maths phd mit

The Mathematics Department at MIT is a world leader in pure and applied mathematical research and education. In pure mathematics we explore exciting current research directions in most of the major fields. 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. Our applied math group focuses on biology, combinatorics, computer science, scientific computing, numerical analysis, and areas of physical applied mathematics.

The Department currently has about 50 faculty members, including many current and former faculty who have received the highest distinctions including an Abel Prize; two National Medals of Science; two MacArthur Awards; two Bôcher, two Cole, five Veblen, one Fulkerson and nine Steele Prizes. Nineteen are members of the National Academies of Science and Engineering and thirty are members of the American Academy of Arts and Sciences. See the award listings for more details.

Post-doctoral Instructors

Instructors in pure and applied mathematics, including CLE Moore Instructors, have the opportunity to reside at MIT for two or three years and work with our faculty. Teaching loads are kept light to provide good conditions for launching research programs. We aim to have roughly 30 Instructors in the Department at any given time.

We are fortunate to attract outstanding students at all levels. Our graduate program trains students for top academic and industrial research positions, and typically has over 100 students enrolled, including many international students. MIT undergraduates are top performers at the Putnam Competition —in the most recent 2023 competition, MIT had the 1st place team and all 5 of the 5 highest ranking individuals. Additionally, 63 MIT students were among those with Honorable Mention or higher. Approximately 360 students are enrolled in one of our two undergraduate programs: Mathematics (course 18) and Mathematics with Computer Science (course 18C).

Environment

Our Department benefits from the lively mathematical community in the Boston area including the outstanding departments at Harvard and other nearby universities.

The Clay Mathematics Institute is a wonderful resource for bringing visitors and hosting important colloquia and workshops. Weekly seminars are held at MIT ; many organized by graduate students. Our annual Simons Lecture series brings distinguished speakers on the latest developments.

Our excellent staff help us run the Department. Most of them can be found in Department Headquarters and in the Math Academic Services office .

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Interdisciplinary Doctoral Program in Statistics

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The Interdisciplinary PhD in Statistics (IDPS) is designed for students currently enrolled in a participating MIT doctoral program who wish to develop their understanding of 21st century statistics, using concepts of computation and data analysis as well as elements of classical statistics and probability within their chosen field of study.

Participating programs: Aeronautics & Astronautics Brain and Cognitive Sciences Economics Mathematics Mechanical Engineering Physics Political Science Social and Engineering Systems

How to join IDPS:

Doctoral students in participating programs may submit a selection form between the end of their second semester and penultimate semester in their doctoral program. Selection forms are due by the current semester add date, and students will be notified of a decision by the drop date.

Required documents include a CV, unofficial transcript, anticipated course plan and thesis proposal or statement of interest in statistics.  For access to the selection form or for further information, please contact the IDSS Academic Office at [email protected]

Graduate Departments:

MIT Statistics + Data Science Center Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139-4307 617-253-1764

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Academic Programs

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MIT Doctoral Programs in Computational Science and Engineering

The Center for Computational Science and Engineering (CCSE) offers two doctoral programs in computational science and engineering (CSE) – one leading to a standalone PhD degree in CSE offered entirely by CCSE (CSE PhD) and the other leading to an interdisciplinary PhD degree offered jointly with participating departments in the School of Engineering and the School of Science (Dept-CSE PhD).

While both programs enable students to specialize at the doctoral level in a computation-related field via focused coursework and a thesis, they differ in essential ways. The standalone CSE PhD program is intended for students who intend to pursue research in cross-cutting methodological aspects of computational science. The resulting doctoral degree in Computational Science and Engineering is awarded by CCSE via the the Schwarzman College of Computing. In contrast, the interdisciplinary CSE PhD program is intended for students who are interested in computation in the context of a specific engineering or science discipline. For this reason, this degree is offered jointly with participating departments across the Institute; the interdisciplinary degree is awarded in a specially crafted thesis field that recognizes the student’s specialization in computation within the chosen engineering or science discipline.

For more information about CCSE’s doctoral programs, please explore the links on the left. Information about our application and admission process is available via the ‘ Admissions ‘ tab in our menu. MIT Registrar’s Office provides graduate tuition and fee rates as set by the MIT Corporation and the Graduate Admissions section of MIT’s Office of Graduate Education (OGE) website contains additional information about costs of attendance and funding .

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Interdisciplinary Doctoral Program in Statistics

The Interdisciplinary Doctoral Program in Statistics is an opportunity for students in a multitude of disciplines to specialize at the doctoral level in a statistics-grounded view of their field. Participating programs include Aeronautics and Astronautics, Brain and Cognitive Sciences, Economics, Mathematics, Mechanical Engineering, Physics, Political Science, and the IDSS Social and Engineering Systems Doctoral Program.

The program is administered jointly by the Statistics and Data Science Center and the participating academic units. Students enrolled in a doctoral program in a participating department may choose to be considered for the Interdisciplinary Doctoral Program in Statistics. Please refer to the program's website for details on the selection process.

Selected students will complete the home department’s degree requirements (including the qualifying exam) along with specified statistics requirements including a doctoral seminar, coursework in probability, statistics, computation and statistics, and data analysis, and a dissertation that utilizes statistical methods in a substantial way. 

For more information about the program, contact the Statistics Academic Administrator .

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Doctoral Degrees

A doctoral degree requires the satisfactory completion of an approved program of advanced study and original research of high quality..

Please note that the Doctor of Philosophy (PhD) and Doctor of Science (ScD) degrees are awarded interchangeably by all departments in the School of Engineering and the School of Science, except in the fields of biology, cognitive science, neuroscience, medical engineering, and medical physics. This means that, excepting the departments outlined above, the coursework and expectations to earn a Doctor of Philosophy and for a Doctor of Science degree from these schools are generally the same. Doctoral students may choose which degree they wish to complete.

Applicants interested in graduate education should apply to the department or graduate program conducting research in the area of interest. Some departments require a doctoral candidate to take a “minor” program outside of the student’s principal field of study; if you wish to apply to one of these departments, please consider additional fields you may like to pursue.

Below is a list of programs and departments that offer doctoral-level degrees.

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Maz Abulnaga is a PhD candidate whose work is changing the way researchers and physicians study the placenta with the goal of improving health outcomes for mothers and babies. Specifically, Maz develops algorithms for studying the placental shape in fetal magnetic resonance imaging, an imaging modality that is being used to track the health and function of the placenta, identify pathology, and support patient outcomes. Maz has made several contributions to the theoretical and algorithmic foundations of volumetric geometry processing and has successfully applied his models to clinical research of the placenta. With the support of his MathWorks Fellowship, Maz will develop geometry processing and machine-learning algorithms to quantify placental shape and functional changes throughout pregnancy, which are necessary to identify pathology and plan pregnancy outcomes. He has recently developed a novel approach to provide a standardized representation of placental shape that is being used by researchers globally. He envisions developing a set of open-source tools to enable clinical studies to improve our understanding of placental and fetal development with the goal of revolutionizing fetal-maternal health.

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Maz Abulnaga

Bernardo is a PhD student in mechanical engineering whose research examines the synthesis of robust strategies for grasping and dexterous manipulation. Specifically, he is examining the problem of caging, a geometric property by which a set of fingers manipulate an object by trapping it rather than immobilizing it. He has developed an explicit optimization approach to formulate the caging condition as a convex mixed-integer optimization problem. This was the first of its kind, since caging had previously been approached as either a topological or computational geometry problem. More recently, he has extended the formulation to synthesizing grasping strategies with certificates of correctness, a problem that is of high interest today both in academia and industry. He uses MATLAB’s Robotics Toolbox, Optimization Toolbox, and Deep Learning Toolbox as the main prototyping interface. In addition, his work has resulted in several tools that have been made open to the robotics and MathWorks communities, including ABB ROS Interface, Contact-TrajectoryOptimization Models, and Certified Grasping Toolbox. Bernando earned a BSc in electronics engineering from Universidad Simón Bolívar, Venezuela

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Bernardo Aceituno - Cabeza

Sayed Saad Afzal is a PhD candidate whose research interests lie in developing novel low-cost and low-power distributed sensor technologies and systems for climate monitoring, environmental sustainability, and food security. Specifically, Sayed is working on ocean sensor technologies to monitor vital signs such as biodiversity and carbon balance. With the support of a MathWorks Fellowship, he will advance his successful work on an underwater sensor utilizing backscatter, a technique that enables communication with 1 million times less power than existing systems. He recently used this technology to build one of the world’s first battery-free underwater cameras, enabling long-duration monitoring in remote ocean locations. MathWorks tools such as MATLAB’s DSP System Toolbox and Signal Processing Toolbox are crucial tools in his research. Sayed has already made notable technological contributions with implications for climate monitoring, food security, and ecology. His future work has tremendous potential to advance sensor technologies, supporting innovative research and a sustainable future.

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Sayed Saad Afzal

Shashank is a PhD student in mechanical engineering whose research primarily focuses on the development of reduced-order, rate-dependent methods for a variety of granular intrusion problems, such as meteorite impacts and animal and vehicular locomotion in sands and deserts (a field referred to as terramechanics). He uses large-scale, detailed numerical simulations to understand the physics of such scenarios, which in turn allows him to develop robust reduced-order models that can be run in real time. He uses MATLAB for various purposes in his research, most importantly for the implementation of reduced-order models for granular intrusion, allowing for the real-time simulation of diverse intrusion scenarios with different system properties and intruder shapes. Shashank earned a BTech in mechanical engineering fromIITGandhinagar, India and an SM in mechanical engineering from MIT.He also worked as a scientist at the Defense Research and Development Organization, India before joining MIT

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Shashank Agarwal

Mohammad is a PhD student in civil and environmental engineering whose research is focused on explaining and predicting biodiversity changes using mathematical models. Specifically, his goal is to provide closed-form solutions using computational methods in order to skip the impossible task of simulating all possible parameter values and combinations and to establish rigorous and testable estimates about the probability of persistence of species. He is developing his own code and integrating libraries from MATLAB with the goal of generating a computational package that will allow researchers to work with different ecological models, derive the range of parameter values (or combinations of parameter values) compatible with the persistence of a given species in an ecological system, and estimate the probabilities (or risks) of species extinctions or species invasions. Mohammad earned a BS in mathematics and a BS in electrical engineering from Rensselaer Polytechnic Institute and an SM in civil and environmental engineering, an SM in electrical engineering and computer science, and an SM in computation for design and optimization from MIT.

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Mohammad AlAdwani

Jasmine Jerry Aloor is a PhD candidate in robotics, aerospace engineering, and artificial intelligence whose research seeks to address the challenge of balancing efficiency and fairness in multi-agent teams using reinforcement learning. As a MathWorks Fellow, Jasmine will pursue research exploring motion planning and control for dynamic multi-agent systems, with a special emphasis on safety, robustness, and human interactivity. Jasmine’s past research endeavors include work in imitation learning for social navigation, control barrier functions, and coverage path planning; recent projects include the development of a safe and socially aware motion planning method using imitation learning for general aviation aircraft in multi-aircraft shared airspace and the design and testing of an improvised box-wing micro air vehicle. MATLAB supports design, modeling, and analysis across her research interests. Jasmine’s innovative work has the could improve machine learning and autonomous systems and help them realize their full potential.

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Jasmine Jerry Aloor

Abtin Ameri is a PhD candidate whose research interests lie at the intersection of quantum computing and plasma physics, with the goal of advancing the use of fusion energy. Specifically, Abtin is working on the development of quantum algorithms for the numerical simulation of fusion reactors by identifying candidate problems suitable for a quantum computing approach and exploring how to mathematically frame those problems to maximize the added efficiency that quantum computers may offer. With the support of a MathWorks Fellowship, Abtin will investigate whether the problem of determining plasma stability in fusion reactors can be efficiently computed on quantum platforms. MATLAB has been an indispensable tool at each step of his work. Abtin’s ambitious research, if successful, will be the first example of a quantum algorithm with immediate relevance to fusion research and could expedite progress toward putting fusion energy on the grid, contributing to sustainable energy solutions and mitigating the impacts of climate change.

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Abtin Ameri

Pasquale is a PhD student in aerospace engineering. He is developing general-purpose tools for estimation that are robust to corrupted information and adversarial attacks, run in real-time, and require minimal tuning. His interests include safe and trustworthy perception with application to single and multi-robot autonomous systems. He uses MATLAB on a regular basis, including the Computer Vision Toolbox and Robotics System Toolbox for his work on robust estimation. He earned his SB in computer science from University of Pisa, Italy, an SM in embedded computing systems from Scuola Superiore Sant’Anna of Pisa, Italy and an SM in aeronautics and astronautics from MIT. Prior to MIT, he was a research scientist at the United Technologies Research Center, Ireland.

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Pasquale Antonante

Nicolas is a PhD student in electrical engineering and computer science whose research focuses on medical imaging technology. He is trying to solve challenging MRI problems by creating and distributing the hardware and computational tools needed for simultaneous optimization of coil geometries, encoding field patterns, and received-signal reconstruction algorithms. He has observed that the steps, or modules, that make up most MRI sequences often have conflicting patient-specific field pattern requirements. Nicolas developed a general MATLAB-based framework for use at scan time that optimizes the field patterns for each sequence module. The framework reads patient rapid-scan data from the imager, exploits the coil pre-characterization and convex formulations to efficiently compute sets of sequence-module optimized coil currents, and then transfers the coil currents sets, along with triggering conditions, to the coil driver array. Nicolas earned an SB in electrical engineering from MIT.

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Nicolas Arango

Attias Ariel is a PhD candidate whose research is focused on designing resilient structures in urban environments that are better able to withstand natural disasters. With the support of a MathWorks Fellowship, he will develop a novel approach, assessing the resilience of urban structures by combining finite elements and statistical physics principles. This strategy characterizes fracture at the element level using spins, describing fracture in a statistical ensemble, effectively bridging the gap between brittle fracture theory and statistical physics. Through his research, which takes extensive use of MathWorks tools, Attias is providing solutions to several computational challenges associated with structural resilience, including fracture initiation, fracture branching, computational efficiency, and in-situ prediction of fracture propagation. His goals include creating a new framework for considering the fragility of structures and the analysis of their life cycle. Attias’s research could have a significant impact on design for the structures of tomorrow, ultimately improving the safety and sustainability of our communities.

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Attias Ariel

Toros Arikan is a PhD candidate whose interdisciplinary work integrates signal processing, machine learning, and computational imaging to significantly expand the capabilities of sensing systems for acoustic, radio frequency (RF), and optical modalities. With the support of a MathWorks Fellowship, Toros will pursue innovative research that takes a new, and potentially highly useful, view of system design. Traditionally, system designs are based on relatively low-dimensional models. Toros’s work, by contrast, embraces high-dimensional models and finds potential in that complexity for better performance and surprising new capabilities. One example is the presence of unknown complex obstacles and reflecting structure in physical environments that can improve object localization and scene imaging performance. MATLAB tools play a central role in Toros’s research, serving as his algorithm prototyping and experimental data analysis environment. The tools, methods, and insights emerging from this work hold important potential for future research and applications, including the acoustic and RF communities. To help realize these new possibilities, Toros will make the resulting tools of his research widely available through the MathWorks File Exchange.

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Toros Arikan

Anantha Narayanan Suresh Babu is a PhD candidate whose research combines computational modeling, uncertainty quantification, Bayesian learning, and scientific machine learning, with applications to sea ice and ocean dynamics. Sea ice is rapidly changing, and submesoscale currents have recently been found to be ubiquitous in the upper ocean. Anantha’s work seeks to broaden our modeling of these complex, nonlinear sea ice and submesoscale dynamics, by melding fundamental physics with limited data. Supported by a MathWorks Fellowship, he is developing probabilistic models and using Bayesian and machine learning to create sea ice rheologies and submesoscale closure models. These new learning methods can discriminate between existing models, detect missing physics, and discover new dynamics formulations utilizing only sparse, indirect, and noisy measurements. Anantha’s work relies on his custom MATLAB framework and could yield useful numerical tools and schemes for fellow researchers. His work has the potential to impact both ocean and data sciences. Ultimately, his results could be valuable for climate modeling, environmental monitoring and forecasting, marine ecosystem protection, resource management, maritime transport, and policy and decision-making for the future of our planet.

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Anantha Narayanan Suresh Babu

Taylor Elise Baum is a PhD candidate whose research interests lie in developing systems that enhance clinician decision-making processes for cardiovascular system management. Currently, Taylor is working on a closed-loop blood pressure control (CLBPC) system to improve blood pressure management in intensive care units and operating rooms. A MathWorks Fellowship will allow her to advance the system toward eventual human testing. Milestone achievements include her development of a physiologically based theoretical framework for the system; a cardiovascular state estimation method to inform control decisions; and an end-to-end hardware system for monitoring blood pressure and controlling vasoactive medications in swine. Notable is her method for cardiovascular state estimation based on the Windkessel model, which has improved accuracy and time resolution compared with current approaches. She is now leading preparations for upcoming animal testing, a crucial step toward human testing. Taylor makes extensive use of MathWorks tools and is pursuing research with strong potential to enhance clinical decision- making in cardiovascular medicine, leading to improved patient outcomes and future innovations in engineering applications for health care.

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Taylor Elise Baum

Maanasa Bhat is a PhD candidate who is integrating interests in health, energy, and materials science to develop innovative new approaches to synthesis of nanomaterials. As a MathWorks Fellow, Maanasa is designing a flame-based synthesis process to manufacture nanomaterials for low-cost sensors and photocatalysts with a wide range of potential environmental and health applications. The goal of Maanasa’s work is to build a broader understanding of the physical and chemical mechanisms involved in the flame synthesis process. The aim is to strengthen connections between the input operating conditions and the composition and morphology of output products to achieve high controllability. MathWorks tools have been indispensable in this research. As her project progresses, Maanasa hopes to tailor the novel flame synthesis process she has developed for specific uses such as the detection of methane leaks, pollutant emissions in air, as well as breath diagnosis for hard-to-detect conditions such as endometriosis and fibroids.

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Maanasa Bhat

Harsh G. Bhundiya is a PhD candidate who is applying structural engineering, computational modeling, and design techniques to purse his interest in space structures. His work is ushering in a new paradigm for constructing large structures in orbit, called in-space manufacturing, with an enormous potential impact on space applications. Drawing on MathWorks tools, Harsh has reached exciting milestones, including the development of a path-planning framework called Bend-Forming, which enables the creation of paraboloids, isogrid columns, and other structural elements that can be linked together to form large structures such as RF reflectors and solar sails. As a MathWorks Fellow, Harsh will characterize the compression behaviors of these structures and study the effects of processing defects on their buckling response through experiments and nonlinear finite element analysis. Simulink and Simscape Multibody will play critical roles in this work. Harsh was pleased to share previous work on MathWorks File Exchange, and will continue to share his research, as he seeks to create a new generation of large space structures for diverse applications.

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Harsh G. Bhundiya

Artittaya (Tiya) Boonkird is a PhD candidate whose primary research interest is in developing machine learning models to predict material properties, with an emphasis on materials containing defects. Specifically, Tiya is examining how the introduction of defects into materials may enhance their mechanical, electrical, and magnetic properties, and how these defects can be controlled to enable useful applications, such as in the semiconductor industry. A MathWorks Fellowship will support Tiya’s work to develop a machine-learning model to extract information about defect type and distribution from x-ray diffraction data collected at various parameter ranges that are typically fixed in conventional measurements. These insights could enable another way to experimentally characterize defects, leading to a better understanding of how they influence material properties. MATLAB is an indispensable tool in Tiya’s work, which has strong potential to advance materials research and offer new avenues for material development in industry.

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Artittaya (Tiya) Boonkird

A PhD student in electrical engineering and computer science, Roberto is a member of the Organic and Nanostructured Electronics Lab. He uses MATLAB’s versatile Partial Differential Equation (PDE), Global Optimization, and ParallelComputingToolboxes to solve, model, and fit the group’s experimental data sets on carrier recombination and diffusion in semiconductors for optoelectronic applications. Leveraging the PDE Toolbox, his most recent work, “AccurateDetermination of Semiconductor Diffusion Coefficient Using Confocal Microscopy,” explores common pitfalls in modeling charge carrier diffusion in semiconductors that were elucidated by simulations carried out using the PDE toolbox. He is now focused on modeling semiconductors for solar applications that present anisotropic diffusion and complex boundaries using the PDE Toolbox coupled with the Parallel Computing and the Global Optimization Toolboxes to optimize multi-variable minimization problems. The Parallel Computing Toolbox has been essential in bringing the computation times down to manageable time frames, allowing him to explore multiple schemes that could represent the complex carrier diffusion anisotropy that is observed. He earned an SB in electrical engineering and physics fromMIT.

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Roberto Brenes

Peng Cao is a PhD candidate who has demonstrated innovative use of MathWorks tools to make fundamental contributions to machine learning and digital health. Specifically, Peng’s research focuses on sensing systems, data analysis, and radio signals. MATLAB is an essential tool in this work, for numerical optimization, complex signal processing, and analysis and manipulation of radio signals and electromagnetic fields. The MathWorks Fellowship will support Peng’s cutting-edge research in several domains. She has developed new approaches to learning from crowdsourcing labels, including robust information-theoretic loss functions that enable learning from multiple labels and noisy labels, as well as semi-supervised, multimodal learning. Peng has also made notable advances in human-motion modeling with a transformer-based deep learning model for generative modeling of 3-D human motion. Another project yielded a new approach to enable robots to move quickly in indoor environments without colliding with people—one of the first such projects to demonstrate indoor localization around corners using radio signals. Most recently, Peng developed the first contactless system to monitor blood oxygen by employing a neural network to assess oxygen levels within 2 to 3 degrees. This work, which was validated with Covid-19 patients, has enormous potential for telehealth applications.

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Yifan (Henry) Cao is a PhD candidate working at the frontier of a new materials science field: high- entropy alloys (HEAs). HEAs offer valuable traits, such as high strength and fracture resistance. As a MathWorks Fellow, Henry will be expanding a new method he has developed to optimize machine learning interatomic potentials (MLP) in cross-scale atomistic simulations, thereby predicting HEA properties with greater efficiency than current methods. Although HEAs are created by mixing many chemical elements, local chemical ordering (LCO), the prominent mechanism underlying many of these properties, remains poorly understood. Henry’s research seeks to close that knowledge gap. Two MathWorks tools will play a central role in the project: Simulink, which will enable the development of a well-rounded MLP training set, and MATLAB, which will be used to extract physically important features and plot relevant data obtained from HEA cross-scale simulations. Henry also plans to explore potential applications of these tools in mesoscale, atomic, and quantum mechanics modeling approaches.

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Yifan (Henry) Cao

Katherine is a master’s student in aeronautical and astronautical engineering and member of the Engineering SystemsLaboratory. A longtime MATLAB and Simulink user, her current research involves population modeling and simulation of pedestrian activity through transit environments. She is implementing agent-based modeling and simulation in order to capture the emergent phenomena of population groups moving across complex sites, both here on Earth and on Mars. She has used MATLAB to numerically simulate pedestrian flows in hallways using a social-force model and is currently developing a Java-based integrated, flexible platform to simulate agent behavior across multiple agent pools and simulation scenarios. Katherine earned a BS in aerospace engineering fromUniversity of Illinois.

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Katherine Carroll

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Those interested in a PhD in Operations Research should visit the Operations Research Center .  

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The goal of the MIT Sloan PhD Program's admissions process is to select a small number of people who are most likely to successfully complete our rigorous and demanding program and then thrive in academic research careers. The admission selection process is highly competitive; we aim for a class size of nineteen students, admitted from a pool of hundreds of applicants.

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The PhD Program is integral to the research of MIT Sloan's world-class faculty. With a reputation as risk-takers who are unafraid to embrace the unconventional, they are engaged in exciting disciplinary and interdisciplinary research that often includes PhD students as key team members.

Research centers across MIT Sloan and MIT provide a rich setting for collaboration and exploration. In addition to exposure to the faculty, PhD students also learn from one another in a creative, supportive research community.

Throughout MIT Sloan's history, our professors have devised theories and fields of study that have had a profound impact on management theory and practice.

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Tomasz Mrowka (MIT)

Myhill lecture series: forty years of four manifolds.

Tomasz Mrowka

Contact: Daniel Sage, [email protected]

April 23, 24 and 25, 2024:  Join us for the Myhill Lecture Series: Forty Years of Four Manifolds, featuring Tomasz Mrowka (MIT). Since the twin breakthroughs in 1982-83 by Freedman and Donaldson the study of four manifolds has been developing rapidly. Freedman’s work showed that the homeomorphism problem for 4-dimensional manifolds was largely under control provided the fundamental group of the 4-manifold was not too complicated. Donaldson’s surprising applications of the Yang-Mills equations hinted that the situation for smooth structures was more complicated and that the tools for studying 4-manifolds would come from diverse parts of mathematics. These three lectures will survey a number of the developments that occurred in the ensuing 40 years.

Many problems have been resolved during this time. An few of highlights include; existence of exotic differentiable structures on ℝ4, the failure of the h-cobordism theorem in dimension four, the Thom conjecture on minimal genus of surfaces in the complex projective plane, the Weinstein conjecture on existence of closed Reeb orbits in dimension three, the disproof of the triangulation conjecture, .... 

Lecture 1 will focus on setting up the problems and basic questions in 4-manifold topology. Lecture 2 will discuss the many tools that have been developed to aid in this study: the Yang-Mills and Seiberg-Witten equations, Ozsváth and Szabó’s Heegaard Floer theory, Embedded Contact Homology and their applications to question in 3- and 4- dimensional topology.

Lecture 3 will try to sketch where the theory is headed, including the study the diffeomorphism groups of four dimensional manifolds. I hope to make the lectures independent but some things may flow from one lecture to another.

Tomasz Mrowka (MIT) April 23, 24 and 25, 2024 Tuesday, Wednesday, Thursday 4:00 P.M. each day 250 Mathematics Building UB North Campus

SPEAKER BIO:   Tomasz Mrowka's  research interests focus on problems in differential geometry and gauge theory. His work combines analysis, geometry, and topology, specializing in the use of partial differential equations, such as the Yang-Mills equations from particle physics to analyze low-dimensional mathematical objects. Jointly with Robert Gompf, he discovered four-dimensional models of space-time topology. 

A graduate of MIT, Mrowka received the Ph.D. from U.C. Berkeley in 1988 under the direction of Clifford Taubes and Robin Kirby. He joined the MIT mathematics faculty as professor in 1996, following faculty appointments at Stanford and at Caltech (professor 1994-96). He chaired the Graduate Student Committee 1999-02, and chaired the Pure Mathematics Committee, 2004-15. From 2014-2017 he served as Department Head. A prior Sloan fellow and Young Presidential Investigator, Mrowka was selected for a Clay Mathematics Visiting Professorship in 1995.

In 2007 he received the Veblen Prize in Geometry by the AMS, jointly with Peter Kronheimer, "for their joint contributions to both three- and four- dimensional topology through the development of deep analytical techniques and applications." Their book, Monopoles and Three Manifolds (Cambridge University Press) also garnered the 2011 Joseph Doob Prize of the AMS. He was appointed Singer Professor of Mathematics from 2007 to 2017. In 2017, Mrowka received a Simons Fellowship in Mathematics. In 2018 delivered a plenary address at ICM18 in Rio de Janeiro. He is a Fellow of the American Academy of Arts & Sciences (2007) and Member of the National Academy of Sciences (2015). Most recently, he was awarded the 2023 Leroy P. Steele Prize for Seminal Contribution to Research for his joint paper with Peter Kronheimer, ‘Gauge theory for embedded surfaces, I’ published in 1993 in Topology. 

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Graduate student Michael Calzadilla awarded 2024 NASA Hubble Fellowship

Michael Calzadilla grew up in Tampa, Florida. As a first-generation college student, he earned his bachelor’s degree in physics from the University of South Florida in 2015. He subsequently crossed the pond to complete a master’s degree in astronomy as a Gates Cambridge scholar under the guidance of Professor Andrew Fabian at the University of Cambridge. Michael will complete his doctorate in physics at the Massachusetts Institute of Technology in May 2024 with his advisor Professor Michael McDonald .

Michael’s work focuses on multiwavelength observations of galaxy clusters to study the baryon cycle that drives the evolution of all galaxies. The largest galaxies residing in these clusters grow via material cooling from their hot atmospheres, which is balanced by feedback from star formation and active galactic nuclei. As part of the South Pole Telescope collaboration, Michael’s work is among the first to leverage recent Sunyaev-Zeldovich-based detections of galaxy clusters to observe this cycling of material out to unprecedented redshifts.

As a Hubble Fellow, Michael will develop machine learning techniques for characterizing the thousands of galaxy clusters being discovered by next-generation cosmological surveys resulting in clean, unbiased samples of the earliest galaxy clusters. Using synergies with large X-ray, optical, and radio datasets, he will seek to answer when galaxy clusters first dynamically relaxed, and how the effectiveness of supermassive black hole feedback has changed over time. He will also use new observatories for more targeted follow-up to investigate the role of feedback-induced turbulence in regulating galaxy growth.

See NASA press release:  https://science.nasa.gov/missions/hubble/nasa-awards-astrophysics-postdoctoral-fellowships-for-2024/ See also   https://www.stsci.edu/stsci-research/fellowships/nasa-hubble-fellowship-program/2024-nhfp-fellows )

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QS World University Rankings rates MIT No. 1 in 11 subjects for 2024

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Programming functional fabrics

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Encouraged by her family, Lavender Tessmer explored various creative pursuits from a young age, particularly textiles, including knitting and crocheting. When she came to MIT, she figured that working with textiles would remain just a hobby; she never expected them to become integral to her career path.

However, when she interviewed for a research assistant position in Self Assembly Lab , it just so happened that the lab had recently received funding from the Advanced Functional Fabrics of America, one of the manufacturing institutes launched during the Obama administration , for a textile-based project.

Tessmer, now a fifth-year doctoral student in design and computation within the School of Architecture and Planning, took on the project, working with Skylar Tibbits , associate professor of design research, and Caitlin Muller , associate professor in building technology. “At MIT, my interest in textiles really exploded and became the center of everything,” Tessmer says.

While textiles may appear commonplace, the Covid-19 pandemic underscored the need for textile products in safeguarding our general health and safety, particularly through the filtration necessary for masks. Recognizing the importance of manufacturing capabilities for textiles, Tessmer’s research has focused on programming textiles with specific functional properties while also considering the feasibility of large-scale manufacturing of such products.

A nonlinear path to MIT

Tessmer studied music as an undergraduate student at Duquesne University, pursuing a passion that bloomed as a high schooler. One assignment opened her eyes to a different career path: She was told to compare a piece of music to some other artistic medium. Through this assignment, she discovered the world of architecture by underscoring the systematic nature of both disciplines, emphasizing the need for repetition and structure to unleash creativity. “I immediately realized that’s what I want to do,” she says.

Tessmer switched gears and decided to devote the year after college to architecture, instead of auditioning for music ensembles. She says, “I always liked making things, and then, with architecture, I realized that you can make things as part of your profession.” She relied on the basic drafting skills that her father had taught her, and channeled these into building her architecture portfolio.

Ultimately, she decided to pursue a master’s degree in architecture at Washington University in St. Louis. She graduated with her master’s at the end of the 2007 economic recession, a time when jobs in architecture were scarce. She eagerly accepted a part-time role teaching at WashU. Over the next five years, this role evolved into a full-time lecturer role, where she taught students while also independently establishing her own design practice and leading various installation design projects. Fittingly, all of the installations were inspired by textiles. “They were these high-performance carbon-fiber braided structures that we hand-made into large-scale braided nets with specific geometries,” Tessmer explains.

“Squeezing everything” out of graduate school

Teaching at WashU was a great experience, but the practice-oriented nature of the architecture department motivated Tessmer to seek complementary perspectives on design. “I wanted a totally new venue that was supportive of research and pushing the boundaries of design. I wanted to see what other approaches were out there,” she says. As her interests continued to grow in that direction, she learned that MIT has some renowned researchers in the field. She decided to apply for a master’s degree in architecture studies, and ultimately a doctorate in design and computation, within the School of Architecture and Planning.

MIT’s program stood out to Tessmer because of the interdisciplinary approach of the architecture department. She says, “If you are an architect or designer, it is not strange to end up in a class full of people who are not architects, and that’s totally normal and even expected.” The integrated nature of her program is a shift from her previous academic experiences, where each discipline had been distinct and separate. She also values the lack of hierarchy between different disciplines within the architecture department here. “There is respect across disciplines for the contribution from each participant,” she says.

As an older student, Tessmer has a slightly different approach to graduate school, compared to her peers. She says, “MIT is amazing because there is so much variety and so many things that you can get involved in. But my style is to be hyperfocused on my interests. For me, there have been huge benefits to focusing on this specific thing and squeezing everything I can out of it, even in the face of all of these other opportunities.”

Tessmer has devoted herself to several projects throughout grad school, but all share a common thread: an emphasis on fiber development and textile programming. As a master’s student in the Self Assembly Lab, she utilized the inherent properties of materials and optimized their configurations for specific functions by integrating computation into the material itself. “At MIT, I learned a much broader definition of computation,” she says. “For example, in the Self Assembly Lab, we believe that material is a storage format of information and that you can program material to behave in certain ways.”  

The first project Tessmer worked on was designing a fiber that could respond to temperature fluctuations. Another project focused on embedding many different properties within a single fabric, potentially for astronauts. “The human body is so varied in the number of properties that you need to match,” she says. In conjunction with collaborators across multiple MIT departments, she designed a spacesuit sleeve with embedded padding, stretchable areas, a compression gradient, and various sensors. Her third project has focused on embedding shapechange behavior into fabric structures to enhance human comfort or fit, as an alternative to manual tailoring. Finally, in a return to her architectural roots, she is also working on designing a reinforced concrete beam using textiles, a more sustainable solution to building with concrete, which has a significant carbon footprint.

Another crucial aspect of Tessmer’s research is her focus on the feasibility of large-scale manufacturing for a product. She regularly relies on industrial-scale machinery and consults with manufacturing partners. She says, “The way research is being conducted in the lab is a close parallel to how it would be made in real life. The potential for a direct bridge between one and the other is a high priority for me and a constraint that I have tried to layer on to all of my projects.”

Dabbling in entrepreneurship

Tessmer says with a laugh, “My entire hobby [textiles] has now been absorbed into my research. So I am in the market for a new hobby.” For now, that hobby has taken the form of entrepreneurship. She has been exploring the commercialization potential of her technologies, having filed multiple patents and completed the Blueprint program with The Engine Accelerator. She hopes that one day her method for embedding properties in textiles, while also reducing manufacturing process steps, will be used for commercial fabrics.

As an example, she points to shoe manufacturing. “Your shoes are normally an assembly of lots of different materials and lots of different layers. Instead, my proposal to The Engine focused on embedding all of these properties in an automated way, eliminating the need for an extensive assembly process.” Tessmer envisions entrepreneurship as one of her potential future paths.

For the time being, however, she plans to remain in academia. “From the outside, being a professor seems like an unattainable position. However, I keep being surprised at my ability to get to the next level of the academic hierarchy.” She aims to integrate all her past experiences into a future research career, designing textiles within an architectural context, while also weaving in the constraints of manufacturing scalability.

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