of multiscale problems in stochastic systems: These projects will involve an analytical study of certain multiscale problems arising in Markov chains and stochastic differential equations. These projects are suited for those interested in both analysis and probability, and will employ tools from differential equations, functional analysis and stochastic processes.
• Numerical methods for sampling constrained distributions: These projects are aimed at sampling problems arising in molecular dynamics. They will deal with designing and analysing numerical schemes to sample constrained probability distributions using stochastic differential equations.  
• Fluid flow in channels with porous walls
• Mathematics education
• Nonlinear differential equations
• Difference equations
• Dynamic equations on time scales
• How many oceans are there? Using novel statistical and machine learning techniques to characterise oceanic zones and provide a blueprint for quantifying the ocean's role in a changing climate.
• How does heat get into the ocean? An investigation of the physical mechanisms that control the ocean's uptake of heat and its effect on climate.
• Making climate models work better: Developing new methods to validate and improve the inner workings of numerical climate models and improve their projections of global warming and its impacts.
• Will it mix? New perspectives on turbulence in rotating fluid flows and how we estimate mixing from observations. 
• Combinatorics
• Graph theory
• Coding theory
• Extremal set theory
• Operator algebras (von Neumann algebras)
• Mathematical physics (quantum field theory)
• Group theory
• Jones subfactor theory
• Vaughan Jones' connection between conformal field theory, Richard Thompson's groups and knot theory.
• Noncommutative algebra
• Algebraic geometry
• Quantum groups/supergroups
• The Schur-Weyl duality
• Representation Theory
• Random graphs
• Asymptotic enumeration
• Randomized combinatorial algorithms

Extremal and probabilistic combinatorics: Possible subjects therein include Ramsey theory, random graphs, positional games and hypergraphs.

• Unlikely Intersection in Number Theory and Diophantine Geometry: These are problems of showing that arithmetic “correlations" between specialisations of algebraic functions are rare unless there is some obvious reason why they happen. These “correlations” may refer to common values or to values factored into essentially the same set of prime ideals and similar. 
• Arithmetic Dynamics: This area is concerned with algebraic and arithmetic aspects of iterations of rational functions over domains of number theoretic interest. 
• Isometries, conformal mappings, and other special mappings on metric Lie groups
• Complex structures on Lie groups and their Lie algebras
• Counting integral and rational solutions to Diophantine equations and congruences. The goal is to obtain upper bounds on the number of integer solutions to some multivariate equations and congruences in variables from a given interval [M, M+N]. Similarly, for rational solutions one restricts both numerators and denominators to certain intervals.
• Kloostermania: Kloosterman and Salie sums and their applications. A classical direction in analytic number theory where the goal is to obtain new bounds on bilinear sums of Kloosterman and Salie sums and apply them to various arithmetic problems, such as the Dirichlet divisor problem in progressions.

Exponential sums and applications. This topic is about understanding the behaviour (e.g. extreme and typical values) of some most important exponential sums, in particular of Weyl sums.  

• Non-commutative functional analysis and its applications to non-commutative geometry, particularly those related to quantised calculus and index theorems.
• Singular (Dixmier) traces and their applications
• Non-commutative integration theory
• Non-commutative probability theory
• Various aspects of Banach space geometry and its applications
• Algebraic geometry (birational geometry and moduli)
• Hodge theory
• Transcendental methods in algebraic geometry 
• Motivic cohomology and algebraic K-theory - an intersection of algebraic geometry and algebraic topology
• Equivariant algebraic topology
• Extreme Value Analysis: Projects available on the modelling of the dependence of multivariate and spatial extremes, spatio-temporal modelling, high-dimensional inference. Interests in environmental/climate applications. 
• Symbolic Data Analysis: Projects available on symbol design, distributional symbols and others. Applications in big and complex data analysis.
• Ancient river systems and landscape dynamics with Bayeslands framework
• Bayesian inference and machine learning for reef modelling 
• Deep learning for the reconstruction of 3D ore-bodies for mineral exploration 
• Bayesian deep learning for protein function detection  
• COVID-19 modelling with deep learning
• Variational Bayes for surrogate assisted deep learning
• Bayesian deep learning for incomplete information
• Computational Statistics
• Event sequence data analysis
• Hidden Markov Models and State-Space Models and their inference and applications
• Financial data analysis and modelling
• Point processes and their inference and applications
• Semi- and non-parametric inference
• Bayesian statistical inference
• Computational statistics and algorithms
• Approximate Bayesian inference
• Quantile regression method
• Statistical text analyses
• Applications to climate science, social science, image analyses

*Yanan Fan is an Adjunct A/Prof in the School and is able to co-supervise students (not as primary supervisor)

• Nonparametric and semiparametric statistics: Nonparametric dependence modelling (copulas) and nonparametric functional data analysis.
• Social network analysis for epidemiology, social sciences, defence, national security, and other areas
• Statistical models for dependent categorical data
• Survey sampling (design and inference), particularly for network data
• Statistical computing, particularly MCMC-based methods
• Dependence measures
• Complex-valued random variables
• Goodness-of-fit tests
• Machine learning (with potential applications in medical imaging)
• Time series analysis
• Real-time analytics with the Raspberry Pi
• For some examples of my current projects, have a look at my  personal webpage .
• Fast and efficient model selection for high-dimensional data
• Development of efficient estimation and sampling algorithms for random graphs and spatial point processes
• Development of model compression methods for deep neural networks.

Topics include regression to the mean, interrupted time series, meta-analysis, and population attributable fractions.

Monte Carlo and Uncertainty Quantification 

• Projects on the stochastic analysis and development of modern Monte Carlo methods for uncertainty quantification, sequential Bayesian inference, high dimensional sampling, particle based Variational Inference (knowledge/experience with stochastic analysis and SDE & PDE theory highly desired).

Machine learning and generative modelling 

• Projects with a focus on (but not restricted to) medical imaging and machine learning methods for uncertainty quantification of image segmentation
• Theoretical analysis of modern machine learning methods (knowledge/experience with functional and stochastic analysis highly desired).

Mathematics of sustainability 

• Projects on stochastic games, agent based models, network science and their applications in sustainability science.
• Automating data analyses via natural language queries
• Bayesian statistics, algorithms and applications
• Building software tools, services and packages
• Data privacy and synthetic data
• Data science, theory and application
• Defence applications (nationality restrictions may apply)
• Extreme value theory and applications
• Machine learning 
• Symbolic data analysis
• Developing statistical methods for point processes
• Financial data modeling
• Computational statistics
• Analysis of capture-recapture data
• Estimation of animal abundance
• Measurement error modelling
• Model selection for multivariate data
• Non-parametric smoothing
• Statistical ecology
• High-dimensional data analysis
• Simulation-based inference
• Eco-Stats project ideas

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Department members engage in cutting-edge research on a wide variety of topics in mathematics and its applications. Topics continually evolve to reflect emerging interests and developments, but can roughly grouped into the following areas.

Algebra, Combinatorics, and Geometry

Algebra, combinatorics, and geometry are areas of very active research at the University of Pittsburgh.

Analysis and Partial Differential Equations

The research of the analysis group covers functional analysis, harmonic analysis, several complex variables, partial differential equations, and analysis on metric and Carnot-Caratheodory spaces.

Applied Analysis

The department is a leader in the analysis of systems of nonlinear differential equations and dynamical systems  that arise in modeling a variety of physical phenomena. They include problems in biology, chemistry, phase transitions, fluid flow, flame propagation, diffusion processes, and pattern formation in nonlinear stochastic partial differential equations.

Mathematical Biology

The biological world stands as the next great frontier for mathematical modeling and analysis. This group studies complex systems and dynamics arising in various biological phenomena.

Mathematical Finance

A rapidly growing area of mathematical finance is Quantitative Behavioral Finance. The high-tech boom and bust of the late 1990s followed by the housing and financial upheavals of 2008 have made a convincing case for the necessity of adopting broader assumptions in finance.

Numerical Analysis and Scientific Computing

The diversity of this group is reflected in its research interests: numerical analysis of partial differential equations , adaptive methods for scientific computing, computational methods of fluid dynamics and turbulence, numerical solution of nonlinear problems arising from porous media flow and transport, optimal control, and simulation of stochastic reaction diffusion systems.

Topology and Differential Geometry

Research in analytic topology continues in the broad area of generalized metric spaces. This group studies relativity theory and differential geometry, with emphasis on twistor methods, as well as geometric and topological aspects of quantum field theory, string theory, and M-theory.

Guide to Graduate Studies

The PhD Program The Ph.D. program of the Harvard Department of Mathematics is designed to help motivated students develop their understanding and enjoyment of mathematics. Enjoyment and understanding of the subject, as well as enthusiasm in teaching it, are greater when one is actively thinking about mathematics in one’s own way. For this reason, a Ph.D. dissertation involving some original research is a fundamental part of the program. The stages in this program may be described as follows:

• Acquiring a broad basic knowledge of mathematics on which to build a future mathematical culture and more detailed knowledge of a field of specialization.
• Choosing a field of specialization within mathematics and obtaining enough knowledge of this specialized field to arrive at the point of current thinking.
• Making a first original contribution to mathematics within this chosen special area.

Students are expected to take the initiative in pacing themselves through the Ph.D. program. In theory, a future research mathematician should be able to go through all three stages with the help of only a good library. In practice, many of the more subtle aspects of mathematics, such as a sense of taste or relative importance and feeling for a particular subject, are primarily communicated by personal contact. In addition, it is not at all trivial to find one’s way through the ever-burgeoning literature of mathematics, and one can go through the stages outlined above with much less lost motion if one has some access to a group of older and more experienced mathematicians who can guide one’s reading, supplement it with seminars and courses, and evaluate one’s first attempts at research. The presence of other graduate students of comparable ability and level of enthusiasm is also very helpful.

University Requirements

The University requires a minimum of two years of academic residence (16 half-courses) for the Ph.D. degree. On the other hand, five years in residence is the maximum usually allowed by the department. Most students complete the Ph.D. in four or five years. Please review the program requirements timeline .

There is no prescribed set of course requirements, but students are required to register and enroll in four courses each term to maintain full-time status with the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences.

Qualifying Exam

The department gives the qualifying examination at the beginning of the fall and spring terms. The qualifying examination covers algebra, algebraic geometry, algebraic topology, complex analysis, differential geometry, and real analysis. Students are required to take the exam at the beginning of the first term. More details about the qualifying exams can be found here .

Students are expected to pass the qualifying exam before the end of their second year. After passing the qualifying exam students are expected to find a Ph.D. dissertation advisor.

Minor Thesis

The minor thesis is complementary to the qualifying exam. In the course of mathematical research, students will inevitably encounter areas in which they have gaps in knowledge. The minor thesis is an exercise in confronting those gaps to learn what is necessary to understand a specific area of math. Students choose a topic outside their area of expertise and, working independently, learns it well and produces a written exposition of the subject.

The topic is selected in consultation with a faculty member, other than the student’s Ph.D. dissertation advisor, chosen by the student. The topic should not be in the area of the student’s Ph.D. dissertation. For example, students working in number theory might do a minor thesis in analysis or geometry. At the end of three weeks time (four if teaching), students submit to the faculty member a written account of the subject and are prepared to answer questions on the topic.

The minor thesis must be completed before the start of the third year in residence.

Language Exam

Mathematics is an international subject in which the principal languages are English, French, German, and Russian. Almost all important work is published in one of these four languages. Accordingly, students are required to demonstrate the ability to read mathematics in French, German, or Russian by passing a two-hour, written language examination. Students are asked to translate one page of mathematics into English with the help of a dictionary. Students may request to substitute the Italian language exam if it is relevant to their area of mathematics. The language requirement should be fulfilled by the end of the second year. For more information on the graduate program requirements, a timeline can be viewed at here .

Non-native English speakers who have received a Bachelor’s degree in mathematics from an institution where classes are taught in a language other than English may request to waive the language requirement.

Upon completion of the language exam and eight upper-level math courses, students can apply for a continuing Master’s Degree.

Teaching Requirement

Most research mathematicians are also university teachers. In preparation for this role, all students are required to participate in the department’s teaching apprenticeship program and to complete two semesters of classroom teaching experience, usually as a teaching fellow. During the teaching apprenticeship, students are paired with a member of the department’s teaching staff. Students attend some of the advisor’s classes and then prepare (with help) and present their own class, which will be videotaped. Apprentices will receive feedback both from the advisor and from members of the class.

Teaching fellows are responsible for teaching calculus to a class of about 25 undergraduates. They meet with their class three hours a week. They have a course assistant (an advanced undergraduate) to grade homework and to take a weekly problem session. Usually, there are several classes following the same syllabus and with common exams. A course head (a member of the department teaching staff) coordinates the various classes following the same syllabus and is available to advise teaching fellows. Other teaching options are available: graduate course assistantships for advanced math courses and tutorials for advanced undergraduate math concentrators.

Final Stages

How students proceed through the second and third stages of the program varies considerably among individuals. While preparing for the qualifying examination or immediately after, students should begin taking more advanced courses to help with choosing a field of specialization. Unless prepared to work independently, students should choose a field that falls within the interests of a member of the faculty who is willing to serve as dissertation advisor. Members of the faculty vary in the way that they go about dissertation supervision; some faculty members expect more initiative and independence than others and some variation in how busy they are with current advisees. Students should consider their own advising needs as well as the faculty member’s field when choosing an advisor. Students must take the initiative to ask a professor if she or he will act as a dissertation advisor. Students having difficulty deciding under whom to work, may want to spend a term reading under the direction of two or more faculty members simultaneously. The sooner students choose an advisor, the sooner they can begin research. Students should have a provisional advisor by the second year.

It is important to keep in mind that there is no technique for teaching students to have ideas. All that faculty can do is to provide an ambiance in which one’s nascent abilities and insights can blossom. Ph.D. dissertations vary enormously in quality, from hard exercises to highly original advances. Many good research mathematicians begin very slowly, and their dissertations and first few papers could be of minor interest. The ideal attitude is: (1) a love of the subject for its own sake, accompanied by inquisitiveness about things which aren’t known; and (2) a somewhat fatalistic attitude concerning “creative ability” and recognition that hard work is, in the end, much more important.

PhD in Mathematics

The PhD in Mathematics provides training in mathematics and its applications to a broad range of disciplines and prepares students for careers in academia or industry. It offers students the opportunity to work with faculty on research over a wide range of theoretical and applied topics.

Degree Requirements

The requirements for obtaining an PhD in Mathematics can be found on the associated page of the BU Bulletin .

• Courses : The courses mentioned on the BU Bulletin page can be chosen from the graduate courses we offer here . Half may be at the MA 500 level or above, but the rest must be at the MA 700 level or above. Students can also request to use courses from other departments to satisfy some of these requirements. Please contact your advisor for more information about which courses can be used in this way. All courses must be passed with a grade of B- or higher.
• Analysis (examples include MA 711, MA 713, and MA 717)
• PDEs and Dynamical Systems (examples include MA 771, MA 775, and MA 776)
• Algebra and Number Theory (examples include MA 741, MA 742, and MA 743)
• Topology (examples include MA 721, MA 722, and MA 727)
• Geometry (examples include MA 725, MA 731, and MA 745)
• Probability and Stochastic Processes (examples include MA 779, MA 780, and MA 783)
• Applied Mathematics (examples include MA 750, MA 751, and MA 770)
• Comprehensive Examination : This exam has both a written and an oral component. The written component consists of an expository paper of typically fifteen to twenty-five pages on which the student works over a period of a few months under the guidance of the advisor. The topic of the expository paper is chosen by the student in consultation with the advisor. On completion of the paper, the student takes an oral exam given by a three-person committee, one of whom is the student’s advisor. The oral exam consists of a presentation by the student on the expository paper followed by questioning by the committee members. A student who does not pass the MA Comprehensive Examination may make a second attempt, but all students are expected to pass the exam no later than the end of the summer following their second year.
• Oral Qualifying Examination: The topics for the PhD oral qualifying exam correspond to the two semester courses taken by the student from one of the 3 subject areas and one semester course each taken by the student from the other two subject areas. In addition, the exam begins with a presentation by the student on some specialized topic relevant to the proposed thesis research. A student who does not pass the qualifying exam may make a second attempt, but all PhD students are expected to pass the exam no later than the end of the summer following their third year.
• Dissertation and Final Oral Examination: This follows the GRS General Requirements for the Doctor of Philosophy Degree .

Admissions information can be found on the BU Arts and Sciences PhD Admissions website .

Financial Aid

Our department funds our PhD students through a combination of University fellowships, teaching fellowships, and faculty research grants. More information will be provided to admitted students.

More Information

Please reach out to us directly at [email protected] if you have further questions.

Ph.D. in Mathematics

General info.

• Faculty working with students: 37
• Students: 52
• Students receiving Financial Aid: 100%
• Part time study available: No
• Application terms: Fall
• Application deadline: December 13

Mark Haskins Director of Graduate Studies Department of Mathematics Duke University Box 90320 Durham, NC 27708-0320

Phone: (919) 660-2800

Email:  [email protected]

Website:  http://www.math.duke.edu/graduate/

Program Description

The graduate program in Mathematics at Duke offers research training in a wide variety of topics in mathematics.  Major areas of research specialization include algebra, number theory, algebraic geometry, analysis, differential geometry and physics, topology, dynamical systems, partial differential equations, scientific computing, and stochastic processes.  There are strong groups in mathematical biology, and in the analysis of high-dimensional data using ideas from topology and harmonic analysis.  These and other research efforts often include collaborations with researchers in the medical school and other departments.  These opportunities for interdisciplinary research and the breadth of research areas in the department make the Duke math department an ideal place to prepare for a job inside or outside of academia.

• Mathematics: PhD Admissions and Enrollment Statistics
• Mathematics: PhD Completion Rate Statistics
• Mathematics: PhD Time to Degree Statistics
• Mathematics: PhD Career Outcomes Statistics

Application Information

Application Terms Available:  Fall

Application Deadline:  December 13

Graduate School Application Requirements See the Application Instructions page for important details about each Graduate School requirement.

• Transcripts: Unofficial transcripts required with application submission; official transcripts required upon admission
• Letters of Recommendation: 3 Required
• Statement of Purpose: Required (see department guidance below)
• Résumé: Required
• GRE General (Optional)
• GRE Subject: Mathematics (Optional)
• English Language Exam: TOEFL, IELTS, or Duolingo English Test required* for applicants whose first language is not English *test waiver may apply for some applicants
• GPA: Undergraduate GPA calculated on 4.0 scale required

Department-Specific Application Requirements (submitted through online application)

Statement of Purpose Guidelines Summarize your mathematical work/background and current interests.  (If your principal academic focus has been outside mathematics, tell us about that work and what brings you to mathematics now.)  Explain why you chose to apply to Duke Math (what makes the department a good fit for you?).  It is generally beneficial to keep the statement under two pages.

Writing Sample None required

We strongly encourage you to review additional department-specific application guidance from the program to which you are applying: Departmental Application Guidance

List of Graduate School Programs and Degrees

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Mathematics PhD theses

A selection of Mathematics PhD thesis titles is listed below, some of which are available online:

2022   2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Melanie Kobras –  Low order models of storm track variability

Ed Clark –  Vectorial Variational Problems in L∞ and Applications to Data Assimilation

Katerina Christou – Modelling PDEs in Population Dynamics using Fixed and Moving Meshes  

Chiara Cecilia Maiocchi –  Unstable Periodic Orbits: a language to interpret the complexity of chaotic systems

Samuel R Harrison – Stalactite Inspired Thin Film Flow

Elena Saggioro – Causal network approaches for the study of sub-seasonal to seasonal variability and predictability

Cathie A Wells – Reformulating aircraft routing algorithms to reduce fuel burn and thus CO 2 emissions  

Jennifer E. Israelsson –  The spatial statistical distribution for multiple rainfall intensities over Ghana

Giulia Carigi –  Ergodic properties and response theory for a stochastic two-layer model of geophysical fluid dynamics

André Macedo –  Local-global principles for norms

Tsz Yan Leung  –  Weather Predictability: Some Theoretical Considerations

Jehan Alswaihli –  Iteration of Inverse Problems and Data Assimilation Techniques for Neural Field Equations

Jemima M Tabeart –  On the treatment of correlated observation errors in data assimilation

Chris Davies –  Computer Simulation Studies of Dynamics and Self-Assembly Behaviour of Charged Polymer Systems

Birzhan Ayanbayev –  Some Problems in Vectorial Calculus of Variations in L∞

Penpark Sirimark –  Mathematical Modelling of Liquid Transport in Porous Materials at Low Levels of Saturation

Adam Barker –  Path Properties of Levy Processes

Hasen Mekki Öztürk –  Spectra of Indefinite Linear Operator Pencils

Carlo Cafaro –  Information gain that convective-scale models bring to probabilistic weather forecasts

Nicola Thorn –  The boundedness and spectral properties of multiplicative Toeplitz operators

James Jackaman  – Finite element methods as geometric structure preserving algorithms

Changqiong Wang - Applications of Monte Carlo Methods in Studying Polymer Dynamics

Jack Kirk - The molecular dynamics and rheology of polymer melts near the flat surface

Hussien Ali Hussien Abugirda - Linear and Nonlinear Non-Divergence Elliptic Systems of Partial Differential Equations

Andrew Gibbs - Numerical methods for high frequency scattering by multiple obstacles (PDF-2.63MB)

Mohammad Al Azah - Fast Evaluation of Special Functions by the Modified Trapezium Rule (PDF-913KB)

Katarzyna (Kasia) Kozlowska - Riemann-Hilbert Problems and their applications in mathematical physics (PDF-1.16MB)

Anna Watkins - A Moving Mesh Finite Element Method and its Application to Population Dynamics (PDF-2.46MB)

Niall Arthurs - An Investigation of Conservative Moving-Mesh Methods for Conservation Laws (PDF-1.1MB)

Samuel Groth - Numerical and asymptotic methods for scattering by penetrable obstacles (PDF-6.29MB)

Katherine E. Howes - Accounting for Model Error in Four-Dimensional Variational Data Assimilation (PDF-2.69MB)

Jian Zhu - Multiscale Computer Simulation Studies of Entangled Branched Polymers (PDF-1.69MB)

Tommy Liu - Stochastic Resonance for a Model with Two Pathways (PDF-11.4MB)

Matthew Paul Edgington - Mathematical modelling of bacterial chemotaxis signalling pathways (PDF-9.04MB)

Anne Reinarz - Sparse space-time boundary element methods for the heat equation (PDF-1.39MB)

Adam El-Said - Conditioning of the Weak-Constraint Variational Data Assimilation Problem for Numerical Weather Prediction (PDF-2.64MB)

Nicholas Bird - A Moving-Mesh Method for High Order Nonlinear Diffusion (PDF-1.30MB)

Charlotta Jasmine Howarth - New generation finite element methods for forward seismic modelling (PDF-5,52MB)

Aldo Rota - From the classical moment problem to the realizability problem on basic semi-algebraic sets of generalized functions (PDF-1.0MB)

Sarah Lianne Cole - Truncation Error Estimates for Mesh Refinement in Lagrangian Hydrocodes (PDF-2.84MB)

Alexander J. F. Moodey - Instability and Regularization for Data Assimilation (PDF-1.32MB)

Dale Partridge - Numerical Modelling of Glaciers: Moving Meshes and Data Assimilation (PDF-3.19MB)

Joanne A. Waller - Using Observations at Different Spatial Scales in Data Assimilation for Environmental Prediction (PDF-6.75MB)

Faez Ali AL-Maamori - Theory and Examples of Generalised Prime Systems (PDF-503KB)

Mark Parsons - Mathematical Modelling of Evolving Networks

Natalie L.H. Lowery - Classification methods for an ill-posed reconstruction with an application to fuel cell monitoring

David Gilbert - Analysis of large-scale atmospheric flows

Peter Spence - Free and Moving Boundary Problems in Ion Beam Dynamics (PDF-5MB)

Timothy S. Palmer - Modelling a single polymer entanglement (PDF-5.02MB)

Mohamad Shukor Talib - Dynamics of Entangled Polymer Chain in a Grid of Obstacles (PDF-2.49MB)

Cassandra A.J. Moran - Wave scattering by harbours and offshore structures

Ashley Twigger - Boundary element methods for high frequency scattering

David A. Smith - Spectral theory of ordinary and partial linear differential operators on finite intervals (PDF-1.05MB)

Stephen A. Haben - Conditioning and Preconditioning of the Minimisation Problem in Variational Data Assimilation (PDF-3.51MB)

Jing Cao - Molecular dynamics study of polymer melts (PDF-3.98MB)

Bonhi Bhattacharya - Mathematical Modelling of Low Density Lipoprotein Metabolism. Intracellular Cholesterol Regulation (PDF-4.06MB)

Tamsin E. Lee - Modelling time-dependent partial differential equations using a moving mesh approach based on conservation (PDF-2.17MB)

Polly J. Smith - Joint state and parameter estimation using data assimilation with application to morphodynamic modelling (PDF-3Mb)

Corinna Burkard - Three-dimensional Scattering Problems with applications to Optical Security Devices (PDF-1.85Mb)

Laura M. Stewart - Correlated observation errors in data assimilation (PDF-4.07MB)

R.D. Giddings - Mesh Movement via Optimal Transportation (PDF-29.1MbB)

G.M. Baxter - 4D-Var for high resolution, nested models with a range of scales (PDF-1.06MB)

C. Spencer - A generalization of Talbot's theorem about King Arthur and his Knights of the Round Table.

P. Jelfs - A C-property satisfying RKDG Scheme with Application to the Morphodynamic Equations (PDF-11.7MB)

L. Bennetts - Wave scattering by ice sheets of varying thickness

M. Preston - Boundary Integral Equations method for 3-D water waves

J. Percival - Displacement Assimilation for Ocean Models (PDF - 7.70MB)

D. Katz - The Application of PV-based Control Variable Transformations in Variational Data Assimilation (PDF- 1.75MB)

S. Pimentel - Estimation of the Diurnal Variability of sea surface temperatures using numerical modelling and the assimilation of satellite observations (PDF-5.9MB)

J.M. Morrell - A cell by cell anisotropic adaptive mesh Arbitrary Lagrangian Eulerian method for the numerical solution of the Euler equations (PDF-7.7MB)

L. Watkinson - Four dimensional variational data assimilation for Hamiltonian problems

M. Hunt - Unique extension of atomic functionals of JB*-Triples

D. Chilton - An alternative approach to the analysis of two-point boundary value problems for linear evolutionary PDEs and applications

T.H.A. Frame - Methods of targeting observations for the improvement of weather forecast skill

C. Hughes - On the topographical scattering and near-trapping of water waves

B.V. Wells - A moving mesh finite element method for the numerical solution of partial differential equations and systems

D.A. Bailey - A ghost fluid, finite volume continuous rezone/remap Eulerian method for time-dependent compressible Euler flows

M. Henderson - Extending the edge-colouring of graphs

K. Allen - The propagation of large scale sediment structures in closed channels

D. Cariolaro - The 1-Factorization problem and same related conjectures

A.C.P. Steptoe - Extreme functionals and Stone-Weierstrass theory of inner ideals in JB*-Triples

D.E. Brown - Preconditioners for inhomogeneous anisotropic problems with spherical geometry in ocean modelling

S.J. Fletcher - High Order Balance Conditions using Hamiltonian Dynamics for Numerical Weather Prediction

C. Johnson - Information Content of Observations in Variational Data Assimilation

M.A. Wakefield - Bounds on Quantities of Physical Interest

M. Johnson - Some problems on graphs and designs

A.C. Lemos - Numerical Methods for Singular Differential Equations Arising from Steady Flows in Channels and Ducts

R.K. Lashley - Automatic Generation of Accurate Advection Schemes on Structured Grids and their Application to Meteorological Problems

J.V. Morgan - Numerical Methods for Macroscopic Traffic Models

M.A. Wlasak - The Examination of Balanced and Unbalanced Flow using Potential Vorticity in Atmospheric Modelling

M. Martin - Data Assimilation in Ocean circulation models with systematic errors

K.W. Blake - Moving Mesh Methods for Non-Linear Parabolic Partial Differential Equations

J. Hudson - Numerical Techniques for Morphodynamic Modelling

A.S. Lawless - Development of linear models for data assimilation in numerical weather prediction .

C.J.Smith - The semi lagrangian method in atmospheric modelling

T.C. Johnson - Implicit Numerical Schemes for Transcritical Shallow Water Flow

M.J. Hoyle - Some Approximations to Water Wave Motion over Topography.

P. Samuels - An Account of Research into an Area of Analytical Fluid Mechnaics. Volume II. Some mathematical Proofs of Property u of the Weak End of Shocks.

M.J. Martin - Data Assimulation in Ocean Circulation with Systematic Errors

P. Sims - Interface Tracking using Lagrangian Eulerian Methods.

P. Macabe - The Mathematical Analysis of a Class of Singular Reaction-Diffusion Systems.

B. Sheppard - On Generalisations of the Stone-Weisstrass Theorem to Jordan Structures.

S. Leary - Least Squares Methods with Adjustable Nodes for Steady Hyperbolic PDEs.

I. Sciriha - On Some Aspects of Graph Spectra.

P.A. Burton - Convergence of flux limiter schemes for hyperbolic conservation laws with source terms.

J.F. Goodwin - Developing a practical approach to water wave scattering problems.

N.R.T. Biggs - Integral equation embedding methods in wave-diffraction methods.

L.P. Gibson - Bifurcation analysis of eigenstructure assignment control in a simple nonlinear aircraft model.

A.K. Griffith - Data assimilation for numerical weather prediction using control theory. .

J. Bryans - Denotational semantic models for real-time LOTOS.

I. MacDonald - Analysis and computation of steady open channel flow .

A. Morton - Higher order Godunov IMPES compositional modelling of oil reservoirs.

S.M. Allen - Extended edge-colourings of graphs.

M.E. Hubbard - Multidimensional upwinding and grid adaptation for conservation laws.

C.J. Chikunji - On the classification of finite rings.

S.J.G. Bell - Numerical techniques for smooth transformation and regularisation of time-varying linear descriptor systems.

D.J. Staziker - Water wave scattering by undulating bed topography .

K.J. Neylon - Non-symmetric methods in the modelling of contaminant transport in porous media. .

D.M. Littleboy - Numerical techniques for eigenstructure assignment by output feedback in aircraft applications .

M.P. Dainton - Numerical methods for the solution of systems of uncertain differential equations with application in numerical modelling of oil recovery from underground reservoirs .

M.H. Mawson - The shallow-water semi-geostrophic equations on the sphere. .

S.M. Stringer - The use of robust observers in the simulation of gas supply networks .

S.L. Wakelin - Variational principles and the finite element method for channel flows. .

E.M. Dicks - Higher order Godunov black-oil simulations for compressible flow in porous media .

C.P. Reeves - Moving finite elements and overturning solutions .

A.J. Malcolm - Data dependent triangular grid generation. .

Applied Mathematics

Faculty and students interested in the applications of mathematics are an integral part of the department of mathematics; there is no formal separation between pure and applied mathematics, and the department takes pride in the many ways in which they enrich each other. we also benefit tremendously from close collaborations with faculty and students in other departments at uc berkeley as well as scientists at  lawrence berkeley national laboratory  and visitors to the  mathematical sciences research institute.

The Department regularly offers courses in ordinary and partial differential equations and their numerical solution, discrete applied mathematics, the methods of mathematical physics, mathematical biology, the mathematical aspects of fluid and solid mechanics, approximation theory, scientific computing, numerical linear algebra, and mathematical aspects of computer science. Courses in probability theory, stochastic processes, data analysis and bioinformatics are offered by the Department of Statistics, while courses in combinatorial and convex optimization are offered by the Department of Industrial Engineering and Operations Research. Our students are encouraged to take courses of mathematical interest in these and many other departments. Topics explored intensively by our faculty and students in recent years include scientific computation and the mathematical aspects of quantum theory, computational genomics, image processing and medical imaging, inverse problems, combinatorial optimization, control, robotics, shape optimization, turbulence, hurricanes, microchip failure, MEMS, biodemography, population genetics, phylogenetics, and computational approaches to historical linguistics. Within the department we also have a  Laboratory for Mathematical and Computational Biology .

Chair:   Per-Olof Persson

Applied Mathematics Faculty, Courses, Dissertations

Senate faculty, graduate students, visiting faculty, meet our faculty, mina aganagic, david aldous, robert m. anderson, sunčica čanić, jennifer chayes, alexandre j. chorin, paul concus, james w. demmel, l. craig evans, steven n. evans, f. alberto grünbaum, venkatesan guruswami, ole h. hald, william m. kahan, richard karp, michael j. klass, hendrik w. lenstra, jr., lin lin (林霖), michael j. lindsey, c. keith miller, john c. neu, beresford n. parlett.

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Pure mathematics articles from across Nature Portfolio

Pure mathematics uses mathematics to explore abstract ideas, mathematics that does not necessarily describe a real physical system. This can include developing the fundamental tools used by mathematicians, such as algebra and calculus, describing multi-dimensional space, or better understanding the philosophical meaning of mathematics and numbers themselves.

Latest Research and Reviews

Classification of artificial intelligence tools for civil engineering under the notion of complex fuzzy rough Frank aggregation operators

• Jabbar Ahmmad

Interval-valued bipolar complex fuzzy soft sets and their applications in decision making

• Abdul Jaleel
• Tahir Mahmood

Max-mixed EWMA control chart for joint monitoring of mean and variance: an application to yogurt packing process

• Seher Malik
• Muhammad Hanif
• Jumanah Ahmed Darwish

Computation and convergence of fixed-point with an RLC-electric circuit model in an extended b-suprametric space

• Sumati Kumari Panda
• Vijayakumar Velusamy
• Shafiullah Niazai

Spectral shifted Chebyshev collocation technique with residual power series algorithm for time fractional problems

• Saad. Z. Rida
• Anas. A. M. Arafa
• Marwa. M. M. Mostafa

A novel group decision making method based on CoCoSo and interval-valued Q-rung orthopair fuzzy sets

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The real value of numbers

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Machine learning to guide mathematicians

• Fernando Chirigati

A different perspective on the history of the proof of Hall conductance quantization

• Matthew B. Hastings

e is everywhere

From determining the compound interest on borrowed money to gauging chances at the roulette wheel in Monte Carlo, Stefanie Reichert explains that there’s no way around Euler’s number.

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Imagination captured

Imaginary numbers have a chequered history, and a sparse — if devoted — following. Abigail Klopper looks at why a concept as beautiful as i gets such a bad rap.

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Prime interference

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Home > College of Natural Sciences > Mathematics > Mathematics Theses, Projects, and Dissertations

Mathematics Theses, Projects, and Dissertations

Theses/projects/dissertations from 2024 2024.

On Cheeger Constants of Knots , Robert Lattimer

Information Based Approach for Detecting Change Points in Inverse Gaussian Model with Applications , Alexis Anne Wallace

Theses/Projects/Dissertations from 2023 2023

DNA SELF-ASSEMBLY OF TRAPEZOHEDRAL GRAPHS , Hytham Abdelkarim

An Exposition of the Curvature of Warped Product Manifolds , Angelina Bisson

Jackknife Empirical Likelihood Tests for Equality of Generalized Lorenz Curves , Anton Butenko

MATHEMATICS BEHIND MACHINE LEARNING , Rim Hammoud

Statistical Analysis of Health Habits for Incoming College Students , Wendy Isamara Lizarraga Noriega

Reverse Mathematics of Ramsey's Theorem , Nikolay Maslov

Distance Correlation Based Feature Selection in Random Forest , Jose Munoz-Lopez

Constructing Hyperbolic Polygons in the Poincaré Disk , Akram Zakaria Samweil

KNOT EQUIVALENCE , Jacob Trubey

Theses/Projects/Dissertations from 2022 2022

SYMMETRIC GENERATIONS AND AN ALGORITHM TO PROVE RELATIONS , Diddier Andrade

The Examination of the Arithmetic Surface (3, 5) Over Q , Rachel J. Arguelles

Error Terms for the Trapezoid, Midpoint, and Simpson's Rules , Jessica E. Coen

de Rham Cohomology, Homotopy Invariance and the Mayer-Vietoris Sequence , Stacey Elizabeth Cox

Symmetric Generation , Ana Gonzalez

SYMMETRIC PRESENTATIONS OF FINITE GROUPS AND RELATED TOPICS , Samar Mikhail Kasouha

Simple Groups and Related Topics , Simrandeep Kaur

Homomorphic Images and Related Topics , Alejandro Martinez

LATTICE REDUCTION ALGORITHMS , Juan Ortega

THE DECOMPOSITION OF THE SPACE OF ALGEBRAIC CURVATURE TENSORS , Katelyn Sage Risinger

Verifying Sudoku Puzzles , Chelsea Schweer

AN EXPOSITION OF ELLIPTIC CURVE CRYPTOGRAPHY , Travis Severns

Theses/Projects/Dissertations from 2021 2021

Non-Abelian Finite Simple Groups as Homomorphic Images , Sandra Bahena

Matroids Determinable by Two Partial Representations , Aurora Calderon Dojaquez

SYMMETRIC REPRESENTATIONS OF FINITE GROUPS AND RELATED TOPICS , Connie Corona

Symmetric Presentation of Finite Groups, and Related Topics , Marina Michelle Duchesne

MEASURE AND INTEGRATION , JeongHwan Lee

A Study in Applications of Continued Fractions , Karen Lynn Parrish

Partial Representations for Ternary Matroids , Ebony Perez

Theses/Projects/Dissertations from 2020 2020

Sum of Cubes of the First n Integers , Obiamaka L. Agu

Permutation and Monomial Progenitors , Crystal Diaz

Tile Based Self-Assembly of the Rook's Graph , Ernesto Gonzalez

Research In Short Term Actuarial Modeling , Elijah Howells

Hyperbolic Triangle Groups , Sergey Katykhin

Exploring Matroid Minors , Jonathan Lara Tejeda

DNA COMPLEXES OF ONE BOND-EDGE TYPE , Andrew Tyler Lavengood-Ryan

Modeling the Spread of Measles , Alexandria Le Beau

Symmetric Presentations and Related Topics , Mayra McGrath

Minimal Surfaces and The Weierstrass-Enneper Representation , Evan Snyder

ASSESSING STUDENT UNDERSTANDING WHILE SOLVING LINEAR EQUATIONS USING FLOWCHARTS AND ALGEBRAIC METHODS , Edima Umanah

Excluded minors for nearly-paving matroids , Vanessa Natalie Vega

Theses/Projects/Dissertations from 2019 2019

Fuchsian Groups , Bob Anaya

Tribonacci Convolution Triangle , Rosa Davila

VANISHING LOCAL SCALAR INVARIANTS ON GENERALIZED PLANE WAVE MANIFOLDS , Brian Matthew Friday

Analogues Between Leibniz's Harmonic Triangle and Pascal's Arithmetic Triangle , Lacey Taylor James

Geodesics on Generalized Plane Wave Manifolds , Moises Pena

Algebraic Methods for Proving Geometric Theorems , Lynn Redman

Pascal's Triangle, Pascal's Pyramid, and the Trinomial Triangle , Antonio Saucedo Jr.

THE EFFECTIVENESS OF DYNAMIC MATHEMATICAL SOFTWARE IN THE INSTRUCTION OF THE UNIT CIRCLE , Edward Simons

CALCULUS REMEDIATION AS AN INDICATOR FOR SUCCESS ON THE CALCULUS AP EXAM , Ty Stockham

Theses/Projects/Dissertations from 2018 2018

PROGENITORS, SYMMETRIC PRESENTATIONS AND CONSTRUCTIONS , Diana Aguirre

Monomial Progenitors and Related Topics , Madai Obaid Alnominy

Progenitors Involving Simple Groups , Nicholas R. Andujo

Simple Groups, Progenitors, and Related Topics , Angelica Baccari

Exploring Flag Matroids and Duality , Zachary Garcia

Images of Permutation and Monomial Progenitors , Shirley Marina Juan

MODERN CRYPTOGRAPHY , Samuel Lopez

Progenitors, Symmetric Presentations, and Related Topics , Joana Viridiana Luna

Symmetric Presentations, Representations, and Related Topics , Adam Manriquez

Toroidal Embeddings and Desingularization , LEON NGUYEN

THE STRUGGLE WITH INVERSE FUNCTIONS DOING AND UNDOING PROCESS , Jesus Nolasco

Tutte-Equivalent Matroids , Maria Margarita Rocha

Symmetric Presentations and Double Coset Enumeration , Charles Seager

MANUAL SYMMETRIC GENERATION , Joel Webster

Theses/Projects/Dissertations from 2017 2017

Investigation of Finite Groups Through Progenitors , Charles Baccari

CONSTRUCTION OF HOMOMORPHIC IMAGES , Erica Fernandez

Making Models with Bayes , Pilar Olid

An Introduction to Lie Algebra , Amanda Renee Talley

SIMPLE AND SEMI-SIMPLE ARTINIAN RINGS , Ulyses Velasco

CONSTRUCTION OF FINITE GROUP , Michelle SoYeong Yeo

Theses/Projects/Dissertations from 2016 2016

Upset Paths and 2-Majority Tournaments , Rana Ali Alshaikh

Regular Round Matroids , Svetlana Borissova

GEODESICS IN LORENTZIAN MANIFOLDS , Amir A. Botros

REALIZING TOURNAMENTS AS MODELS FOR K-MAJORITY VOTING , Gina Marie Cheney

Solving Absolute Value Equations and Inequalities on a Number Line , Melinda A. Curtis

BIO-MATHEMATICS: INTRODUCTION TO THE MATHEMATICAL MODEL OF THE HEPATITIS C VIRUS , Lucille J. Durfee

ANALYSIS AND SYNTHESIS OF THE LITERATURE REGARDING ACTIVE AND DIRECT INSTRUCTION AND THEIR PROMOTION OF FLEXIBLE THINKING IN MATHEMATICS , Genelle Elizabeth Gonzalez

LIFE EXPECTANCY , Ali R. Hassanzadah

PLANAR GRAPHS, BIPLANAR GRAPHS AND GRAPH THICKNESS , Sean M. Hearon

A Dual Fano, and Dual Non-Fano Matroidal Network , Stephen Lee Johnson

Mathematical Reasoning and the Inductive Process: An Examination of The Law of Quadratic Reciprocity , Nitish Mittal

The Kauffman Bracket and Genus of Alternating Links , Bryan M. Nguyen

Probabilistic Methods In Information Theory , Erik W. Pachas

THINKING POKER THROUGH GAME THEORY , Damian Palafox

Indicators of Future Mathematics Proficiency: Literature Review & Synthesis , Claudia Preciado

Ádám's Conjecture and Arc Reversal Problems , Claudio D. Salas

AN INTRODUCTION TO BOOLEAN ALGEBRAS , Amy Schardijn

The Evolution of Cryptology , Gwendolyn Rae Souza

Theses/Projects/Dissertations from 2015 2015

SYMMETRIC PRESENTATIONS AND RELATED TOPICS , Mashael U. Alharbi

Homomorphic Images And Related Topics , Kevin J. Baccari

Geometric Constructions from an Algebraic Perspective , Betzabe Bojorquez

Discovering and Applying Geometric Transformations: Transformations to Show Congruence and Similarity , Tamara V. Bonn

Symmetric Presentations and Generation , Dustin J. Grindstaff

HILBERT SPACES AND FOURIER SERIES , Terri Joan Harris Mrs.

SYMMETRIC PRESENTATIONS OF NON-ABELIAN SIMPLE GROUPS , Leonard B. Lamp

Simple Groups and Related Topics , Manal Abdulkarim Marouf Ms.

Elliptic Curves , Trinity Mecklenburg

A Fundamental Unit of O_K , Susana L. Munoz

CONSTRUCTIONS AND ISOMORPHISM TYPES OF IMAGES , Jessica Luna Ramirez

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181 Mathematics Research Topics From PhD Experts

If you are reading this blog post, it means you are looking for some exceptional math research topics. You want them to be original, unique even. If you manage to find topics like this, you can be sure your professor will give you a top grade (if you write a decent paper, that is). The good news is that you have arrived at just the right place – at the right time. We have just finished updating our list of topics, so you will find plenty of original ideas right on this page. All our topics are 100 percent free to use as you see fit. You can reword them and you don’t need to give us any credit.

And remember: if you need assistance from a professional, don’t hesitate to reach out to us. We are not just the best place for math research topics for high school students; we are also the number one choice for students looking for top-notch research paper writing services.

Our Newest Research Topics in Math

We know you probably want the best and most recent research topics in math. You want your paper to stand out from all the rest. After all, this is the best way to get some bonus points from your professor. On top of this, finding some great topics for your next paper makes it easier for you to write the essay. As long as you know at least something about the topic, you’ll find that writing a great paper or buy phd thesis isn’t as difficult as you previously thought.

So, without further ado, here are the 181 brand new topics for your next math research paper:

Cool Math Topics to Research

Are you looking for some cool math topics to research? We have a list of original topics for your right here. Pick the one you like and start writing now:

• Roll two dice and calculate a probability
• Discuss ancient Greek mathematics
• Is math really important in school?
• Discuss the binomial theorem
• The math behind encryption
• Game theory and its real-life applications
• Analyze the Bernoulli scheme
• What are holomorphic functions and how do they work?
• Describe big numbers
• Solving the Tower of Hanoi problem

Undergraduate Math Research Topics

If you are an undergraduate looking for some research topics for your next math paper, you will surely appreciate our list of interesting undergraduate math research topics:

• Methods to count discrete objects
• The origins of Greek symbols in mathematics
• Methods to solve simultaneous equations
• Real-world applications of the theorem of Pythagoras
• Discuss the limits of diffusion
• Use math to analyze the abortion data in the UK over the last 100 years
• Discuss the Knot theory
• Analyze predictive models (take meteorology as an example)
• In-depth analysis of the Monte Carlo methods for inverse problems
• Squares vs. rectangles (compare and contrast)

Number Theory Topics to Research

Interested in writing about number theory? It is not an easy subject to discuss, we know. However, we are sure you will appreciate these number theory topics:

• Discuss the greatest common divisor
• Explain the extended Euclidean algorithm
• What are RSA numbers?
• Discuss Bézout’s lemma
• In-depth analysis of the square-free polynomial
• Discuss the Stern-Brocot tree
• Analyze Fermat’s little theorem
• What is a discrete logarithm?
• Gauss’s lemma in number theory
• Analyze the Pentagonal number theorem

Math Research Topics for High School

High school students shouldn’t be too worried about their math papers because we have some unique, and quite interesting, math research topics for high school right here:

• Discuss Brun’s constant
• An in-depth look at the Brahmagupta–Fibonacci identity
• What is derivative algebra?
• Describe the Symmetric Boolean function
• Discuss orders of approximation in limits
• Solving Regiomontanus’ angle maximization problem
• What is a Quadratic integral?
• Define and describe complementary angles
• Analyze the incircle and excircles of a triangle
• Analyze the Bolyai–Gerwien theorem in geometry
• Math in our everyday life

Complex Math Topics

If you want to give some complex math topics a try, we have the best examples below. Remember, these topics should only be attempted by students who are proficient in mathematics:

• Mathematics and its appliance in Artificial Intelligence
• Try to solve an unsolved problem in math
• Discuss Kolmogorov’s zero-one law
• What is a discrete random variable?
• Analyze the Hewitt–Savage zero-one law
• What is a transferable belief model?
• Discuss 3 major mathematical theorems
• Describe and analyze the Dempster-Shafer theory
• An in-depth analysis of a continuous stochastic process
• Identify and analyze Gauss-Markov processes

Easy Math Research Paper Topics

Perhaps you don’t want to spend too much time working on your next research paper. Who can blame you? Check out these easy math research paper topics:

• Define the hyperbola
• Do we need to use a calculator during math class?
• The binomial theorem and its real-world applications
• What is a parabola in geometry?
• How do you calculate the slope of a curve?
• Define the Jacobian matrix
• Solving matrix problems effectively
• Why do we need differential equations?
• Should math be mandatory in all schools?
• What is a Hessian matrix?

Logic Topics to Research

We have some interesting logical topics for research papers. These are perfect for students interested in writing about math logic. Pick one right now:

• Discuss the reductio ad absurdum approach
• Discuss Boolean algebra
• What is consistency proof?
• Analyze Trakhtenbrot’s theorem (the finite model theory)
• Discuss the Gödel completeness theorem
• An in-depth analysis of Morley’s categoricity theorem
• How does the Back-and-forth method work?
• Discuss the Ehrenfeucht–Fraïssé game technique
• Discuss Aleph numbers (Aleph-null and Aleph-one)
• Solving the Suslin problem

Algebra Topics for a Research Paper

Would you like to write about an algebra topic? No problem, our seasoned writers have compiled a list of the best algebra topics for a research paper:

• Discuss the differential equation
• Analyze the Jacobson density theorem
• The 4 properties of a binary operation in algebra
• Analyze the unary operator in depth
• Analyze the Abel–Ruffini theorem
• Epimorphisms vs. monomorphisms: compare and contrast
• Discuss the Morita duality in algebraic structures
• Idempotent vs. nilpotent in Ring theory
• Discuss the Artin-Wedderburn theorem
• What is a commutative ring in algebra?
• Analyze and describe the Noetherian ring

Math Education Research Topics

There is nothing wrong with writing about math education, especially if your professor did not give you writing prompts. Here are some very nice math education research topics:

• What are the goals a mathematics professor should have?
• What is math anxiety in the classroom?
• Teaching math in UK schools: the difficulties
• Computer programming or math in high school?
• Is math education in Europe at a high enough level?
• Common Core Standards and their effects on math education
• Culture and math education in Africa
• What is dyscalculia and how does it manifest itself?
• When was algebra first thought in schools?
• Math education in the United States versus the United Kingdom

Computability Theory Topics to Research

Writing about computability theory can be a very interesting adventure. Give it a try! Here are some of our most interesting computability theory topics to research:

• What is a multiplication table?
• Analyze the Scholz conjecture
• Explain exponentiating by squaring
• Analyze the Myhill-Nerode theorem
• What is a tree automaton?
• Compare and contrast the Pushdown automaton and the Büchi automaton
• Discuss the Markov algorithm
• What is a Turing machine?
• Analyze the post correspondence problem
• Discuss the linear speedup theorem
• Discuss the Boolean satisfiability problem

Interesting Math Research Topics

We know you want topics that are interesting and relatively easy to write about. This is why we have a separate list of our most interesting math research topics:

• What is two-element Boolean algebra?
• The life of Gauss
• The life of Isaac Newton
• What is an orthodiagonal quadrilateral?
• Tessellation in Euclidean plane geometry
• Describe a hyperboloid in 3D geometry
• What is a sphericon?
• Discuss the peculiarities of Borel’s paradox
• Analyze the De Finetti theorem in statistics
• What are Martingales?
• The basics of stochastic calculus

Applied Math Research Topics

Interested in writing about applied mathematics? Our team managed to create a list of awesome applied math research topics from scratch for you:

• Discuss Newton’s laws of motion
• Analyze the perpendicular axes rule
• How is a Galilean transformation done?
• The conservation of energy and its applications
• Discuss Liouville’s theorem in Hamiltonian mechanics
• Analyze the quantum field theory
• Discuss the main components of the Lorentz symmetry
• An in-depth look at the uncertainty principle

Geometry Topics for a Research Paper

Geometry can be a very captivating subject, especially when you know plenty about it. Check out our list of geometry topics for a research paper and pick the best one today:

• Most useful trigonometry functions in math
• The life of Archimedes and his achievements
• Trigonometry in computer graphics
• Using Vincenty’s formulae in geodesy
• Define and describe the Heronian tetrahedron
• The math behind the parabolic microphone
• Discuss the Japanese theorem for concyclic polygons
• Analyze Euler’s theorem in geometry

Math Research Topics for Middle School

Yes, even middle school children can write about mathematics. We have some original math research topics for middle school right here:

• Finding critical points in a graph
• The basics of calculus
• What makes a graph ultrahomogeneous?
• How do you calculate the area of different shapes?
• What contributions did Euclid have to the field of mathematics?
• What is Diophantine geometry?
• What makes a graph regular?
• Analyze a full binary tree

Math Research Topics for College Students

As you’ve probably already figured out, college students should pick topics that are a bit more complex. We have some of the best math research topics for college students right here:

• What are extremal problems and how do you solve them?
• Discuss an unsolvable math problem
• How can supercomputers solve complex mathematical problems?
• An in-depth analysis of fractals
• Discuss the Boruvka’s algorithm (related to the minimum spanning tree)
• Discuss the Lorentz–FitzGerald contraction hypothesis in relativity
• An in-depth look at Einstein’s field equation
• The math behind computer vision and object recognition

Calculus Topics for a Research Paper

Let’s face it: calculus is not a very difficult field. So, why don’t you pick one of our excellent calculus topics for a research paper and start writing your essay right away:

• When do we need to apply the L’Hôpital rule?
• Discuss the Leibniz integral rule
• Calculus in ancient Egypt
• Discuss and analyze linear approximations
• The applications of calculus in real life
• The many uses of Stokes’ theorem
• Discuss the Borel regular measure
• An in-depth analysis of Lebesgue’s monotone convergence theorem

Simple Math Research Paper Topics for High School

This is the place where you can find some pretty simple topics if you are a high school student. Check out our simple math research paper topics for high school:

• The life and work of the famous Pierre de Fermat
• What are limits and why are they useful in calculus?
• Explain the concept of congruency
• The life and work of the famous Jakob Bernoulli
• Analyze the rhombicosidodecahedron and its applications
• Calculus and the Egyptian pyramids
• The life and work of the famous Jean d’Alembert
• Discuss the hyperplane arrangement in combinatorial computational geometry
• The smallest enclosing sphere method in combinatorics

Business Math Topics

If you want to surprise your professor, why don’t you write about business math? We have some exceptional topics that nobody has thought about right here:

• Is paying a loan with another loan a good approach?
• Discuss the major causes of a stock market crash
• Best debt amortization methods in the US
• How do bank loans work in the UK?
• Calculating interest rates the easy way
• Discuss the pros and cons of annuities
• Basic business math skills everyone should possess
• Business math in United States schools
• Analyze the discount factor

Probability and Statistics Topics for Research

Probability and statistics are not easy fields. However, you can impress your professor with one of our unique probability and statistics topics for research:

• What is the autoregressive conditional duration?
• Applying the ANOVA method to ranks
• Discuss the practical applications of the Bates distribution
• Explain the principle of maximum entropy
• Discuss Skorokhod’s representation theorem in random variables
• What is the Factorial moment in the Theory of Probability?
• Compare and contrast Cochran’s C test and his Q test
• Analyze the De Moivre-Laplace theorem
• What is a negative probability?

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• Arthur Bakker   ORCID: orcid.org/0000-0002-9604-3448 1 ,
• Jinfa Cai   ORCID: orcid.org/0000-0002-0501-3826 2 &
• Linda Zenger 1  

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Before the pandemic (2019), we asked: On what themes should research in mathematics education focus in the coming decade? The 229 responses from 44 countries led to eight themes plus considerations about mathematics education research itself. The themes can be summarized as teaching approaches, goals, relations to practices outside mathematics education, teacher professional development, technology, affect, equity, and assessment. During the pandemic (November 2020), we asked respondents: Has the pandemic changed your view on the themes of mathematics education research for the coming decade? If so, how? Many of the 108 respondents saw the importance of their original themes reinforced (45), specified their initial responses (43), and/or added themes (35) (these categories were not mutually exclusive). Overall, they seemed to agree that the pandemic functions as a magnifying glass on issues that were already known, and several respondents pointed to the need to think ahead on how to organize education when it does not need to be online anymore. We end with a list of research challenges that are informed by the themes and respondents’ reflections on mathematics education research.

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1 An international survey in two rounds

Around the time when Educational Studies in Mathematics (ESM) and the Journal for Research in Mathematics Education (JRME) were celebrating their 50th anniversaries, Arthur Bakker (editor of ESM) and Jinfa Cai (editor of JRME) saw a need to raise the following future-oriented question for the field of mathematics education research:

Q2019: On what themes should research in mathematics education focus in the coming decade?

To that end, we administered a survey with just this one question between June 17 and October 16, 2019.

When we were almost ready with the analysis, the COVID-19 pandemic broke out, and we were not able to present the results at the conferences we had planned to attend (NCTM and ICME in 2020). Moreover, with the world shaken up by the crisis, we wondered if colleagues in our field might think differently about the themes formulated for the future due to the pandemic. Hence, on November 26, 2020, we asked a follow-up question to those respondents who in 2019 had given us permission to approach them for elaboration by email:

Q2020: Has the pandemic changed your view on the themes of mathematics education research for the coming decade? If so, how?

In this paper, we summarize the responses to these two questions. Similar to Sfard’s ( 2005 ) approach, we start by synthesizing the voices of the respondents before formulating our own views. Some colleagues put forward the idea of formulating a list of key themes or questions, similar to the 23 unsolved mathematical problems that David Hilbert published around 1900 (cf. Schoenfeld, 1999 ). However, mathematics and mathematics education are very different disciplines, and very few people share Hilbert’s formalist view on mathematics; hence, we do not want to suggest that we could capture the key themes of mathematics education in a similar way. Rather, our overview of themes drawn from the survey responses is intended to summarize what is valued in our global community at the time of the surveys. Reasoning from these themes, we end with a list of research challenges that we see worth addressing in the future (cf. Stephan et al., 2015 ).

2 Methodological approach

2.1 themes for the coming decade (2019).

We administered the 1-question survey through email lists that we were aware of (e.g., Becker, ICME, PME) and asked mathematics education researchers to spread it in their national networks. By October 16, 2019, we had received 229 responses from 44 countries across 6 continents (Table 1 ). Although we were happy with the larger response than Sfard ( 2005 ) received (74, with 28 from Europe), we do not know how well we have reached particular regions, and if potential respondents might have faced language or other barriers. We did offer a few Chinese respondents the option to write in Chinese because the second author offered to translate their emails into English. We also received responses in Spanish, which were translated for us.

Ethical approval was given by the Ethical Review Board of the Faculties of Science and Geo-science of Utrecht University (Bèta L-19247). We asked respondents to indicate if they were willing to be quoted by name and if we were allowed to approach them for subsequent information. If they preferred to be named, we mention their name and country; otherwise, we write “anonymous.” In our selection of quotes, we have focused on content, not on where the response came from. On March 2, 2021, we approached all respondents who were quoted to double-check if they agreed to be quoted and named. One colleague preferred the quote and name to be deleted; three suggested small changes in wording; the others approved.

On September 20, 2019, the three authors met physically at Utrecht University to analyze the responses. After each individual proposal, we settled on a joint list of seven main themes (the first seven in Table 2 ), which were neither mutually exclusive nor exhaustive. The third author (Zenger, then still a student in educational science) next color coded all parts of responses belonging to a category. These formed the basis for the frequencies and percentages presented in the tables and text. The first author (Bakker) then read all responses categorized by a particular code to identify and synthesize the main topics addressed within each code. The second author (Cai) read all of the survey responses and the response categories, and commented. After the initial round of analysis, we realized it was useful to add an eighth theme: assessment (including evaluation).

Moreover, given that a large number of respondents made comments about mathematics education research itself, we decided to summarize these separately. For analyzing this category of research, we used the following four labels to distinguish types of comments on our discipline of mathematics education research: theory, methodology, self-reflection (including ethical considerations), interdisciplinarity, and transdisciplinarity. We then summarized the responses per type of comment.

It has been a daunting and humbling experience to study the huge coverage and diversity of topics that our colleagues care about. Any categorization felt like a reduction of the wealth of ideas, and we are aware of the risks of “sorting things out” (Bowker & Star, 2000 ), which come with foregrounding particular challenges rather than others (Stephan et al., 2015 ). Yet the best way to summarize the bigger picture seemed by means of clustering themes and pointing to their relationships. As we identified these eight themes of mathematics education research for the future, a recurring question during the analysis was how to represent them. A list such as Table 2 does not do justice to the interrelations between the themes. Some relationships are very clear, for example, educational approaches (theme 2) working toward educational or societal goals (theme 1). Some themes are pervasive; for example, equity and (positive) affect are both things that educators want to achieve but also phenomena that are at stake during every single moment of learning and teaching. Diagrams we considered to represent such interrelationships were either too specific (limiting the many relevant options, e.g., a star with eight vertices that only link pairs of themes) or not specific enough (e.g., a Venn diagram with eight leaves such as the iPhone symbol for photos). In the end, we decided to use an image and collaborated with Elisabeth Angerer (student assistant in an educational sciences program), who eventually made the drawing in Fig. 1 to capture themes in their relationships.

Artistic impression of the future themes

2.2 Has the pandemic changed your view? (2020)

On November 26, 2020, we sent an email to the colleagues who responded to the initial question and who gave permission to be approached by email. We cited their initial response and asked: “Has the pandemic changed your view on the themes of mathematics education research for the coming decade? If so, how?” We received 108 responses by January 12, 2021. The countries from which the responses came included China, Italy, and other places that were hit early by the COVID-19 virus. The length of responses varied from a single word response (“no”) to elaborate texts of up to 2215 words. Some people attached relevant publications. The median length of the responses was 87 words, with a mean length of 148 words and SD = 242. Zenger and Bakker classified them as “no changes” (9 responses) or “clearly different views” (8); the rest of the responses saw the importance of their initial themes reinforced (45), specified their initial responses (43), or added new questions or themes (35). These last categories were not mutually exclusive, because respondents could first state that they thought the initial themes were even more relevant than before and provide additional, more specified themes. We then used the same themes that had been identified in the first round and identified what was stressed or added in the 2020 responses.

3 The themes

The most frequently mentioned theme was what we labeled approaches to teaching (64% of the respondents, see Table 2 ). Next was the theme of goals of mathematics education on which research should shed more light in the coming decade (54%). These goals ranged from specific educational goals to very broad societal ones. Many colleagues referred to mathematics education’s relationships with other practices (communities, institutions…) such as home, continuing education, and work. Teacher professional development is a key area for research in which the other themes return (what should students learn, how, how to assess that, how to use technology and ensure that students are interested?). Technology constitutes its own theme but also plays a key role in many other themes, just like affect. Another theme permeating other ones is what can be summarized as equity, diversity, and inclusion (also social justice, anti-racism, democratic values, and several other values were mentioned). These values are not just societal and educational goals but also drivers for redesigning teaching approaches, using technology, working on more just assessment, and helping learners gain access, become confident, develop interest, or even love for mathematics. To evaluate if approaches are successful and if goals have been achieved, assessment (including evaluation) is also mentioned as a key topic of research.

In the 2020 responses, many wise and general remarks were made. The general gist is that the pandemic (like earlier crises such as the economic crisis around 2008–2010) functioned as a magnifying glass on themes that were already considered important. Due to the pandemic, however, systemic societal and educational problems were said to have become better visible to a wider community, and urge us to think about the potential of a “new normal.”

3.1 Approaches to teaching

We distinguish specific teaching strategies from broader curricular topics.

3.1.1 Teaching strategies

There is a widely recognized need to further design and evaluate various teaching approaches. Among the teaching strategies and types of learning to be promoted that were mentioned in the survey responses are collaborative learning, critical mathematics education, dialogic teaching, modeling, personalized learning, problem-based learning, cross-curricular themes addressing the bigger themes in the world, embodied design, visualization, and interleaved learning. Note, however, that students can also enhance their mathematical knowledge independently from teachers or parents through web tutorials and YouTube videos.

Many respondents emphasized that teaching approaches should do more than promote cognitive development. How can teaching be entertaining or engaging? How can it contribute to the broader educational goals of developing students’ identity, contribute to their empowerment, and help them see the value of mathematics in their everyday life and work? We return to affect in Section 3.7 .

In the 2020 responses, we saw more emphasis on approaches that address modeling, critical thinking, and mathematical or statistical literacy. Moreover, respondents stressed the importance of promoting interaction, collaboration, and higher order thinking, which are generally considered to be more challenging in distance education. One approach worth highlighting is challenge-based education (cf. Johnson et al. 2009 ), because it takes big societal challenges as mentioned in the previous section as its motivation and orientation.

3.1.2 Curriculum

Approaches by which mathematics education can contribute to the aforementioned goals can be distinguished at various levels. Several respondents mentioned challenges around developing a coherent mathematics curriculum, smoothing transitions to higher school levels, and balancing topics, and also the typical overload of topics, the influence of assessment on what is taught, and what teachers can teach. For example, it was mentioned that mathematics teachers are often not prepared to teach statistics. There seems to be little research that helps curriculum authors tackle some of these hard questions as well as how to monitor reform (cf. Shimizu & Vithal, 2019 ). Textbook analysis is mentioned as a necessary research endeavor. But even if curricula within one educational system are reasonably coherent, how can continuity between educational systems be ensured (cf. Jansen et al., 2012 )?

In the 2020 responses, some respondents called for free high-quality curriculum resources. In several countries where Internet access is a problem in rural areas, a shift can be observed from online resources to other types of media such as radio and TV.

3.2 Goals of mathematics education

The theme of approaches is closely linked to that of the theme of goals. For example, as Fulvia Furinghetti (Italy) wrote: “It is widely recognized that critical thinking is a fundamental goal in math teaching. Nevertheless it is still not clear how it is pursued in practice.” We distinguish broad societal and more specific educational goals. These are often related, as Jane Watson (Australia) wrote: “If Education is to solve the social, cultural, economic, and environmental problems of today’s data-driven world, attention must be given to preparing students to interpret the data that are presented to them in these fields.”

3.2.1 Societal goals

Respondents alluded to the need for students to learn to function in the economy and in society more broadly. Apart from instrumental goals of mathematics education, some emphasized goals related to developing as a human being, for instance learning to see the mathematics in the world and develop a relation with the world. Mathematics education in these views should empower students to combat anti-expertise and post-fact tendencies. Several respondents mentioned even larger societal goals such as avoiding extinction as a human species and toxic nationalism, resolving climate change, and building a sustainable future.

In the second round of responses (2020), we saw much more emphasis on these bigger societal issues. The urgency to orient mathematics education (and its research) toward resolving these seemed to be felt more than before. In short, it was stressed that our planet needs to be saved. The big question is what role mathematics education can play in meeting these challenges.

3.2.2 Educational goals

Several respondents expressed a concern that the current goals of mathematics education do not reflect humanity’s and societies’ needs and interests well. Educational goals to be stressed more were mathematical literacy, numeracy, critical, and creative thinking—often with reference to the changing world and the planet being at risk. In particular, the impact of technology was frequently stressed, as this may have an impact on what people need to learn (cf. Gravemeijer et al., 2017 ). If computers can do particular things much better than people, what is it that students need to learn?

Among the most frequently mentioned educational goals for mathematics education were statistical literacy, computational and algorithmic thinking, artificial intelligence, modeling, and data science. More generally, respondents expressed that mathematics education should help learners deploy evidence, reasoning, argumentation, and proof. For example, Michelle Stephan (USA) asked:

What mathematics content should be taught today to prepare students for jobs of the future, especially given growth of the digital world and its impact on a global economy? All of the mathematics content in K-12 can be accomplished by computers, so what mathematical procedures become less important and what domains need to be explored more fully (e.g., statistics and big data, spatial geometry, functional reasoning, etc.)?

One challenge for research is that there is no clear methodology to arrive at relevant and feasible learning goals. Yet there is a need to choose and formulate such goals on the basis of research (cf. Van den Heuvel-Panhuizen, 2005 ).

Several of the 2020 responses mentioned the sometimes problematic way in which numbers, data, and graphs are used in the public sphere (e.g., Ernest, 2020 ; Kwon et al., 2021 ; Yoon et al., 2021 ). Many respondents saw their emphasis on relevant educational goals reinforced, for example, statistical and data literacy, modeling, critical thinking, and public communication. A few pandemic-specific topics were mentioned, such as exponential growth.

3.3 Relation of mathematics education to other practices

Many responses can be characterized as highlighting boundary crossing (Akkerman & Bakker, 2011 ) with disciplines or communities outside mathematics education, such as in science, technology, engineering, art, and mathematics education (STEM or STEAM); parents or families; the workplace; and leisure (e.g., drama, music, sports). An interesting example was the educational potential of mathematical memes—“humorous digital objects created by web users copying an existing image and overlaying a personal caption” (Bini et al., 2020 , p. 2). These boundary crossing-related responses thus emphasize the movements and connections between mathematics education and other practices.

In the 2020 responses, we saw that during the pandemic, the relationship between school and home has become much more important, because most students were (and perhaps still are) learning at home. Earlier research on parental involvement and homework (Civil & Bernier, 2006 ; de Abreu et al., 2006 ; Jackson, 2011 ) proves relevant in the current situation where many countries are still or again in lockdown. Respondents pointed to the need to monitor students and their work and to promote self-regulation. They also put more stress on the political, economic, and financial contexts in which mathematics education functions (or malfunctions, in many respondents’ views).

3.4 Teacher professional development

Respondents explicitly mentioned teacher professional development as an important domain of mathematics education research (including teacher educators’ development). For example, Loide Kapenda (Namibia) wrote, “I am supporting UNESCO whose idea is to focus on how we prepare teachers for the future we want.” (e.g., UNESCO, 2015 ) And, Francisco Rojas (Chile) wrote:

Although the field of mathematics education is broad and each time faced with new challenges (socio-political demands, new intercultural contexts, digital environments, etc.), all of them will be handled at school by the mathematics teacher, both in primary as well as in secondary education. Therefore, from my point of view, pre-service teacher education is one of the most relevant fields of research for the next decade, especially in developing countries.

It is evident from the responses that teaching mathematics is done by a large variety of people, not only by people who are trained as primary school teachers, secondary school mathematics teachers, or mathematicians but also parents, out-of-field teachers, and scientists whose primary discipline is not mathematics but who do use mathematics or statistics. How teachers of mathematics are trained varies accordingly. Respondents frequently pointed to the importance of subject-matter knowledge and particularly noted that many teachers seem ill-prepared to teach statistics (e.g., Lonneke Boels, the Netherlands).

Key questions were raised by several colleagues: “How to train mathematics teachers with a solid foundation in mathematics, positive attitudes towards mathematics teaching and learning, and wide knowledge base linking to STEM?” (anonymous); “What professional development, particularly at the post-secondary level, motivates changes in teaching practices in order to provide students the opportunities to engage with mathematics and be successful?” (Laura Watkins, USA); “How can mathematics educators equip students for sustainable, equitable citizenship? And how can mathematics education equip teachers to support students in this?” (David Wagner, Canada)

In the 2020 responses, it was clear that teachers are incredibly important, especially in the pandemic era. The sudden change to online teaching means that

higher requirements are put forward for teachers’ educational and teaching ability, especially the ability to carry out education and teaching by using information technology should be strengthened. Secondly, teachers’ ability to communicate and cooperate has been injected with new connotation. (Guangming Wang, China)

It is broadly assumed that education will stay partly online, though more so in higher levels of education than in primary education. This has implications for teachers, for instance, they will have to think through how they intend to coordinate teaching on location and online. Hence, one important focus for professional development is the use of technology.

3.5 Technology

Technology deserves to be called a theme in itself, but we want to emphasize that it ran through most of the other themes. First of all, some respondents argued that, due to technological advances in society, the societal and educational goals of mathematics education need to be changed (e.g., computational thinking to ensure employability in a technological society). Second, responses indicated that the changed goals have implications for the approaches in mathematics education. Consider the required curriculum reform and the digital tools to be used in it. Students do not only need to learn to use technology; the technology can also be used to learn mathematics (e.g., visualization, embodied design, statistical thinking). New technologies such as 3D printing, photo math, and augmented and virtual reality offer new opportunities for learning. Society has changed very fast in this respect. Third, technology is suggested to assist in establishing connections with other practices , such as between school and home, or vocational education and work, even though there is a great disparity in how successful these connections are.

In the 2020 responses, there was great concern about the current digital divide (cf. Hodgen et al., 2020 ). The COVID-19 pandemic has thus given cause for mathematics education research to understand better how connections across educational and other practices can be improved with the help of technology. Given the unequal distribution of help by parents or guardians, it becomes all the more important to think through how teachers can use videos and quizzes, how they can monitor their students, how they can assess them (while respecting privacy), and how one can compensate for the lack of social, gestural, and embodied interaction that is possible when being together physically.

Where mobile technology was considered very innovative before 2010, smartphones have become central devices in mathematics education in the pandemic with its reliance on distance learning. Our direct experience showed that phone applications such as WhatsApp and WeChat have become key tools in teaching and learning mathematics in many rural areas in various continents where few people have computers (for a report on podcasts distributed through WhatsApp, community loudspeakers, and local radio stations in Colombia, see Saenz et al., 2020 ).

3.6 Equity, diversity, and inclusion

Another cross-cutting theme can be labeled “equity, diversity, and inclusion.” We use this triplet to cover any topic that highlights these and related human values such as equality, social and racial justice, social emancipation, and democracy that were also mentioned by respondents (cf. Dobie & Sherin, 2021 ). In terms of educational goals , many respondents stressed that mathematics education should be for all students, including those who have special needs, who live in poverty, who are learning the instruction language, who have a migration background, who consider themselves LGBTQ+, have a traumatic or violent history, or are in whatever way marginalized. There is broad consensus that everyone should have access to high-quality mathematics education. However, as Niral Shah (USA) notes, less attention has been paid to “how phenomena related to social markers (e.g., race, class, gender) interact with phenomena related to the teaching and learning of mathematical content.”

In terms of teaching approaches , mathematics education is characterized by some respondents from particular countries as predominantly a white space where some groups feel or are excluded (cf. Battey, 2013 ). There is a general concern that current practices of teaching mathematics may perpetuate inequality, in particular in the current pandemic. In terms of assessment , mathematics is too often used or experienced as a gatekeeper rather than as a powerful resource (cf. Martin et al., 2010 ). Steve Lerman (UK) “indicates that understanding how educational opportunities are distributed inequitably, and in particular how that manifests in each end every classroom, is a prerequisite to making changes that can make some impact on redistribution.” A key research aim therefore is to understand what excludes students from learning mathematics and what would make mathematics education more inclusive (cf. Roos, 2019 ). And, what does professional development of teachers that promotes equity look like?

In 2020, many respondents saw their emphasis on equity and related values reinforced in the current pandemic with its risks of a digital divide, unequal access to high-quality mathematics education, and unfair distribution of resources. A related future research theme is how the so-called widening achievement gaps can be remedied (cf. Bawa, 2020 ). However, warnings were also formulated that thinking in such deficit terms can perpetuate inequality (cf. Svensson et al., 2014 ). A question raised by Dor Abrahamson (USA) is, “What roles could digital technology play, and in what forms, in restoring justice and celebrating diversity?”

Though entangled with many other themes, affect is also worth highlighting as a theme in itself. We use the term affect in a very broad sense to point to psychological-social phenomena such as emotion, love, belief, attitudes, interest, curiosity, fun, engagement, joy, involvement, motivation, self-esteem, identity, anxiety, alienation, and feeling of safety (cf. Cobb et al., 2009 ; Darragh, 2016 ; Hannula, 2019 ; Schukajlow et al., 2017 ). Many respondents emphasized the importance of studying these constructs in relation to (and not separate from) what is characterized as cognition. Some respondents pointed out that affect is not just an individual but also a social phenomenon, just like learning (cf. Chronaki, 2019 ; de Freitas et al., 2019 ; Schindler & Bakker, 2020 ).

Among the educational goals of mathematics education, several participants mentioned the need to generate and foster interest in mathematics. In terms of approaches , much emphasis was put on the need to avoid anxiety and alienation and to engage students in mathematical activity.

In the 2020 responses, more emphasis was put on the concern about alienation, which seems to be of special concern when students are socially distanced from peers and teachers as to when teaching takes place only through technology . What was reiterated in the 2020 responses was the importance of students’ sense of belonging in a mathematics classroom (cf. Horn, 2017 )—a topic closely related to the theme of equity, diversity, and inclusion discussed before.

3.8 Assessment

Assessment and evaluation were not often mentioned explicitly, but they do not seem less important than the other related themes. A key challenge is to assess what we value rather than valuing what we assess. In previous research, the assessment of individual students has received much attention, but what seems to be neglected is the evaluation of curricula. As Chongyang Wang (China) wrote, “How to evaluate the curriculum reforms. When we pay much energy in reforming our education and curriculum, do we imagine how to ensure it will work and there will be pieces of evidence found after the new curricula are carried out? How to prove the reforms work and matter?” (cf. Shimizu & Vithal, 2019 )

In the 2020 responses, there was an emphasis on assessment at a distance. Distance education generally is faced with the challenge of evaluating student work, both formatively and summatively. We predict that so-called e-assessment, along with its privacy challenges, will generate much research interest in the near future (cf. Bickerton & Sangwin, 2020 ).

4 Mathematics education research itself

Although we only asked for future themes, many respondents made interesting comments about research in mathematics education and its connections with other disciplines and practices (such as educational practice, policy, home settings). We have grouped these considerations under the subheadings of theory, methodology, reflection on our discipline, and interdisciplinarity and transdisciplinarity. As with the previous categorization into themes, we stress that these four types are not mutually exclusive as theoretical and methodological considerations can be intricately intertwined (Radford, 2008 ).

Several respondents expressed their concern about the fragmentation and diversity of theories used in mathematics education research (cf. Bikner-Ahsbahs & Prediger, 2014 ). The question was raised how mathematics educators can “work together to obtain valid, reliable, replicable, and useful findings in our field” and “How, as a discipline, can we encourage sustained research on core questions using commensurable perspectives and methods?” (Keith Weber, USA). One wish was “comparing theoretical perspectives for explanatory power” (K. Subramaniam, India). At the same time, it was stressed that “we cannot continue to pretend that there is just one culture in the field of mathematics education, that all the theoretical framework may be applied in whichever culture and that results are universal” (Mariolina Bartolini Bussi, Italy). In addition, the wish was expressed to deepen theoretical notions such as numeracy, equity, and justice as they play out in mathematics education.

4.2 Methodology

Many methodological approaches were mentioned as potentially useful in mathematics education research: randomized studies, experimental studies, replication, case studies, and so forth. Particular attention was paid to “complementary methodologies that bridge the ‘gap’ between mathematics education research and research on mathematical cognition” (Christian Bokhove, UK), as, for example, done in Gilmore et al. ( 2018 ). Also, approaches were mentioned that intend to bridge the so-called gap between educational practice and research, such as lesson study and design research. For example, Kay Owens (Australia) pointed to the challenge of studying cultural context and identity: “Such research requires a multi-faceted research methodology that may need to be further teased out from our current qualitative (e.g., ethnographic) and quantitative approaches (‘paper and pencil’ (including computing) testing). Design research may provide further possibilities.”

Francisco Rojas (Chile) highlighted the need for more longitudinal and cross-sectional research, in particular in the context of teacher professional development:

It is not enough to investigate what happens in pre-service teacher education but understand what effects this training has in the first years of the professional career of the new teachers of mathematics, both in primary and secondary education. Therefore, increasingly more longitudinal and cross-sectional studies will be required to understand the complexity of the practice of mathematics teachers, how the professional knowledge that articulates the practice evolves, and what effects have the practice of teachers on the students’ learning of mathematics.

4.3 Reflection on our discipline

Calls were made for critical reflection on our discipline. One anonymous appeal was for more self-criticism and scientific modesty: Is research delivering, or is it drawing away good teachers from teaching? Do we do research primarily to help improve mathematics education or to better understand phenomena? (cf. Proulx & Maheux, 2019 ) The general gist of the responses was a sincere wish to be of value to the world and mathematics education more specifically and not only do “research for the sake of research” (Zahra Gooya, Iran). David Bowers (USA) expressed several reflection-inviting views about the nature of our discipline, for example:

We must normalize (and expect) the full taking up the philosophical and theoretical underpinnings of all of our work (even work that is not considered “philosophical”). Not doing so leads to uncritical analysis and implications.

We must develop norms wherein it is considered embarrassing to do “uncritical” research.

There is no such thing as “neutral.” Amongst other things, this means that we should be cultivating norms that recognize the inherent political nature of all work, and norms that acknowledge how superficially “neutral” work tends to empower the oppressor.

We must recognize the existence of but not cater to the fragility of privilege.

In terms of what is studied, some respondents felt that the mathematics education research “literature has been moving away from the original goals of mathematics education. We seem to have been investigating everything but the actual learning of important mathematics topics.” (Lyn English, Australia) In terms of the nature of our discipline, Taro Fujita (UK) argued that our discipline can be characterized as a design science, with designing mathematical learning environments as the core of research activities (cf. Wittmann, 1995 ).

A tension that we observe in different views is the following: On the one hand, mathematics education research has its origin in helping teachers teach particular content better. The need for such so-called didactical, topic-specific research is not less important today but perhaps less fashionable for funding schemes that promote innovative, ground-breaking research. On the other hand, over time it has become clear that mathematics education is a multi-faceted socio-cultural and political endeavor under the influence of many local and global powers. It is therefore not surprising that the field of mathematics education research has expanded so as to include an increasingly wide scope of themes that are at stake, such as the marginalization of particular groups. We therefore highlight Niral Shah’s (USA) response that “historically, these domains of research [content-specific vs socio-political] have been decoupled. The field would get closer to understanding the experiences of minoritized students if we could connect these lines of inquiry.”

Another interesting reflective theme was raised by Nouzha El Yacoubi (Morocco): To what extent can we transpose “research questions from developed to developing countries”? As members of the plenary panel at PME 2019 (e.g., Kazima, 2019 ; Kim, 2019 ; Li, 2019 ) conveyed well, adopting interventions that were successful in one place in another place is far from trivial (cf. Gorard, 2020 ).

Juan L. Piñeiro (Spain in 2019, Chile in 2020) highlighted that “mathematical concepts and processes have different natures. Therefore, can it be characterized using the same theoretical and methodological tools?” More generally, one may ask if our theories and methodologies—often borrowed from other disciplines—are well suited to the ontology of our own discipline. A discussion started by Niss ( 2019 ) on the nature of our discipline, responded to by Bakker ( 2019 ) and Cai and Hwang ( 2019 ), seems worth continuing.

An important question raised in several comments is how close research should be to existing curricula. One respondent (Benjamin Rott, Germany) noted that research on problem posing often does “not fit into school curricula.” This makes the application of research ideas and findings problematic. However, one could argue that research need not always be tied to existing (local) educational contexts. It can also be inspirational, seeking principles of what is possible (and how) with a longer-term view on how curricula may change in the future. One option is, as Simon Zell (Germany) suggests, to test designs that cover a longer timeframe than typically done. Another way to bridge these two extremes is “collaboration between teachers and researchers in designing and publishing research” (K. Subramaniam, India) as is promoted by facilitating teachers to do PhD research (Bakx et al., 2016 ).

One of the responding teacher-researchers (Lonneke Boels, the Netherlands) expressed the wish that research would become available “in a more accessible form.” This wish raises the more general questions of whose responsibility it is to do such translation work and how to communicate with non-researchers. Do we need a particular type of communication research within mathematics education to learn how to convey particular key ideas or solid findings? (cf. Bosch et al., 2017 )

4.4 Interdisciplinarity and transdisciplinarity

Many respondents mentioned disciplines which mathematics education research can learn from or should collaborate with (cf. Suazo-Flores et al., 2021 ). Examples are history, mathematics, philosophy, psychology, psychometry, pedagogy, educational science, value education (social, emotional), race theory, urban education, neuroscience/brain research, cognitive science, and computer science didactics. “A big challenge here is how to make diverse experts approach and talk to one another in a productive way.” (David Gómez, Chile)

One of the most frequently mentioned disciplines in relation to our field is history. It is a common complaint in, for instance, the history of medicine that historians accuse medical experts of not knowing historical research and that medical experts accuse historians of not understanding the medical discipline well enough (Beckers & Beckers, 2019 ). This tension raises the question who does and should do research into the history of mathematics or of mathematics education and to what broader purpose.

Some responses go beyond interdisciplinarity, because resolving the bigger issues such as climate change and a more equitable society require collaboration with non-researchers (transdisciplinarity). A typical example is the involvement of educational practice and policy when improving mathematics education (e.g., Potari et al., 2019 ).

Let us end this section with a word of hope, from an anonymous respondent: “I still believe (or hope?) that the pandemic, with this making-inequities-explicit, would help mathematics educators to look at persistent and systemic inequalities more consistently in the coming years.” Having learned so much in the past year could indeed provide an opportunity to establish a more equitable “new normal,” rather than a reversion to the old normal, which one reviewer worried about.

5 The themes in their coherence: an artistic impression

As described above, we identified eight themes of mathematics education research for the future, which we discussed one by one. The disadvantage of this list-wise discussion is that the entanglement of the themes is backgrounded. To compensate for that drawback, we here render a brief interpretation of the drawing of Fig. 1 . While doing so, we invite readers to use their own creative imagination and perhaps use the drawing for other purposes (e.g., ask researchers, students, or teachers: Where would you like to be in this landscape? What mathematical ideas do you spot?). The drawing mainly focuses on the themes that emerged from the first round of responses but also hints at experiences from the time of the pandemic, for instance distance education. In Appendix 1 , we specify more of the details in the drawing and we provide a link to an annotated image (available at https://www.fisme.science.uu.nl/toepassingen/28937/ ).

The boat on the river aims to represent teaching approaches. The hand drawing of the boat hints at the importance of educational design: A particular approach is being worked out. On the boat, a teacher and students work together toward educational and societal goals, further down the river. The graduation bridge is an intermediate educational goal to pass, after which there are many paths leading to other goals such as higher education, citizenship, and work in society. Relations to practices outside mathematics education are also shown. In the left bottom corner, the house and parents working and playing with children represent the link of education with the home situation and leisure activity.

The teacher, represented by the captain in the foreground of the ship, is engaged in professional development, consulting a book, but also learning by doing (cf. Bakkenes et al., 2010 , on experimenting, using resources, etc.). Apart from graduation, there are other types of goals for teachers and students alike, such as equity, positive affect, and fluent use of technology. During their journey (and partially at home, shown in the left bottom corner), students learn to orient themselves in the world mathematically (e.g., fractal tree, elliptical lake, a parabolic mountain, and various platonic solids). On their way toward various goals, both teacher and students use particular technology (e.g., compass, binoculars, tablet, laptop). The magnifying glass (representing research) zooms in on a laptop screen that portrays distance education, hinting at the consensus that the pandemic magnifies some issues that education was already facing (e.g., the digital divide).

Equity, diversity, and inclusion are represented with the rainbow, overarching everything. On the boat, students are treated equally and the sailing practice is inclusive in the sense that all perform at their own level—getting the support they need while contributing meaningfully to the shared activity. This is at least what we read into the image. Affect is visible in various ways. First of all, the weather represents moods in general (rainy and dark side on the left; sunny bright side on the right). Second, the individual students (e.g., in the crow’s nest) are interested in, anxious about, and attentive to the things coming up during their journey. They are motivated to engage in all kinds of tasks (handling the sails, playing a game of chance with a die, standing guard in the crow’s nest, etc.). On the bridge, the graduates’ pride and happiness hints at positive affect as an educational goal but also represents the exam part of the assessment. The assessment also happens in terms of checks and feedback on the boat. The two people next to the house (one with a camera, one measuring) can be seen as assessors or researchers observing and evaluating the progress on the ship or the ship’s progress.

More generally, the three types of boats in the drawing represent three different spaces, which Hannah Arendt ( 1958 ) would characterize as private (paper-folded boat near the boy and a small toy boat next to the girl with her father at home), public/political (ships at the horizon), and the in-between space of education (the boat with the teacher and students). The students and teacher on the boat illustrate school as a special pedagogic form. Masschelein and Simons ( 2019 ) argue that the ancient Greek idea behind school (σχολή, scholè , free time) is that students should all be treated as equal and should all get equal opportunities. At school, their descent does not matter. At school, there is time to study, to make mistakes, without having to work for a living. At school, they learn to collaborate with others from diverse backgrounds, in preparation for future life in the public space. One challenge of the lockdown situation as a consequence of the pandemic is how to organize this in-between space in a way that upholds its special pedagogic form.

6 Research challenges

Based on the eight themes and considerations about mathematics education research itself, we formulate a set of research challenges that strike us as deserving further discussion (cf. Stephan et al., 2015 ). We do not intend to suggest these are more important than others or that some other themes are less worthy of investigation, nor do we suggest that they entail a research agenda (cf. English, 2008 ).

6.1 Aligning new goals, curricula, and teaching approaches

There seems to be relatively little attention within mathematics education research for curricular issues, including topics such as learning goals, curriculum standards, syllabi, learning progressions, textbook analysis, curricular coherence, and alignment with other curricula. Yet we feel that we as mathematics education researchers should care about these topics as they may not necessarily be covered by other disciplines. For example, judging from Deng’s ( 2018 ) complaint about the trends in the discipline of curriculum studies, we cannot assume scholars in that field to address issues specific to the mathematics-focused curriculum (e.g., the Journal of Curriculum Studies and Curriculum Inquiry have published only a limited number of studies on mathematics curricula).

Learning goals form an important element of curricula or standards. It is relatively easy to formulate important goals in general terms (e.g., critical thinking or problem solving). As a specific example, consider mathematical problem posing (Cai & Leikin, 2020 ), which curriculum standards have specifically pointed out as an important educational goal—developing students’ problem-posing skills. Students should be provided opportunities to formulate their own problems based on situations. However, there are few problem-posing activities in current mathematics textbooks and classroom instruction (Cai & Jiang, 2017 ). A similar observation can be made about problem solving in Dutch primary textbooks (Kolovou et al., 2009 ). Hence, there is a need for researchers and educators to align problem posing in curriculum standards, textbooks, classroom instruction, and students’ learning.

The challenge we see for mathematics education researchers is to collaborate with scholars from other disciplines (interdisciplinarity) and with non-researchers (transdisciplinarity) in figuring out how the desired societal and educational goals can be shaped in mathematics education. Our discipline has developed several methodological approaches that may help in formulating learning goals and accompanying teaching approaches (cf. Van den Heuvel-Panhuizen, 2005 ), including epistemological analyses (Sierpinska, 1990 ), historical and didactical phenomenology (Bakker & Gravemeijer, 2006 ; Freudenthal, 1986 ), and workplace studies (Bessot & Ridgway, 2000 ; Hoyles et al., 2001 ). However, how should the outcomes of such research approaches be weighed against each other and combined to formulate learning goals for a balanced, coherent curriculum? What is the role of mathematics education researchers in relation to teachers, policymakers, and other stakeholders (Potari et al., 2019 )? In our discipline, we seem to lack a research-informed way of arriving at the formulation of suitable educational goals without overloading the curricula.

6.2 Researching mathematics education across contexts

Though methodologically and theoretically challenging, it is of great importance to study learning and teaching mathematics across contexts. After all, students do not just learn at school; they can also participate in informal settings (Nemirovsky et al., 2017 ), online forums, or affinity networks (Ito et al., 2018 ) where they may share for instance mathematical memes (Bini et al., 2020 ). Moreover, teachers are not the only ones teaching mathematics: Private tutors, friends, parents, siblings, or other relatives can also be involved in helping children with their mathematics. Mathematics learning could also be situated on streets or in museums, homes, and other informal settings. This was already acknowledged before 2020, but the pandemic has scattered learners and teachers away from the typical central school locations and thus shifted the distribution of labor.

In particular, physical and virtual spaces of learning have been reconfigured due to the pandemic. Issues of timing also work differently online, for example, if students can watch online lectures or videos whenever they like (asynchronously). Such reconfigurations of space and time also have an effect on the rhythm of education and hence on people’s energy levels (cf. Lefebvre, 2004 ). More specifically, the reconfiguration of the situation has affected many students’ levels of motivation and concentration (e.g., Meeter et al., 2020 ). As Engelbrecht et al. ( 2020 ) acknowledged, the pandemic has drastically changed the teaching and learning model as we knew it. It is quite possible that some existing theories about teaching and learning no longer apply in the same way. An interesting question is whether and how existing theoretical frameworks can be adjusted or whether new theoretical orientations need to be developed to better understand and promote productive ways of blended or online teaching, across contexts.

6.3 Focusing teacher professional development

Professional development of teachers and teacher educators stands out from the survey as being in need of serious investment. How can teachers be prepared for the unpredictable, both in terms of beliefs and actions? During the pandemic, teachers have been under enormous pressure to make quick decisions in redesigning their courses, to learn to use new technological tools, to invent creative ways of assessment, and to do what was within their capacity to provide opportunities to their students for learning mathematics—even if technological tools were limited (e.g., if students had little or no computer or internet access at home). The pressure required both emotional adaption and instructional adjustment. Teachers quickly needed to find useful information, which raises questions about the accessibility of research insights. Given the new situation, limited resources, and the uncertain unfolding of education after lockdowns, focusing teacher professional development on necessary and useful topics will need much attention. In particular, there is a need for longitudinal studies to investigate how teachers’ learning actually affects teachers’ classroom instruction and students’ learning.

In the surveys, respondents mainly referred to teachers as K-12 school mathematics teachers, but some also stressed the importance of mathematics teacher educators (MTEs). In addition to conducting research in mathematics education, MTEs are acting in both the role of teacher educators and of mathematics teachers. There has been increased research on MTEs as requiring professional development (Goos & Beswick, 2021 ). Within the field of mathematics education, there is an emerging need and interest in how mathematics teacher educators themselves learn and develop. In fact, the changing situation also provides an opportunity to scrutinize our habitual ways of thinking and become aware of what Jullien ( 2018 ) calls the “un-thought”: What is it that we as educators and researchers have not seen or thought about so much about that the sudden reconfiguration of education forces us to reflect upon?

6.4 Using low-tech resources

Particular strands of research focus on innovative tools and their applications in education, even if they are at the time too expensive (even too labor intensive) to use at large scale. Such future-oriented studies can be very interesting given the rapid advances in technology and attractive to funding bodies focusing on innovation. Digital technology has become ubiquitous, both in schools and in everyday life, and there is already a significant body of work capitalizing on aspects of technology for research and practice in mathematics education.

However, as Cai et al. ( 2020 ) indicated, technology advances so quickly that addressing research problems may not depend so much on developing a new technological capability as on helping researchers and practitioners learn about new technologies and imagine effective ways to use them. Moreover, given the millions of students in rural areas who during the pandemic have only had access to low-tech resources such as podcasts, radio, TV, and perhaps WhatsApp through their parents’ phones, we would like to see more research on what learning, teaching, and assessing mathematics through limited tools such as Whatsapp or WeChat look like and how they can be improved. In fact, in China, a series of WeChat-based mini-lessons has been developed and delivered through the WeChat video function during the pandemic. Even when the pandemic is under control, mini-lessons are still developed and circulated through WeChat. We therefore think it is important to study the use and influence of low-tech resources in mathematics education.

6.5 Staying in touch online

With the majority of students learning at home, a major ongoing challenge for everyone has been how to stay in touch with each other and with mathematics. With less social interaction, without joint attention in the same physical space and at the same time, and with the collective only mediated by technology, becoming and staying motivated to learn has been a widely felt challenge. It is generally expected that in the higher levels of education, more blended or distant learning elements will be built into education. Careful research on the affective, embodied, and collective aspects of learning and teaching mathematics is required to overcome eventually the distance and alienation so widely experienced in online education. That is, we not only need to rethink social interactions between students and/or teachers in different settings but must also rethink how to engage and motivate students in online settings.

6.6 Studying and improving equity without perpetuating inequality

Several colleagues have warned, for a long time, that one risk of studying achievement gaps, differences between majority and minority groups, and so forth can also perpetuate inequity. Admittedly, pinpointing injustice and the need to invest in particular less privileged parts of education is necessary to redirect policymakers’ and teachers’ attention and gain funding. However, how can one reorient resources without stigmatizing? For example, Svensson et al. ( 2014 ) pointed out that research findings can fuel political debates about groups of people (e.g., parents with a migration background), who then may feel insecure about their own capacities. A challenge that we see is to identify and understand problematic situations without legitimizing problematic stereotyping (Hilt, 2015 ).

Furthermore, the field of mathematics education research does not have a consistent conceptualization of equity. There also seem to be regional differences: It struck us that equity is the more common term in the responses from the Americas, whereas inclusion and diversity were more often mentioned in the European responses. Future research will need to focus on both the conceptualization of equity and on improving equity and related values such as inclusion.

6.7 Assessing online

A key challenge is how to assess online and to do so more effectively. This challenge is related to both privacy, ethics, and performance issues. It is clear that online assessment may have significant advantages to assess student mathematics learning, such as more flexibility in test-taking and fast scoring. However, many teachers have faced privacy concerns, and we also have the impression that in an online environment it is even more challenging to successfully assess what we value rather than merely assessing what is relatively easy to assess. In particular, we need to systematically investigate any possible effect of administering assessments online as researchers have found a differential effect of online assessment versus paper-and-pencil assessment (Backes & Cowan, 2019 ). What further deserves careful ethical attention is what happens to learning analytics data that can and are collected when students work online.

6.8 Doing and publishing interdisciplinary research

When analyzing the responses, we were struck by a discrepancy between what respondents care about and what is typically researched and published in our monodisciplinary journals. Most of the challenges mentioned in this section require interdisciplinary or even transdisciplinary approaches (see also Burkhardt, 2019 ).

An overarching key question is: What role does mathematics education research play in addressing the bigger and more general challenges mentioned by our respondents? The importance of interdisciplinarity also raises a question about the scope of journals that focus on mathematics education research. Do we need to broaden the scope of monodisciplinary journals so that they can publish important research that combines mathematics education research with another disciplinary perspective? As editors, we see a place for interdisciplinary studies as long as there is one strong anchor in mathematics education research. In fact, there are many researchers who do not identify themselves as mathematics education researchers but who are currently doing high-quality work related to mathematics education in fields such as educational psychology and the cognitive and learning sciences. Encouraging the reporting of high-quality mathematics education research from a broader spectrum of researchers would serve to increase the impact of the mathematics education research journals in the wider educational arena. This, in turn, would serve to encourage further collaboration around mathematics education issues from various disciplines. Ultimately, mathematics education research journals could act as a hub for interdisciplinary collaboration to address the pressing questions of how mathematics is learned and taught.

7 Concluding remarks

In this paper, based on a survey conducted before and during the pandemic, we have examined how scholars in the field of mathematics education view the future of mathematics education research. On the one hand, there are no major surprises about the areas we need to focus on in the future; the themes are not new. On the other hand, the responses also show that the areas we have highlighted still persist and need further investigation (cf. OECD, 2020 ). But, there are a few areas, based on both the responses of the scholars and our own discussions and views, that stand out as requiring more attention. For example, we hope that these survey results will serve as propelling conversation about mathematics education research regarding online assessment and pedagogical considerations for virtual teaching.

The survey results are limited in two ways. The set of respondents to the survey is probably not representative of all mathematics education researchers in the world. In that regard, perhaps scholars in each country could use the same survey questions to survey representative samples within each country to understand how the scholars in that country view future research with respect to regional needs. The second limitation is related to the fact that mathematics education is a very culturally dependent field. Cultural differences in the teaching and learning of mathematics are well documented. Given the small numbers of responses from some continents, we did not break down the analysis for regional comparison. Representative samples from each country would help us see how scholars from different countries view research in mathematics education; they will add another layer of insights about mathematics education research to complement the results of the survey presented here. Nevertheless, we sincerely hope that the findings from the surveys will serve as a discussion point for the field of mathematics education to pursue continuous improvement.

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Acknowledgments

We thank Anna Sfard for her advice on the survey, based on her own survey published in Sfard ( 2005 ). We are grateful for Stephen Hwang’s careful copyediting for an earlier version of the manuscript. Thanks also to Elisabeth Angerer, Elske de Waal, Paul Ernest, Vilma Mesa, Michelle Stephan, David Wagner, and anonymous reviewers for their feedback on earlier drafts.

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Appendix 1: Explanation of Fig. 1

We have divided Fig. 1 in 12 rectangles called A1 (bottom left) up to C4 (top right) to explain the details (for image annotation go to https://www.fisme.science.uu.nl/toepassingen/28937 )

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Bakker, A., Cai, J. & Zenger, L. Future themes of mathematics education research: an international survey before and during the pandemic. Educ Stud Math 107 , 1–24 (2021). https://doi.org/10.1007/s10649-021-10049-w

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+ - Algebraic geometry Click to collapse

Geometric Invariant Theory : Faculty: Santosha Pattanayak

The geometry of algebraic varieties: :

The geometry of algebraic varieties with reductive group actions, including flag varieties, toric varieties, torus actions on algebraic varieties, and spherical varieties. I am also interested in studying the structure of algebraic groups and GIT (Geometric Invariant Theory) quotients of algebraic varieties and exploring toric degenerations of algebraic varieties. Moreover, my interest extends to areas such as the Seshadri constants of algebraic varieties and the symplectic invariants of smooth projective algebraic varieties, including the Gromov width. Faculty: Narasimha Chary Bonala

+ - Commutative Algebra Click to collapse

The research areas are Derivations, Higher Derivations, Differential Ideals, Multiplication Modules and the Radical Formula. Faculty: A. K. Maloo

+ - Complex Analysis & Operator Theory Click to collapse

I mainly consider various analytic function spaces defined on the unit disk or on some half plane of the complex plane and various operators on these spaces such as multiplication operators, composition operators, Cesaro operators. Also, I work on similar operators on some discrete function spaces defined on an infinite rooted tree (graph), in particular, on the discrete analogue of Hardy spaces. I deal with number of other problems which connects geometric function theory with function spaces and operator theory. Faculty: P. Muthukumar

+ - Computational Acoustics and Electromagnetics Click to collapse

The study of interaction of electromagnetic fields with physical objects and the environment constitutes the main subject matter of Computational Electromagnetics. One of the major challenges in this area of research is in the development of efficient, accurate and rapidly-convergent algorithms for the simulation of propagation and scattering of acoustic and electromagnetic fields within and around structures that possess complex geometrical characteristics. These problems are of fundamental importance in diverse fields, with applications ranging from space exploration, medical imaging and oil exploration on the civilian side to aircraft design and decoy detection on the military side - just to name a few. Computational modeling of electromagnetic scattering problems has typically been attempted on the basis of classical, low-order Finite-Difference-Time-Domain (FDTD) or Finite-Element-Method (FEM) approaches. An important computational alternative to these approaches is provided by boundary integral-equation formulations that we have adopted owing to a number of excellent properties that they enjoy. Listed below are some of the key areas of interest in related research: 1. Design of high-order integrators for boundary integral equations arising from surface and volumetric scattering of acoustic and electromagnetic waves from complex engineering structures including from open surfaces and from geometries with singular features like edges and corners. 2. Accurate representation of complex surfaces in three dimensions with applications to enhancement of low quality CAD models and in the development of direct CAD-to-EM tools. 3. High frequency scattering methods in three dimensions with frequency independent cost in the context of multiple scattering configurations. A related field of interest in this regard includes high-order geometrical optics simulator for inverse ray tracing. 4. High performance computing. Faculty: Akash Anand , B. V. Rathish Kumar

+ - Computational Fluid Dynamics Click to collapse

Development of Numerical Schemes for Incompressible Newtonian and Non-Newtonian Fluid Flows based on FDM, FEM, FVM, Wavelets, SEM, BEM etc. Development of Parallel Numerical Methods for Heat & Fluid Flow Analysis on Large Scale Parallel Computing systems based on MPI-OpenMP-Cuda programming concepts, ANN/ML methods for Flow Analysis. Global Climate Modelling on Very Large Scale Parallel Systems. Faculty: B. V. Rathish Kumar , Saktipada Ghorai

+ - Differential Equations Click to collapse

Semigroups of linear operators and their applications, Functional differential equations, Galerkin approximations

Many unsteady state physical problems are governed by partial differential equations of parabolic or hyperbolic types. These problems are mostly prototypes since they represent as members of large classes of such similar problems. So, to make a useful study of these problems we concentrate on their invariant properties which are satisfied by each member of the class. We reformulate these problems as evolution equations in abstract spaces such as Hilbert or more generally Banach spaces. The operators appearing in these equations have the property that they are the generators of semigroups. The theory of semigroups then plays an important role of establishing the well-posedness of these evolution equations. The analysis of functional differential equations enhances the applicability of evolution equations as these include the equations involving finite as well as infinite delays. Equations involving integrals can also be tackled using the techniques of functional differential equations. The Galerkin method and its nonlinear variants are fundamental tools to obtain the approximate solutions of the evolution and functional differential equations.

  Faculty: D. Bahuguna

Homogenization and Variational methods for partial differential equation

The main interest is on Aysmptotic Analysis of partial differential equations. This is a technique to understand the macroscopic behaviour of a composite medium through its microscopic properties. The technique is commonly used for PDE with highly oscillating coefficients. The idea is to replace a given heterogeneous medium by a fictitious homogeneous one (the `homogenized' material) for numerical computations. The technique is also known as ``Multi scale analysis''. The known and unknown quantities in the study of physical or mechanical processes in a medium with micro structure depend on a small parameter $\varepsilon$. The study of the limit as $ \varepsilon \rightarrow0 $, is the aim of the mathematical theory of homogenization. The notion of $G$-convergence, $H$-convergence, two-scale convergence are some examples of the techniques employed for specific cases. The variational characterization of the technique for problems in calculus of variations is given by $\Gamma$- convergence.

  Faculty: T. Muthukumar , B.V. Rathish Kumar

Functional inequalities on Sobolev space

Sobolev spaces are the natural spaces where one looks for solutions of Partial differential equations (PDEs). Functional inequalities on this spaces ( for example Moser-Trudinger Inequality, Poincare Inequality, Hardy- Sobolev Inequality and many other) plays a very significant role in establishing existence of solutions for various PDEs. Existence of extremal function for such inequalities is another key aspect that is investigated

Asymptotic analysis on changing domains

Study of differential equations on long cylinders appears naturally in various branches of Physics, Engineering applications and real life problems. Problems (not necessarily PDEs, can be purely variational in nature) set on cylindrical domains whose length tends to infinity, is analysed. Faculty: Prosenjit Roy , Kaushik Bal , Indranil Chowdhury

Analysis of Nonlinear PDEs involving fractional/nonlocal operator:

Fully nonlinear elliptic and parabolic equation involving nonlocal operators. Equations motivated from stochastic control/ Game problems including Hamilton Jacobi Bellman Equations, Isaacs Equations, Mean Field Games problems -

• Viscosity Solution theory, Comparison Principle, Wellposedness theory, Stability, Continuous Dependence.
• Numerical Analysis –Wellposedness, Convergence, Error estimates of Finite difference method, Semi-lagrangian Method.

  Faculty: Indranil Chowdhury

Control Theory and its applications:

We study several aspects of controllability, say exact controllability, null controllability, approximate controllability, controllability to the trajectories of a given system of ordinary and partial differential equations (both linear and nonlinear). We study stabilizability (exponential, asymptotic) of a system of differential equations and construct feedback control for that system. Currently we are studying controllability of reaction diffusion systems of partial equations using Carleman inequalities and fixed point technique. Multiplier techniques are also used to show controllability of system of hyperbolic partial differential equations. Several mixed systems (hyperbolic and parabolic) are also been studied.

  Faculty: Mrinmay Biswas

+ - Functional Analysis & Operator Theory Click to collapse

Banach space theory

Geometric and proximinality aspects in Banach spaces. Faculty: P. Shunmugaraj

Function-theoretic and graph-theoretic operator theory

The primary goal is to implement methods from the complex function theory and the graph theory into the multivariable operator theory. The topics of interests include de Branges-Rovnyak spaces and weighted shifts on directed graphs. Faculty: Sameer Chavan

Non-commutative geometry

The main emphasis is on the metric aspect of noncommutative geometry. Faculty: Satyajit Guin

Bounded linear operators

A central theme in operator theory is the study of B(H), the algebra of bounded linear operators on a separable complex Hilbert space. We focus on operator ideals, subideals and commutators of compact operators in B(H). There is also a continuing interest in semigroups of operators in B(H) from different perspectives. We work in operator semigroups involve characterization of special classes of semigroups which relate to solving certain operator equations. Faculty: Sasmita Patnaik + - Harmonic Analysis Click to collapse

Operator spaces

The main emphasis is on operator space techniques in abstract Harmonic Analysis.

In the Euclidean setting

Analysis, boundedness and weighted boundedness of singular integral operators are major thrust areas in the department. In abstract Harmonic analysis we do work in studying Lacunary sets in the noncommutative Lp spaces.

  Faculty: Parasar Mohanty  

On Lie groups

Problems related to integral geometry on Lie groups are being studied.

  Faculty: Rama Rawat  

  + - Homological Algebra Click to collapse

Cohomology and Deformation theory of algebraic structures

Research work in this area encompasses cohomology and deformation theory of algebraic structures, mainly focusing on Lie and Leibniz algebras arising out of topology and geometry. In particular, one is interested in the cohomology and Versal deformation for Lie and Leibniz brackets on the space of sections of vector bundles e.g. Lie algebroids and Courant algebroids.

This study naturally relate questions about other algebraic structures which include Lie-Rinehart algebras, hom-Lie-Rinehart algebras, Hom-Gerstenhaber algebras, homotopy algebras associated to Courant algebras, higher categories and related fields.

  Faculty: Ashis Mandal + - Image Processing Click to collapse

TPDE based Image processing for Denoising, Inpainting, Classification, Compression, Registration, Optical flow analysis etc. Bio-Medical Image Analysis based on CT/MRI/US clinical data, ANN/ML methods in Image Analysis, Wavelet methods for Image processing.

  Faculty: B. V. Rathish Kumar + - Mathematical Biology Click to collapse

There is an active group working in the area of Mathematical Biology. The research is carried out in the following directions.

Mathematical ecology

1. Research in this area is focused on the local and global stability analysis, detection of possible bifurcation scenario and derivation of normal form, chaotic dynamics for the ordinary as well as delay differential equation models, stochastic stability analysis for stochastic differential equation model systems and analysis of noise induced phenomena. Also the possible spatio-temporal pattern formation is studied for the models of interacting populations dispersing over two dimensional landscape.

2. Mathematical Modeling of the survival of species in polluted water bodies; depletion of dissolved oxygen in water bodies due to organic pollutants.

Mathematical epidemiology

1. Mathematical Modeling of epidemics using stability analysis; effects of environmental, demographic and ecological factors.

2. Mathematical Modeling of HIV Dynamics in vivo

Bioconvection

Bioconvection is the process of spontaneous pattern formation in a suspension of swimming microorganisms. These patterns are associated with up- and down-welling of the fluid. Bioconvection is due to the individual and collective behaviours of the micro-organisms suspended in a fluid. The physical and biological mechanisms of bioconvection are investigated by developing mathematical models and analysing them using a variety of linear, nonlinear and computational techniques.

Bio-fluid dynamics

Mathematical Models for blood flow in cardiovascular system; renal flows; Peristaltic transport; mucus transport; synovial joint lubrication.

  Faculty: Malay Banerjee , Saktipada Ghorai , B.V. Rathish Kumar  

Cardiac electrophysiology

Theory, Modeling & Simulation of Cardiac Electrical Activity (CEA) in Human Cardiac Tissue based on PDEODE models such as Monodomain Model, Biodomain model, Cardiac Arrhythmia, pace makers etc

  Faculty: B.V. Rathish Kumar + - Number Theory & Arithmetic Geometry Click to collapse

Algebraic number theory and Arithmetic geometry

Iwasawa Theory of elliptic curves and modular forms, Galois representations, Congruences between special values of L-functions.

  Faculty: Sudhanshu Shekhar

Analytic number theory

L-functions, sub-convexity problems, Sieve method

  Faculty: Saurabh Kumar Singh

Number theory and Arithmetic geometry

Iwasawa Theory of elliptic curves and modular forms, Selmer groups

  Faculty: Somnath Jha

Number theory, Dynamical systems, Random walks on groups

During the last four decades, it has been realized that some problems in number theory and, in particular, in Diophantine approximation, can be solved using techniques from the theory of homogeneous dynamics, random walks on homogeneous spaces etc. Indeed, one translates such problems into a problem on the behavior of certain trajectories in homogeneous spaces of Lie groups under flows or random walks; and subsequently resolves using very powerful techniques from the theory of dynamics on homogeneous spaces, random walk etc. I undertake this theme.

  Faculty: Arijit Ganguly + - Numerical Analysis and Scientific Computing Click to collapse

The faculty group in the area of Numerical Analysis & Scientific Computing are very actively engaged in high-quality research in the areas that include (but are not limited to): Singular Perturbation problems, Multiscale Phenomena, Hyperbolic Conservation Laws, Elliptic and Parabolic PDEs, Integral Equations, Computational Acoustics and Electromagnetics, Computational Fluid Dynamics, Computer-Aided Tomography and Parallel Computing. The faculty group is involved in the development, analysis, and application of efficient and robust algorithms for solving challenging problems arising in several applied areas. There is expertise in several discretization methods that include: Finite Difference Methods, Finite Element Methods, Spectral Element Methods, Boundary Element Methods, Nyström Method, Spline and Wavelet approximations, etc. This encompasses a very high level of computation that requires software skills of the highest order and parallel computing as well.

  Faculty: B. V. Rathish Kumar , Akash Anand , Indranil Chowdhury + - Operator Algebra Click to collapse

Broadly speaking, I work with topics in C*-algebras and von Neumann algebras. More precisely, my work involves Jones theory of subfactors and planar algebras.

  Faculty: Keshab Chandra Bakshi + - Representation Theory Click to collapse

Representation of Lie and linear algebraic groups over local fields, Representation-theoretic methods, automorphic representations over local and global fields, Linear algebraic groups and related topics MSC classification (22E50, 11F70, 20Gxx:)

  Faculty: Santosh Nadimpalli

Representations of finite and arithmetic groups

Current research interests: Representations of Linear groups over local rings, Projective representations of finite and arithmetic groups, Applications of representation theory.

  Faculty: Pooja Singla

Representation theory of Lie algebras and algebraic groups

 Faculty: Santosha Pattanayak

Representation theory of infinite dimensional Lie algebras

Current research interest: Representation theory of Kac-Moody algebras; Toroidal Lie algebras and extended affine Lie algebras.

  Faculty: Sachin S. Sharma

Representation theory and Invariant theory

Current research interest: Representation and structure theory of algebraic groups, Classical invariant theory of reductive algebraic groups and associated Weyl groups.

  Faculty: Preena Samuel

Combinatorial representation theory

String algebras form a class of tame representation type algebras that are presented combinatorially using quivers and relations. Currently I am interested in studying the combinatorics of strings to understand the Auslander-Reiten quiver that encodes the generators for the category of finite length R-modules as well as the Ziegler spectrum associated with string algebras whose topology is described model-theoretically

  Faculty: Amit Kuber

Representation Theory of Algebraic groups:

Representation theory of Algebraic groups and Lie algebras, and its applications to Invariant theory and Algebraic geometry.

  Faculty: Narasimha Chary Bonala + - Set Theory and Logic Click to collapse

Set theory (MSC Classification 03Exx)

We apply tools from set theory to problems from other areas of mathematics like measure theory and topology. Most of these applications involve the use of forcing to establish independence results. For examples of such results see https://home.iitk.ac.in/~krashu/

  Faculty: Ashutosh Kumar

Rough set theory and Modal logic

Algebraic studies of structures and corresponding logics that have arisen in the course of investigations in Rough Set Theory (RST) constitute a primary part of my research. Currently, we are working on algebras and logics stemming from a combination of formal concept analysis and RST, and also from different approaches to paraconsistency.

  Faculty: Mohua Banerjee + - Several Complex Variables Click to collapse

Broadly speaking, my work lies in the theory of functions of several complex variables. Two major themes of my work till now are related to _Pick-Nevanlinna interpolation problem_ and on the _Kobayashi geometry of bounded domains_. I am also interested in complex potential theory and complex dynamics in one variable setting.

  Faculty: Vikramjeet Singh Chandel + - Topology and Geometry Click to collapse

Algebraic topology and Homotopy theory

The primary interest is in studying equivariant algebraic topology and homotopy theory with emphasis on unstable homotopy. Specific topics include higher operations such as Toda bracket, pi-algebras, Bredon cohomology, simplicial/ cosimplicial methods, homotopical algebra.

  Faculty: Debasis Sen

Algebraic topology, Combinatorial topology

I apply tools from algebraic topology and combinatorics to address problems in topology and graph theory.

  Faculty: Nandini Nilakantan

Differential geometry

Geometric Analysis and Geometric PDEs. Interested in geometry of the eigenvalues of Laplace operator, Geometry of geodesics.

  Faculty: G. Santhanam

Low dimensional topology

The main interest is in Knot Theory and its Applications. This includes the study of amphicheirality, the study of closed braids, and the knot polynomials, specially the Jones polynomial.

  Faculty: Aparna Dar

Geometric group theory and Hyperbolic geometry

Work in this area involves relatively hyperbolic groups and Cannon-Thurston maps between relatively hyperbolic boundaries. Mapping Class Groups are also explored.

  Faculty: Abhijit Pal

Manifolds and Characteristic classes

We are interested in the construction of new examples of non-Kahler complex manifolds. We aim also at answering the question of existence of almost-complex structures on certain even dimension real manifolds. Characteristic classes of vector bundles over certain spaces are also studied.

  Faculty: Ajay Singh Thakur

Moduli spaces of hyperbolic surfaces

The central question we study here to find combinatorial descriptions of moduli spaces of closed and oriented hyperbolic surfaces. Also, we study isometric embedding of metric graphs on surfaces of following types: (a) quasi-essential on closed and oriented hyperbolic surfaces (b) non-compact surfaces, where complementary regions are punctured discs, (c) on half-translation surfaces etc.

  Faculty: Bidyut Sanki

Systolic topology and Geometry

We are interested to study the configuration of systolic geodesics (i.e., shortest closed geodesics) on oriented hyperbolic surfaces. Also, we are interested in studying the maximal surfaces and deformations on hyperbolic surfaces of finite type to increase systolic lengths.

Topological graph theory

We study configuration of graphs, curves, arcs on surfaces, fillings, action of mapping class groups on graphs on surfaces, minimal graphs of higher genera.

  Faculty: Bidyut Sanki + - Tribology Click to collapse

Active work has been going on in the area of "Tribology". Tribology deals with the issues related to lubrication, friction and wear in moving machine parts. Work is going in the direction of hydrodynamic and elastohydrodynamic lubrication, including thermal, roughness and non-newtonian effects. The work is purely theoretical in nature leading to a system on non-linear partial differential equations, which are solved using high speed computers.

  Faculty: B. V. Rathish Kumar

Research Areas in Statistics and Probability Theory

Here are the areas of Statistics in which research is being done currently.

+ - Bayesian Nonparametric Methods Click to collapse

Exponential growth in computing power in the past few decades has made Bayesian methods for infinitedimensional models possible, which is termed as the Bayesian nonparametric (BN) methods. BN is a vast area dealing with modelling and making inference in various fields of Statistics, including, and not restricted to density estimation, regression, variable selection, classification, clustering. Irrespective of the field of execution, a BN method deals with prior construction on an infinite-dimensional parameter space, posterior computation and thereby making posterior predictive inference. Finally, the method is validated by supportive asymptotic properties to show the closeness of the proposed method to the true underlying data generating process.

Faculty: Minerva Mukhopadhyay

+ - Data Mining in Finance Click to collapse

Economic globalization and evolution of information technology has in recent times accounted for huge volume of financial data being generated and accumulated at an unprecedented pace. Effective and efficient utilization of massive amount of financial data using automated data driven analysis and modelling to help in strategic planning, investment, risk management and other decision-making goals is of critical importance. Data mining techniques have been used to extract hidden patterns and predict future trends and behaviours in financial markets. Data mining is an interdisciplinary field bringing together techniques from machine learning, pattern recognition, statistics, databases and visualization to address the issue of information extraction from such large databases. Advanced statistical, mathematical and artificial intelligence techniques are typically required for mining such data, especially the high frequency financial data. Solving complex financial problems using wavelets, neural networks, genetic algorithms and statistical computational techniques is thus an active area of research for researchers and practitioners.

Faculty: Amit Mitra , Sharmishtha Mitra

+ - Econometric Modelling Click to collapse

Econometric modelling involves analytical study of complex economic phenomena with the help of sophisticated mathematical and statistical tools. The size of a model typically varies with the number of relationships and variables it is applying to replicate and simulate in a regional, national or international level economic system. On the other hand, the methodologies and techniques address the issues of its basic purpose – understanding the relationship, forecasting the future horizon and/or building "what-if" type scenarios. Econometric modelling techniques are not only confined to macro-economic theory, but also are widely applied to model building in micro-economics, finance and various other basic and social sciences. The successful estimation and validation part of the model-building relies heavily on the proper understanding of the asymptotic theory of statistical inference. A challenging area of econometric

Faculty: Shalabh , Sharmishtha Mitra

+ - Entropy Estimation and Applications Click to collapse

Estimation of entropies of molecules is an important problem in molecular sciences. A commonly used method by molecular scientist is based on the assumption of a multivariate normal distribution for the internal molecular coordinates. For the multivariate normal distribution, we have proposed various estimators of entropy and established their optimum properties. The assumption of a multivariate normal distribution for the internal coordinates of molecules is adequate when the temperature at which the molecule is studied is low, and thus the fluctuations in internal coordinates are small. However, at higher temperatures, the multivariate normal distribution is inadequate as the dihedral angles at higher temperatures exhibit multimodes and skewness in their distribution. Moreover the internal coordinates of molecules are circular variables and thus the assumption of multivariate normality is inappropriate. Therefore a nonparametric and circular statistic approach to the problem of estimation of entropy is desirable. We have adopted a circular nonparametric approach for estimating entropy of a molecule. This approach is getting a lot of attention among molecular scientists.

Faculty: Neeraj Misra

+ - Environmental Statistics Click to collapse

The main goal of environmental statistics is to build sophisticated modelling techniques that are necessary for analysing temperature, precipitation, ozone concentration in air, salinity in seawater, fire weather index, etc. There are multiple sources of such observations, like weather stations, satellites, ships, and buoys, as well as climate models. While station-based data are generally available for long time periods, the geographical coverage of such stations is mostly sparse. On the other hand, satellite-derived data are available only for the last few decades, but they are generally of much higher spatial resolution. While the current statistical literature has already explored various techniques for station-based data, methods available for modelling high-resolution satellite-based datasets are relatively scarce and there is ample opportunity for building statistical methods to handle such datasets. Here, the data are not only huge in volume, but they are also spatially dependent. Modelling such complex dependencies is challenging also due to the high nonstationary often present in the data. The sophisticated methods also need suitable computational tools and thus provide scopes for novel research directions in computational statistics. Apart from real datasets, statistical modelling of climate model outputs is a new area of research, particularly keeping in mind the issue of climate change. Under different representative concentration pathways (RCPs) of the Intergovernmental Panel for Climate Change (IPCC), different carbon emission

Faculty: Arnab Hazra

+ - Estimation in Restricted Parameter Space Click to collapse

In many practical situations, it is natural to restrict the parameter space. This additional information of restricted parameter space can be intelligently used to derive estimators that improve upon the standard (natural) estimators, meant for the case of unrestricted parameter space. We deal with the problems of estimation parameters of one or more populations when it is known apriori that some or all of them satisfy certain restrictions, leading to the consideration of restricted parameter space. The goal is to find estimators that improve upon the standard (natural) estimators, meant for the case of unrestricted parameter space. We also deal with the decision theoretic aspects of this problem.

+ - Game Theory Click to collapse

The mathematical discipline of Game theory models and analyses interactions between competing and cooperative players. Some research areas in game theory are choice theory, mechanism design, differential games, stochastic games, graphon games, combinatorial games, evolutionary games, cooperative games, Bayesian games, algorithmic games - and this list is certainly not exhaustive. Gametheoretic models are used in many real-life problems such as decision making, voting, matching, auctioning, bargaining/negotiating, queuing, distributing/dividing wealth, dealing with cheap talks, the evolution of living organisms, disease propagation, cancer treatment, and many more. Game Theory is also a popular research area in computer science where equilibrium structures are explored using computer algorithms. Mathematical topics such as combinatorics, graph theory, probability (discrete and measure-theoretic), analysis (real and functional), algebra (linear and abstract), etc., are used in solving game-theoretic problems.

Faculty: Soumyarup Sadhukhan

+ - Machine Learning and Statistical Pattern Recognition Click to collapse

Build machine learning algorithms based on statistical modeling of data. With a statistical model in hand, we apply probability theory to get a sound understanding of the algorithms.

Faculty: Subhajit Dutta

+ - Markov chain Monte Carlo Click to collapse

Markov chain Monte Carlo (MCMC) algorithms produce correlated samples from a desired target distribution, using an ergodic Markov chain. Due to the lack of independence of the samples, and the challenges of working with Markov chains, many theoretical and practical questions arise. Much of the research in this area can be divided into three broad topics: (1) development of new sampling algorithms for complicated target distributions, (2) studying rates of convergence of the Markov chains employed in various applications like variable selection, regression, survival analysis etc, and (3) measuring the quality of MCMC samples in an effort to quantify the variability in the final estimators of the features of the target.

Faculty: Dootika Vats

+ - Non-Parametric and Robust Statistical methods Click to collapse

Detection of different features (in terms of shape) of non-parametric regression functions are studied; asymptotic distributions of the proposed estimators (along with their robustness properties) of the shaperestricted regression function are also investigated. Apart from this, work on the test of independence for more than two random variables is pursued. Statistical Signal Processing and Statistical Pattern Recognition are the other areas of interest.

Faculty: Subhra Sankar Dhar

+ - Optimal Experimental Design Click to collapse

The area of optimal experimental design has been an integral part of many scientific investigation including agriculture and animal husbandry, biology, medicine, physical and chemical sciences, and industrial research. A well-designed experiment utilizes the limited recourse (cost, time, experimental units, etc) optimally to answer the underlying scientific question. For example, optimal cluster/crossover designs may be applied to cluster/cross randomized trials to efficiently estimates the treatment effects. Optimal standard ANOVA designs can be utilized to test the equality of several experimental groups. Most popular categories of optimal designs include Bayesian designs, longitudinal designs, designs for ordered experiments and factorial designs to name a few.

Faculty: Satya Prakash Singh

+ - Ranking and Selection Problems Click to collapse

About fifty years ago statistical inference problems were first formulated in the now-familiar "Ranking and Selection" framework. Ranking and selection problems broadly deal with the goal of ordering of different populations in terms of unknown parameters associated with them. We deal with the following aspects of Ranking and Selection Problems:1. Obtaining optimal ranking and selection procedures using decision theoretic approach;2. Obtaining optimal ranking and selection procedures under heteroscedasticity;3. Simultaneous confidence intervals for all distances from the best and/or worst populations, where the best (worst) population is the one corresponding to the largest (smallest) value of the parameter;4. Estimation of ranked parameters when the ranking between parameters is not known apriori;5. Estimation of (random) parameters of the populations selected using a given decision rule for ranking and selection problems.

+ - Regression Modelling Click to collapse

The outcome of any experiment depends on several variables and such dependence involves some randomness which can be characterized by a statistical model. The statistical tools in regression analysis help in determining such relationships based on the sample experimental data. This helps further in describing the behaviour of the process involved in experiment. The tools in regression analysis can be applied in social sciences, basic sciences, engineering sciences, medical sciences etc. The unknown and unspecified form of relationship among the variables can be linear as well as nonlinear which is to be determined on the basis of a sample of experimental data only. The tools in regression analysis help in the determination of such relationships under some standard statistical assumptions. In many experimental situations, the data do not satisfy the standard assumptions of statistical tools, e.g. the input variables may be linearly related leading to the problem of multicollinearity, the output data may not have constant variance giving rise to the hetroskedasticity problem, parameters of the model may have some restrictions, the output data may be auto correlated, some data on input and/or output variables may be missing, the data on input and output variables may not be correctly observable but contaminated with measurement errors etc. Different types of models including the econometric models, e.g., multiple regression models, restricted regression models, missing data models, panel data models, time series models, measurement error models, simultaneous equation models, seemingly unrelated regression equation models etc. are employed in such situations. So the need of development of new statistical tools arises for the detection of problem, analysis of such non-standard data in different models and to find the relationship among different variables under nonstandard statistical conditions. The development of such tools and the study of their theoretical statistical properties using finite sample theory and asymptotic theory supplemented with numerical studies based on simulation and real data are the objectives of the research work in this area.

Faculty: Shalabh

+ - Robust Estimation in Nonlinear Models Click to collapse

Efficient estimation of parameters of nonlinear regression models is a fundamental problem in applied statistics. Isolated large values in the random noise associated with model, which is referred to as an outliers or an atypical observation, while of interest, should ideally not influence estimation of the regular pattern exhibited by the model and the statistical method of estimation should be robust against outliers. The nonlinear least squares estimators are sensitive to presence of outliers in the data and other departures from the underlying distributional assumptions. The natural choice of estimation technique in such a scenario is the robust M-estimation approach. Study of the asymptotic theoretical properties of Mestimators under different possibilities of the M-estimation function and noise distribution assumptions is an interesting problem. It is further observed that a number of important nonlinear models used to model real life phenomena have a nested superimposed structure. It is thus desirable also to have robust order estimation techniques and study the corresponding theoretical asymptotic properties. Theoretical asymptotic properties of robust model selection techniques for linear regression models are well established in the literature, it is an important and challenging problem to design robust order estimation techniques for nonlinear nested models and establish their asymptotic optimality properties. Furthermore, study of the asymptotic properties of robust M-estimators as the number of nested superimposing terms increase is also an important problem. Huber and Portnoy established asymptotic behavior of the M-estimators when the number of components in a linear regression model is large and established conditions under which consistency and asymptotic normality results are valid. It is possible to derive conditions under which similar results hold for different nested nonlinear models.

Faculty: Debasis Kundu , Amit Mitra

+ - Rough Paths and Regularity structures Click to collapse

The seminal works of Terry Lyons on extensions of Young integration, the latter being an extension of Riemann integration, to functions with Holder regular paths (or those with finite p-variation for some 0 < p < 1) lead to the study of Rough Paths and Rough Differential Equations. Martin Hairer, Massimiliano Gubinelli and their collaborators developed fundamental results in this area of research. Extensions of these ideas to functions with negative regularity (read as "distributions") opened up the area of Regularity structures. Important applications of these topics include constructions of `pathwise' solutions of stochastic differential equations and stochastic partial differential equations.

Faculty: Suprio Bhar

Numerical analysis of differential equation driven by rough noise:

Developing numerical scheme for differential equations driven by rough noise and studying its convergence, rate of convergence etc.

Faculty: Mrinmay Biswas and Suprio Bhar

+ - Spatial statistics Click to collapse

The branch of statistics that focuses on the methods for analysing data observed across some spatial locations in 2-D or 3-D (most common), is called spatial statistics. The spatial datasets can be broadly divided into three types: point-referenced data, areal data, and point patterns. Temperature data collected by a few monitoring stations spread across a city on some specific day is an example of the first type. When data are obtained as summaries of some geographical regions, they are of the second type, crime rate dataset from the different states of India on a specific year is an example. An example of the third type is the IED attack locations in Afghanistan during a year, where the geographical coordinates are themselves the data. Because of the natural dependence among the observations obtained from two close locations, the data cannot be assumed to be independent. When the study domain is large, often we have a large number of observational sites and at the same time, those sites are possibly distributed across a nonhomogeneous area. This leads to the necessity of models that can handle a large number of sites as well as the nonstationary dependence structure and this is a very active area of research. Apart from common geostatistical models, a very active area of research is focused on spatial extreme value theory where max-stable stochastic processes are the natural models to explain the tail-dependence. While the available methods for such spatial extremes are highly scarce, specifically for moderately highdimensional problems, different future research directions are being explored currently in the literature. For better uncertainty quantification and computational flexibility using hierarchically defined models, the Bayesian paradigm is often a natural choice.

+ - Statistical Signal Processing Click to collapse

Signal processing may broadly be considered to involve the recovery of information from physical observations. The received signals are usually disturbed by thermal, electrical, atmospheric or intentional interferences. Due to the random nature of the signal, statistical techniques play an important role in signal processing. Statistics is used in the formulation of appropriate models to describe the behaviour of the system, the development of appropriate techniques for estimation of model parameters, and the assessment of model performances. Statistical Signal Processing basically refers to the analysis of random signals using appropriate statistical techniques. Different one and multidimensional models have been used in analyzing various one and multidimensional signals. For example ECG and EEG signals, or different grey and white or colour textures can be modelled quite effectively, using different non-linear models. Effective modelling are very important for compression as well as for prediction purposes. The important issues are to develop efficient estimation procedures and to study their properties. Due to non-linearity, finite sample properties of the estimators cannot be derived; most of the results are asymptotic in nature. Extensive Monte Carlo simulations are generally used to study the finite sample behaviour of the different estimators.

+ - Step-Stress Modelling Click to collapse

Traditionally, life-data analysis involves analysing the time-to-failure data obtained under normal operating conditions. However, such data are difficult to obtain due to long durability of modern days. products, lack of time-gap in designing, manufacturing and actually releasing such products in market, etc. Given these difficulties as well as the ever-increasing need to observe failures of products to better understand their failure modes and their life characteristics in today's competitive scenario, attempts have been made to devise methods to force these products to fail more quickly than they would under normal use conditions. Various methods have been developed to study this type of "accelerated life testing" (ALT) models. Step-stress modelling is a special case of ALT, where one or more stress factors are applied in a life-testing experiment, which are changed according to pre-decided design. The failure data observed as order statistics are used to estimate parameters of the distribution of failure times under normal operating conditions. The process requires a model relating the level of stress and the parameters of the failure distribution at that stress level. The difficulty level of estimation procedure depends on several factors like, the lifetime distribution and number of parameters thereof, the uncensored or various censoring (Type I, Type II, Hybrid, Progressive, etc.) schemes adopted, the application of non-Bayesian or Bayesian estimation procedures, etc.

Faculty: Debasis Kundu , Sharmishtha Mitra

+ - Stochastic Partial Differential Equations Click to collapse

The study of Stochastic calculus, more specifically, that of stochastic differential equations and stochastic partial differential equations, has a broad range of applications across various disciplines or branches of Mathematics, such as Partial Differential Equations, Evolution systems, Interacting particle systems, Finance, Mathematical Biology. Theoretical understanding for such equations was first obtained in finite dimensional Euclidean spaces. Later on, to describe various natural phenomena, models were constructed (and analyzed) with values in Banach spaces, Hilbert spaces and in the duals of nuclear spaces. Important topics/questions in this area of research include existence and uniqueness of solutions, Stability, Stationarity, Stochastic flows, Stochastic Filtering theory and Stochastic Control Theory, to name a few.

+ - Theory of Stochastic Orders and Aging and Applications Click to collapse

The manner in which a component (or system) improves or deteriorates with time can be described by concepts of aging. Various aging notions have been proposed in the literature. Similarly lifetimes of two different systems can be compared using the concepts of stochastic orders between the probability distributions of corresponding (random) lifetimes. Various stochastic orders between probability distributions have been defined in the literature. We study the concepts of aging and stochastic orders for various coherent systems. In many situations, the performance of a system can be improved by introducing some kind of redundancy into the system. The problem of allocating redundant components to the components of a coherent system, in order to optimize its reliability or some other system performance characteristic, is of considerable interest in reliability engineering. These problems often lead to interesting theoretical results in Probability Theory. We study the problem of optimally allocating spares to the components of various coherent systems, in order to optimize their reliability or some other system performance characteristic. Performances of systems arising out of different allocations are studied using concepts of aging and stochastic orders.

DEPARTMENT OF Mathematics & Statistics

INDIAN INSTITUTE OF TECHNOLOGY KANPUR

Kanpur, UP 208016 | Phone: 0512-259-xxxx | Fax: 0512-259-xxxx

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Ph.d. in education mathematics education specialization .

Doctoral students in the Mathematics Education specialization work closely and collaboratively with faculty, engaging in research designed to enrich mathematics teaching and improve mathematics learning for all students. A distinctive feature of the specialization area in mathematics education is the integration of teaching and research experiences.

Students in this specialization

• Examine critical issues of mathematics teaching and learning across K-12 and undergraduate levels.
• Teach undergraduate mathematics content and methods courses for prospective K-8 teachers.
• Participate in course instructor groups to study and revise components of these courses.
• Conduct research in your area of interest with the support of internationally-recognized faculty.
• Engage in multiple opportunities to present research in progress and receive feedback from faculty and peers.

Program Coordinator: Dr. Teomara Rutherford

Specialization Coordinators: Dr. Charles Hohensee

Admission Requirements

In addition to the  admission requirement of all applicants to Ph.D. programs in the School of Education, applicants to the program in Mathematics Education are expected to hold a Bachelor’s degree in mathematics or equivalent and/or a Master’s degree in mathematics, mathematics education, or a related field. School teaching experience is preferred. Applicants with special strengths and somewhat different profiles, such as educators who have an interest in further exploring the teaching and learning of mathematics, will also be considered.

Specialization Coursework

In addition to the Doctoral Core Courses, the following specialization courses are required of all Ph.D. students in mathematics education.

• EDUC 833: Research and Theory of Mathematics Learning
• EDUC 834: Research and Theory of Mathematics Teaching
• EDUC 835: Research and Theory of Mathematics Curriculum
• EDUC 836: Research and Theory of Mathematics Teacher Education and School Improvement

Note: the fourth course fits into the curriculum as a course needed for the specialization.

Sample Course Schedules

Sample Mathematics Education course schedules for students who enter the Ph.D. program in the following semesters are available through the links below.

• Students entering in the Fall of odd years
• Students entering in the Fall of even years

Specialization Requirements

All PhD students in the mathematics education specialization must complete the following additional requirements.

• Enroll in the seminar course EDUC 838: Research Issues in Mathematics Education (1 credit) in the first 3 semesters for credit and in the remaining semesters as a listener.
• Complete a  Qualifying Study  that will be submitted as part of your Qualifying Examination.
• Pass the  Qualifying Examination  in mathematics education, taken after completion of the four mathematics education courses (usually taken during the summer of Year Two).
• Develop a portfolio demonstrating expertise in the common activities of the profession (e.g. ,  presenting a paper at a professional conference).

Alterations in the program require approval of the full faculty in mathematics education.

Recent Graduates

Our graduates accept academic positions in research universities, departments of education, and school districts, as well as industry positions in educational organizations.

For example, recent PhD in Education graduates with a mathematics education specialization have accepted positions at Washington State University, Western Michigan University, Iona College, University of Southern Mississippi, University of California Chico, Delaware State University, University of Delaware, The College of New Jersey, Towson University, Central Michigan University, West Chester State University, University of Missouri, Rowan University, Salisbury University, Kennesaw State University, University of Northern Iowa, University of Pittsburgh, California State University in Fullerton, Stockton University, Montclair State University, Shippensburg University of Pennsylvania, Purdue University Fort Wayne, Virginia Tech University, Illinois State University and Manhattanville College.

Program Faculty

Our faculty hold grants from the National Science Foundation, the Institute of Education Sciences and private foundations. They have been recognized for their work by the American Educational Research Association, American Association of Colleges of Teacher Education and National Council of Teachers of Mathematics.

Doctoral Students

Student Spotlight

Amanda Mohammad Mirzaei

“I have experienced a rigorous program in pursuing my Ph.D. in Education at UD, and it certainly hasn’t been a walk in the park. However, the support from the faculty and other graduate students has guaranteed that even at its most difficult, juggling the demands of this degree while pursuing a personal life isn’t impossible. I am happy that I chose the University of Delaware for my Ph.D. program and I would choose it again without hesitation.”

Research Projects

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Home > Physical and Mathematical Sciences > Mathematics Education > Theses and Dissertations

Mathematics Education Theses and Dissertations

Theses/dissertations from 2024 2024.

New Mathematics Teachers' Goals, Orientations, and Resources that Influence Implementation of Principles Learned in Brigham Young University's Teacher Preparation Program , Caroline S. Gneiting

Theses/Dissertations from 2023 2023

Impact of Applying Visual Design Principles to Boardwork in a Mathematics Classroom , Jennifer Rose Canizales

Practicing Mathematics Teachers' Perspectives of Public Records in Their Classrooms , Sini Nicole White Graff

Parents' Perceptions of the Importance of Teaching Mathematics: A Q-Study , Ashlynn M. Holley

Engagement in Secondary Mathematics Group Work: A Student Perspective , Rachel H. Jorgenson

Theses/Dissertations from 2022 2022

Understanding College Students' Use of Written Feedback in Mathematics , Erin Loraine Carroll

Identity Work to Teach Mathematics for Social Justice , Navy B. Dixon

Developing a Quantitative Understanding of U-Substitution in First-Semester Calculus , Leilani Camille Heaton Fonbuena

The Perception of At-Risk Students on Caring Student-Teacher Relationships and Its Impact on Their Productive Disposition , Brittany Hopper

Variational and Covariational Reasoning of Students with Disabilities , Lauren Rigby

Structural Reasoning with Rational Expressions , Dana Steinhorst

Student-Created Learning Objects for Mathematics Renewable Assignments: The Potential Value They Bring to the Broader Community , Webster Wong

Theses/Dissertations from 2021 2021

Emotional Geographies of Beginning and Veteran Reformed Teachers in Mentor/Mentee Relationships , Emily Joan Adams

You Do Math Like a Girl: How Women Reason Mathematically Outside of Formal and School Mathematics Contexts , Katelyn C. Pyfer

Developing the Definite Integral and Accumulation Function Through Adding Up Pieces: A Hypothetical Learning Trajectory , Brinley Nichole Stevens

Theses/Dissertations from 2020 2020

Mathematical Identities of Students with Mathematics Learning Dis/abilities , Emma Lynn Holdaway

Teachers' Mathematical Meanings: Decisions for Teaching Geometric Reflections and Orientation of Figures , Porter Peterson Nielsen

Student Use of Mathematical Content Knowledge During Proof Production , Chelsey Lynn Van de Merwe

Theses/Dissertations from 2019 2019

Making Sense of the Equal Sign in Middle School Mathematics , Chelsea Lynn Dickson

Developing Understanding of the Chain Rule, Implicit Differentiation, and Related Rates: Towards a Hypothetical Learning Trajectory Rooted in Nested Multivariation , Haley Paige Jeppson

Secondary Preservice Mathematics Teachers' Curricular Reasoning , Kimber Anne Mathis

“Don’t Say Gay. We Say Dumb or Stupid”: Queering ProspectiveMathematics Teachers’ Discussions , Amy Saunders Ross

Aspects of Engaging Problem Contexts From Students' Perspectives , Tamara Kay Stark

Theses/Dissertations from 2018 2018

Addressing Pre-Service Teachers' Misconceptions About Confidence Intervals , Kiya Lynn Eliason

How Teacher Questions Affect the Development of a Potential Hybrid Space in a Classroom with Latina/o Students , Casandra Helen Job

Teacher Graphing Practices for Linear Functions in a Covariation-Based College Algebra Classroom , Konda Jo Luckau

Principles of Productivity Revealed from Secondary Mathematics Teachers' Discussions Around the Productiveness of Teacher Moves in Response to Teachable Moments , Kylie Victoria Palsky

Theses/Dissertations from 2017 2017

Curriculum Decisions and Reasoning of Middle School Teachers , Anand Mikel Bernard

Teacher Response to Instances of Student Thinking During Whole Class Discussion , Rachel Marie Bernard

Kyozaikenkyu: An In-Depth Look into Japanese Educators' Daily Planning Practices , Matthew David Melville

Analysis of Differential Equations Applications from the Coordination Class Perspective , Omar Antonio Naranjo Mayorga

Theses/Dissertations from 2016 2016

The Principles of Effective Teaching Student Teachershave the Opportunity to Learn in an AlternativeStudent Teaching Structure , Danielle Rose Divis

Insight into Student Conceptions of Proof , Steven Daniel Lauzon

Theses/Dissertations from 2015 2015

Teacher Participation and Motivation inProfessional Development , Krystal A. Hill

Student Evaluation of Mathematical Explanations in anInquiry-Based Mathematics Classroom , Ashley Burgess Hulet

English Learners' Participation in Mathematical Discourse , Lindsay Marie Merrill

Mathematical Interactions between Teachers and Students in the Finnish Mathematics Classroom , Paula Jeffery Prestwich

Parents and the Common Core State Standards for Mathematics , Rebecca Anne Roberts

Examining the Effects of College Algebra on Students' Mathematical Dispositions , Kevin Lee Watson

Problems Faced by Reform Oriented Novice Mathematics Teachers Utilizing a Traditional Curriculum , Tyler Joseph Winiecke

Academic and Peer Status in the Mathematical Life Stories of Students , Carol Ann Wise

Theses/Dissertations from 2014 2014

The Effect of Students' Mathematical Beliefs on Knowledge Transfer , Kristen Adams

Language Use in Mathematics Textbooks Written in English and Spanish , Kailie Ann Bertoch

Teachers' Curricular Reasoning and MKT in the Context of Algebra and Statistics , Kolby J. Gadd

Mathematical Telling in the Context of Teacher Interventions with Collaborative Groups , Brandon Kyle Singleton

An Investigation of How Preservice Teachers Design Mathematical Tasks , Elizabeth Karen Zwahlen

Theses/Dissertations from 2013 2013

Student Understanding of Limit and Continuity at a Point: A Look into Four Potentially Problematic Conceptions , Miriam Lynne Amatangelo

Exploring the Mathematical Knowledge for Teaching of Japanese Teachers , Ratu Jared R. T. Bukarau

Comparing Two Different Student Teaching Structures by Analyzing Conversations Between Student Teachers and Their Cooperating Teachers , Niccole Suzette Franc

Professional Development as a Community of Practice and Its Associated Influence on the Induction of a Beginning Mathematics Teacher , Savannah O. Steele

Types of Questions that Comprise a Teacher's Questioning Discourse in a Conceptually-Oriented Classroom , Keilani Stolk

Theses/Dissertations from 2012 2012

Student Teachers' Interactive Decisions with Respect to Student Mathematics Thinking , Jonathan J. Call

Manipulatives and the Growth of Mathematical Understanding , Stacie Joyce Gibbons

Learning Within a Computer-Assisted Instructional Environment: Effects on Multiplication Math Fact Mastery and Self-Efficacy in Elementary-Age Students , Loraine Jones Hanson

Mathematics Teacher Time Allocation , Ashley Martin Jones

Theses/Dissertations from 2011 2011

How Student Positioning Can Lead to Failure in Inquiry-based Classrooms , Kelly Beatrice Campbell

Teachers' Decisions to Use Student Input During Class Discussion , Heather Taylor Toponce

A Conceptual Framework for Student Understanding of Logarithms , Heather Rebecca Ambler Williams

Theses/Dissertations from 2010 2010

Growth in Students' Conceptions of Mathematical Induction , John David Gruver

Contextualized Motivation Theory (CMT): Intellectual Passion, Mathematical Need, Social Responsibility, and Personal Agency in Learning Mathematics , Janelle Marie Hart

Thinking on the Brink: Facilitating Student Teachers' Learning Through In-the-Moment Interjections , Travis L. Lemon

Understanding Teachers' Change Towards a Reform-Oriented Mathematics Classroom , Linnae Denise Williams

Theses/Dissertations from 2009 2009

A Comparison of Mathematical Discourse in Online and Face-to-Face Environments , Shawn D. Broderick

The Influence of Risk Taking on Student Creation of Mathematical Meaning: Contextual Risk Theory , Erin Nicole Houghtaling

Uncovering Transformative Experiences: A Case Study of the Transformations Made by one Teacher in a Mathematics Professional Development Program , Rachelle Myler Orsak

Theses/Dissertations from 2008 2008

Student Teacher Knowledge and Its Impact on Task Design , Tenille Cannon

How Eighth-Grade Students Estimate with Fractions , Audrey Linford Hanks

Similar but Different: The Complexities of Students' Mathematical Identities , Diane Skillicorn Hill

Choose Your Words: Refining What Counts as Mathematical Discourse in Students' Negotiation of Meaning for Rate of Change of Volume , Christine Johnson

Mathematics Student Teaching in Japan: A Multi-Case Study , Allison Turley Shwalb

Theses/Dissertations from 2007 2007

Applying Toulmin's Argumentation Framework to Explanations in a Reform Oriented Mathematics Class , Jennifer Alder Brinkerhoff

What Are Some of the Common Traits in the Thought Processes of Undergraduate Students Capable of Creating Proof? , Karen Malina Duff

Probing for Reasons: Presentations, Questions, Phases , Kellyn Nicole Farlow

One Problem, Two Contexts , Danielle L. Gigger

The Main Challenges that a Teacher-in-Transition Faces When Teaching a High School Geometry Class , Greg Brough Henry

Discovering the Derivative Can Be "Invigorating:" Mark's Journey to Understanding Instantaneous Velocity , Charity Ann Gardner Hyer

Theses/Dissertations from 2006 2006

How a Master Teacher Uses Questioning Within a Mathematical Discourse Community , Omel Angel Contreras

Determining High School Geometry Students' Geometric Understanding Using van Hiele Levels: Is There a Difference Between Standards-based Curriculum Students and NonStandards-based Curriculum Students? , Rebekah Loraine Genz

The Nature and Frequency of Mathematical Discussion During Lesson Study That Implemented the CMI Framework , Andrew Ray Glaze

Second Graders' Solution Strategies and Understanding of a Combination Problem , Tiffany Marie Hessing

What Does It Mean To Preservice Mathematics Teachers To Anticipate Student Responses? , Matthew M. Webb

Theses/Dissertations from 2005 2005

Fraction Multiplication and Division Image Change in Pre-Service Elementary Teachers , Jennifer J. Cluff

An Examination of the Role of Writing in Mathematics Instruction , Amy Jeppsen

Theses/Dissertations from 2004 2004

Reasoning About Motion: A Case Study , Tiffini Lynn Glaze

Theses/Dissertations from 2003 2003

An Analysis of the Influence of Lesson Study on Preservice Secondary Mathematics Teachers' View of Self-As Mathematics Expert , Julie Stafford

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For everyone whose relationship with mathematics is distant or broken, Jo Boaler , a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start – to approach the subject with playfulness and curiosity, not anxiety or dread.

“Most people have only ever experienced what I call narrow mathematics – a set of procedures they need to follow, at speed,” Boaler says. “Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience.”

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed , a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced How to Learn Math , the first massive open online course (MOOC) on mathematics education. She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users.

In her new book, Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics , Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls “ishing” a math problem can help students make better sense of the answer.

What do you mean by “math-ish” thinking?

It’s a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport – these are generally answered with what I call “ish” numbers, and that’s very different from the way we use and learn numbers in school.

In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They’re given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2 – but the most common answer 13-year-olds gave was 19. The second most common was 21.

I’m not surprised, because when students learn fractions, they often don’t learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don’t you also risk sending the message that mathematical precision isn’t important?

I’m not saying precision isn’t important. What I’m suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they’ll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, “Isn’t ‘ishing’ just estimating?” It is, but when we ask students to estimate, they often groan, thinking it’s yet another mathematical method. But when we ask them to “ish” a number, they're more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mindset. I think we all need a little more ish in our lives.

You also argue that mathematics should be taught in more visual ways. What do you mean by that?

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things – all of that contributes to our understanding of how it works.

There’s an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik’s Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking.

Some years back we were interviewing students a year after they’d done that activity in our summer camp and asked what had stayed with them. One student said, “I’m in geometry class now, and I still remember that sugar cube, what it looked like and felt like.” His class had been asked to estimate the volume of their shoes, and he said he’d imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don’t get to see and manipulate them. When we learn about square roots, we don’t take squares and look at their diagonals. We just manipulate numbers.

I wonder if people consider the physical representations more appropriate for younger kids.

That’s the thing – elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it’s all symbolic. There’s a myth that there’s a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.

There’s an example in the book where you’ve asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn’t it be easier to teach students one standard method?

A depiction of various ways to calculate 38 x 5, numerically and visually. | Courtesy Jo Boaler

That narrow, rigid version of mathematics where there’s only one right approach is what most students experience, and it’s a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they’re not developing number sense. They don’t learn how to use numbers flexibly in different situations. It also makes students who think differently believe there’s something wrong with them.

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics. When we bring those forms of diversity together, it’s powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.

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May 21, 2024

This article has been reviewed according to Science X's editorial process and policies . Editors have highlighted the following attributes while ensuring the content's credibility:

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The case for 'math-ish' thinking

by Stanford University

For everyone whose relationship with mathematics is distant or broken, Jo Boaler, a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start—to approach the subject with playfulness and curiosity, not anxiety or dread.

"Most people have only ever experienced what I call narrow mathematics—a set of procedures they need to follow, at speed," Boaler says. "Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience."

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed, a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced How to Learn Math, the first massive open online course (MOOC) on mathematics education . She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users.

In her new book, " Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics ," Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls "ishing" a math problem can help students make better sense of the answer.

What do you mean by 'math-ish' thinking?

It's a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport—these are generally answered with what I call "ish" numbers, and that's very different from the way we use and learn numbers in school.

In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They're given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2—but the most common answer 13-year-olds gave was 19. The second most common was 21.

I'm not surprised, because when students learn fractions, they often don't learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don't you also risk sending the message that mathematical precision isn't important?

I'm not saying precision isn't important. What I'm suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they'll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, "Isn't 'ishing' just estimating?" It is, but when we ask students to estimate, they often groan, thinking it's yet another mathematical method. But when we ask them to "ish" a number, they're more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mindset. I think we all need a little more ish in our lives.

You also argue that mathematics should be taught in more visual ways. What do you mean by that?

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things—all of that contributes to our understanding of how it works.

There's an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik's Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking.

Some years back we were interviewing students a year after they'd done that activity in our summer camp and asked what had stayed with them. One student said, "I'm in geometry class now, and I still remember that sugar cube, what it looked like and felt like." His class had been asked to estimate the volume of their shoes, and he said he'd imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don't get to see and manipulate them. When we learn about square roots, we don't take squares and look at their diagonals. We just manipulate numbers.

I wonder if people consider the physical representations more appropriate for younger kids.

That's the thing—elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it's all symbolic. There's a myth that there's a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.

There's an example in the book where you've asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn't it be easier to teach students one standard method?

That narrow, rigid version of mathematics where there's only one right approach is what most students experience, and it's a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they're not developing number sense.

They don't learn how to use numbers flexibly in different situations. It also makes students who think differently believe there's something wrong with them.

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics.

When we bring those forms of diversity together, it's powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.

Provided by Stanford University

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Applied Mathematics Research

In applied mathematics, we look for important connections with other disciplines that may inspire interesting and useful mathematics, and where innovative mathematical reasoning may lead to new insights and applications.

Applied Mathematics Fields

• Combinatorics
• Computational Biology
• Physical Applied Mathematics
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Applied Math Committee

9 Undergraduate Research Projects That Wowed Us This Year

The telegraph. The polio vaccine. The bar code. Light beer. Throughout its history, NYU has been known for innovation, with faculty and alumni in every generation contributing to some of the most notable inventions and scientific breakthroughs of their time. But you don’t wind up in the history books—or peer-reviewed journals—by accident; academic research, like any specialized discipline, takes hard work and lots of practice. 

And at NYU, for students who are interested, that training can start early—including during an undergraduate's first years on campus. Whether through assistantships in faculty labs, summer internships, senior capstones, or independent projects inspired by coursework, undergrad students have many opportunities to take what they’re learning in the classroom and apply it to create original scholarship throughout their time at NYU. Many present their work at research conferences, and some even co-author work with faculty and graduate students that leads to publication. 

As 2023-2024 drew to a close, the NYU News team coordinated with the Office of the Provost to pull together a snapshot of the research efforts that students undertook during this school year. The nine featured here represent just a small fraction of the impressive work we encountered in fields ranging from biology, chemistry, and engineering to the social sciences, humanities, and the arts. 

These projects were presented at NYU research conferences for undergrads, including Migration and Im/Mobility , Pathways for Discovery: Undergraduate Research and Writing Symposium , Social Impact: NYU’s Applied Undergraduate Research Conference , Arts-Based Undergraduate Research Conference , Gallatin Student Research Conference ,  Dreammaker’s Summit , Tandon’s Research Excellence Exhibit , and Global Engagement Symposium . Learn more about these undergrad research opportunities and others.

Jordan Janowski (CAS '24)

Sade Chaffatt (NYU Abu Dhabi '24)

Elsa Nyongesa (GPH, CAS ’24 )

Anthony Offiah (Gallatin ’26)

Kimberly Sinchi (Tandon ’24) and Sarah Moughal (Tandon ’25)

Rohan Bajaj (Stern '24)

Lizette Saucedo (Liberal Studies ’24)

Eva Fuentes (CAS '24)

Andrea Durham (Tandon ’26)

Jordan Janowski (CAS ’24) Major: Biochemistry Thesis title: “Engineering Chirality for Functionality in Crystalline DNA”

Jordan Janowski (CAS '24). Photo by Tracey Friedman

I work in the Structural DNA Nanotechnology Lab, which was founded by the late NYU professor Ned Seeman, who is known as the father of the field. My current projects are manipulating DNA sequences to self-assemble into high order structures.

Essentially, we’re using DNA as a building material, instead of just analyzing it for its biological functions. It constantly amazes me that this is possible.

I came in as a pre-med student, but when I started working in the lab I realized that I was really interested in continuing my research there. I co-wrote a paper with postdoc Dr. Simon Vecchioni who has been a mentor to me and helped me navigate applying to grad school. I’m headed to Scripps Research in the fall. This research experience has led me to explore some of the molecules that make up life and how they could be engineered into truly unnatural curiosities and technologies.

My PI, Prof. Yoel Ohayon , has been super supportive of my place on the  NYU women’s basketball team, which I’m a  member of. He’s been coming to my games since sophomore year, and he’ll text me with the score and “great game!”— it’s been so nice to have that support for my interests beyond the lab.

Anthony Offiah (Gallatin ’26) Concentration: Fashion design and business administration MLK Scholars research project title: “project: DREAMER”

Anthony Offiah (Gallatin '26). Photo by Tracey Friedman

In “project: DREAMER,” I explored how much a person’s sense of fashion is a result of their environment or societal pressures based on their identity. Certain groups are pressured or engineered to present a certain way, and I wanted to see how much of the opposing force—their character, their personality—affected their sense of style. 

This was a summer research project through the MLK Scholars Program . I did ethnographic interviews with a few people, and asked them to co-design their ideal garments with me. They told me who they are, how they identify, and what they like in fashion, and we synthesized that into their dream garments. And then we had a photo shoot where they were empowered to make artistic choices. 

Some people told me they had a hard time conveying their sense of style because they were apprehensive about being the center of attention or of being dissimilar to the people around them. So they chose to conform to protect themselves. And then others spoke about wanting to safeguard the artistic or vulnerable—or one person used the word “feminine”—side of them so they consciously didn’t dress how they ideally would. 

We ended the interviews by stating an objective about how this co-designing process didn’t end with them just getting new clothes—it was about approaching fashion differently than how they started and unlearning how society might put them in a certain box without their approval.  

My concentration in Gallatin is fashion design and business administration. In the industry some clothing is critiqued and some clothing is praised—and navigating that is challenging, because what you like might not be well received. So doing bespoke fashion for just one person is freeing in a sense because you don’t have to worry about all that extra stuff. It’s just the art. And I like being an artist first and thinking about the business second.

Lizette Saucedo (Global Liberal Studies ’24) Major: Politics, rights, and development Thesis title: “Acknowledging and Remembering Deceased Migrants Crossing the U.S.-Mexican Border”

Lizette Saucedo (Global Liberal Studies '24). Photo by Tracey Friedman

My thesis project is on commemorating migrants who are dying on their journey north to cross the U.S.–Mexican border. I look at it through different theoretical lenses, and one of the terms is necropolitics—how politics shapes the way the State governs life and especially death. And then of the main issues aside from the deaths is that a lot of people in the U.S. don’t know about them, due to the government trying to eschew responsibility for migrant suffering. In the final portion of the thesis, I argue for presenting what some researchers call “migrant artifacts”—the personal belongings left behind by people trying to cross over—to the public, so that people can become aware and have more of a human understanding of what’s going on. 

This is my senior thesis for Liberal Studies, but the idea for it started in an International Human Rights course I took with professor Joyce Apsel . We read a book by Jason De León called The Land of the Open Graves , which I kept in the back of my mind. And then when I studied abroad in Germany during my junior year, I noticed all the different memorials and museums, and wondered why we didn’t have the equivalent in the U.S. My family comes from Mexico—my parents migrated—and ultimately all of these interests came together.

I came into NYU through the Liberal Studies program and I loved it. It’s transdisciplinary, which shaped how I view my studies. My major is politics, rights, and development and my minor is social work, but I’ve also studied museum studies, and I’ve always loved the arts. The experience of getting to work one-on-one on this thesis has really fortified my belief that I can combine all those things.

Sade Chaffatt (Abu Dhabi ’24) Major: Biology Thesis title: “The Polycomb repressive component, EED in mouse hepatocytes regulates liver homeostasis and survival following partial hepatectomy.”

Sade Chaffatt (NYU Abu Dhabi '24). Photo courtesy of NYUAD

Imagine your liver as a room. Within the liver there are epigenetic mechanisms that control gene expression. Imagine these epigenetic mechanisms as a dimmer switch, so that you could adjust the light in the room. If we remove a protein that is involved in regulating these mechanisms, there might be dysregulation—as though the light is too bright or too dim. One such protein, EED, plays a crucial role in regulating gene expression. And so my project focuses on investigating whether EED is required in mouse hepatocytes to regulate liver homeostasis and to regulate survival following surgical resection.

Stepping into the field of research is very intimidating when you’re an undergraduate student and know nothing. But my capstone mentor, Dr. Kirsten Sadler , encourages students to present their data at lab meetings and to speak with scientists. Even though this is nerve-wracking, it helps to promote your confidence in communicating science to others in the field.

If you’d asked 16-year-old me, I never would’ve imagined that I’d be doing research at this point. Representation matters a lot, and you often don't see women—especially not Black women—in research. Being at NYUAD has really allowed me to see more women in these spaces. Having had some experience in the medical field through internships, I can now say I’m more interested in research and hope to pursue a PhD in the future.

Kimberly Sinchi (Tandon ’24) Major: Computer Science Sarah Moughal (Tandon ’25) Major: Computer Science Project: Robotic Design Team's TITAN

Sarah Moughal (Tandon '25, left) and Kimberly Sinchi (Tandon '24). Photo by Tracey Friedman

Kimberly: The Robotic Design Team has been active at NYU for at least five years. We’re 60-plus undergrad and grad students majoring in electrical engineering, mechanical engineering, computer science, and integrated design. We’ve named our current project TITAN because of how huge it is. TITAN stands for “Tandon’s innovation in terraforming and autonomous navigation.”

Sarah: We compete in NASA’s lunatics competition every year, which means we build a robot from scratch to be able to compete in lunar excavation and construction. We make pretty much everything in house in the Tandon MakerSpace, and everyone gets a little experience with machining, even if you're not mechanical. A lot of it is about learning how to work with other people—communicating across majors and disciplines and learning how to explain our needs to someone who may not be as well versed in particular technologies as we are. 

Kimberly: With NYU’s Vertically Integrated Project I’ve been able to take what I was interested in and actually have a real world impact with it. NASA takes notes on every Rover that enters this competition. What worked and what didn’t actually influences their designs for rovers they send to the moon and to Mars.

Eva Fuentes (CAS ’24) Major: Anthropology Thesis title: “Examining the relationship between pelvic shape and numbers of lumbar vertebrae in primates”

Eva Fuentes (CAS '24). Photo by Tracey Friedman

I came into NYU thinking I wanted to be an art history major with maybe an archeology minor. To do the archeology minor, you have to take the core classes in anthropology, and so I had to take an intro to human evolution course. I was like, this is the coolest thing I’ve learned—ever. So I emailed people in the department to see if I could get involved. 

Since my sophomore year, I’ve been working in the Evolutionary Morphology Lab with Scott Williams, who is primarily interested in the vertebral column of primates in the fossil record because of how it can inform the evolution of posture and locomotion in humans.

For my senior thesis, I’m looking at the number of lumbar vertebrae—the vertebrae that are in the lower back specifically—and aspects of pelvic shape to see if it is possible to make inferences about the number of lumbar vertebrae a fossil may have had. The bones of the lower back are important because they tell us about posture and locomotion.

I committed to a PhD program at Washington University in St. Louis a few weeks ago to study biological anthropology. I never anticipated being super immersed in the academic world. I don’t come from an academic family. I had no idea what I was doing when I started, but Scott Williams, and everyone in the lab, is extremely welcoming and easy to talk to. It wasn't intimidating to come into this lab at all.

Elsa Nyongesa (GPH, CAS ’24 ) Major: Global Public Health and Biology Project: “Diversity in Breast Oncological Studies: Impacts on Black Women’s Health Outcomes”

Elsa Nyongesa (GPH, CAS '24). Photo by Tracey Friedman

I interned at Weill Cornell Medicine through their Travelers Summer Research Fellowship Program where I worked with my mentor, Dr. Lisa Newman, who is the head of the International Center for the Study of Breast Cancer Subtypes. I analyzed data on the frequency of different types of breast cancer across racial and ethnic groups in New York. At the same time, I was also working with Dr. Rachel Kowolsky to study minority underrepresentation in clinical research. 

In an experiential learning course taught by Professor Joyce Moon Howard in the GPH department, I created a research question based on my internship experience. I thought about how I could combine my experiences from the program which led to my exploration of the correlation between minority underrepresentation in breast oncological studies, and how it affects the health outcomes of Black women with breast cancer.

In my major, we learn about the large scope of health disparities across different groups. This opportunity allowed me to learn more about these disparities in the context of breast cancer research. As a premedical student, this experience broadened my perspective on health. I learned more about the social, economic, and environmental factors influencing health outcomes. It also encouraged me to examine literature more critically to find gaps in knowledge and to think about potential solutions to health problems. Overall, this experience deepened my philosophy of service, emphasizing the importance of health equity and advocacy at the research and clinical level.

Rohan Bajaj (Stern ’24) Major: Finance and statistics Thesis title: “Measuring Socioeconomic Changes and Investor Attitude in Chicago’s Post-Covid Economic Recovery”

Rohan Bajaj (Stern '24). Photo by Tracey Friedman

My thesis is focused on understanding the effects of community-proposed infrastructure on both the socioeconomic demographics of cities and on fiscal health. I’m originally from Chicago, so it made a lot of sense to pay tribute back to the place that raised me. I’m compiling a list of characteristics of infrastructure that has been developed since 2021 as a part of the Chicago Recovery Plan and then assessing how neighborhoods have changed geographically and economically. 

I’m looking at municipal bond yields in Chicago as a way of evaluating the fiscal health of the city. Turns out a lot of community-proposed infrastructure is focused in lower income areas within Chicago rather than higher income areas. So that makes the research question interesting, to see if there’s a correlation between the proposed and developed infrastructure projects, and if these neighborhoods are being gentrified alongside development.

I kind of stumbled into the impact investing industry accidentally from an internship I had during my time at NYU. I started working at a renewable energies brokerage in midtown, where my main job was collecting a lot of market research trends and delivering insights on how these different energy markets would come into play. I then worked with the New York State Insurance Fund, where I helped construct and execute their sustainable investment strategy from the ground up. 

I also took a class called “Design with Climate Change” with Peter Anker in Gallatin during my junior year, and a lot of that class was focused on how to have climate resilient and publicly developed infrastructure, and understanding the effects it has on society. It made me start thinking about the vital role that physical surroundings play in steering communities.

In the short term I want to continue diving into impact-focused investing and help identify urban planners and city government to develop their communities responsibly and effectively.

Andrea Durham (Tandon, ’26)  Major: Biomolecular science Research essay title: “The Rise and Fall of Aduhelm”

Andrea Durham (Tandon '26). Photo by Tracey Friedman

This is an essay I wrote last year in an advanced college essay writing class with Professor Lorraine Doran on the approval of a drug for Alzheimer’s disease called Aduhelm—a monoclonal antibody therapy developed by Biogen in 2021, which was described as being momentous and groundbreaking. But there were irregularities ranging from the design of its clinical trials to government involvement that led to the resignation of three scientists on an advisory panel, because not everybody in the scientific community agreed that it should be approved.

When I was six years old, my grandmother was diagnosed. Seeing the impact that it had over the years broke my heart and ignited a passion in me to pursue research. 

When I started at NYU, I wasn’t really sure what I was going to do in the future, or what opportunities I would go after. This writing class really gave me an opportunity to reflect on the things that were important to me in my life. The September after I wrote this paper, I started volunteering in a lab at Mount Sinai for Alzheimer's disease research, and that’s what I’m doing now—working as a volunteer at the Center for Molecular Integrative Neuroresilience under Dr. Giulio Pasinetti. I have this opportunity to be at the forefront, and because of the work I did in my writing class I feel prepared going into these settings with an understanding of the importance of conducting ethical research and working with integrity.

• English (Current language)

INFO: [email protected]

PEC: [email protected]

TEL: (+39)0403787111

Edward Witten - Black Hole Thermodynamics: Then and Now - Sciama Colloquium 2024

The traditional SISSA Sciama Colloquium returns on Wednesday June 12th with a very special guest: Ed Witten, from the Institute for Advanced Study, Princeton, who will hold a special seminar titled "Black Hole Thermodynamics: Then and Now". The event will be held in Aula Magna Budinich starting from 17:00.

Professor Witten will discuss the fundamental concepts of black hole thermodynamics as initially developed in the 1970s, and give insights into the current advancements of the field.

Edward Witten is a renowned American theoretical physicist celebrated for his groundbreaking contributions to string theory, topological quantum field theory, and other areas of mathematical physics. Soon after receiving his PhD from Princeton University, Witten became a junior fellow in the Harvard Society of Fellows. He then held an appointment at Princeton University, before moving to the Institute for Advanced Study (Princeton) in 1987, where he has remained ever since.

His work has significantly influenced topics such as string theory, quantum gravity, and supersymmetric quantum field theories, bridging the gap between physics and pure mathematics. Notably, Witten was the first physicist awarded the Fields Medal in 1990, recognizing his proof of the positive energy theorem in general relativity and his interpretation of the Jones invariants of knots as Feynman integrals. Among his numerous other accolades are the Dirac Medal, the Lorentz Medal, and the Breakthrough Prize. In 2004, he was also listed among Time magazine's 100 most influential people.

The Sciama Colloquium is named in memory of Dennis Sciama, one of the most important cosmologists of the 20th century, and for many years a pillar of the astrophysics field at SISSA.

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Scaling up AI-based professional development for math teachers everywhere

New study positively reports the impact of AI-powered teacher professional development on mathematics instruction.

The United States has struggled to fill and retain math teacher jobs for years. In the 2011–2012 school year, 19% of public schools were unable to fill a teaching position for math, and by the 2020–2021 school year, 32% of public schools were unable to fill a teaching position for math. Compounding this math teacher shortage is the reality that many math teachers do not have positive experiences with the subject and do not receive adequate support. Fortunately, scalable and accessible online professional development (PD) for math instructors providing just-in-time feedback based on teachers’ understanding of content can improve the quality of math instruction. Just-in-time feedback provides teachers with training and development the moment they need it, instead of waiting for a PD trainer to assess and provide the feedback for the teacher. It also allows the teacher to apply new knowledge quickly.

Associate Professor Yasemin Copur-Gencturk’s newly published article positively reports the impact of artificial intelligence (AI)-powered PD on mathematics instruction with a prototype of a scalable and accessible program that creates an active learning environment for math teachers.

While evidence suggests that PD programs can improve teachers’ knowledge, instructional practices and learning outcomes for students, one problem that has not been addressed is whether effective programs are scalable so teachers in any location can access them. The barriers to accessing high-quality PD can be significant for teachers from rural and high-needs areas. Other factors such as lack of time and travel distances can limit teachers’ ability to participate in high-quality training.

Copur-Gencturk started her career as a math instructor in her native Turkey and understands the challenges that math teachers face. This on-the-ground experience inspired her research to examine alternative forms of learning environments, such as asynchronous online PD programs.

“While there has been online instruction in teacher education for some time, people have quit these programs because of the lack of interaction. We need to find a way to reach teachers anywhere and anytime to provide quality feedback, and this is where AI comes in,” said Copur-Gencturk.

To overcome the limitations of asynchronous PD programs by incorporating effective elements of in-person PD, Copur-Gencturk used intelligent tutoring systems to create an interactive, personalized learning environment by analyzing the performance of several instructors and interacting with them through a sequence of feedback cycles. The program that Copur-Gencturk and her research team developed utilizes a virtual facilitator that interacts with teachers through multiple activities. Math teachers from across the country worked through targeted content at their own pace, in their own space and on their own schedule.

Findings: The data reported in Copur-Gencturk’s new article shows that the teachers who participated in the AI-based PD program utilized mathematically richer tasks and created a more coherent and connected learning environment for students to build conceptual understandings than teachers who did not have access to the training. This piece builds off of Copur-Gencturk’s previous work over the past several years looking at how AI can be used to create effective, scalable teacher PD. The findings in this recently published article point to how AI-based PD programs that provide just-in-time feedback based on instructors’ understanding could improve the quality of their instruction. Tailored, just-in-time feedback from a virtual facilitator could be a substitute when human facilitators are unavailable. Reshaping how practitioners think about and conduct PD is one strategy that could be used to retain math instructors and support their continued development in the field.

Yasemin Copur-Gencturk

• Associate Professor of Education

Article Type

Article topics.

• Teaching resources

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