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Ph.D. Program

Introduction.

These guidelines are intended to help familiarize graduate students with the policies governing the graduate program leading to the degrees of Doctor of Philosophy (Ph.D.) in Applied Mathematics. This material supplements the graduate school requirements found on the  Graduate Student Resources  page and the  Doctoral Degree Policies  of the graduate school. Students are expected to be familiar with these procedures and regulations.

The Doctor of Philosophy program

The Doctor of Philosophy (Ph.D.) Degree in Applied Mathematics is primarily a research degree, and is not conferred as a result of course work. The granting of the degree is based on proficiency in Applied Mathematics, and the ability to carry out an independent investigation as demonstrated by the completion of a doctoral dissertation. This dissertation must exhibit original mathematical contributions that are relevant to a significant area of application.

Course requirements for the Ph.D. program

• AMATH 561, 562, 563
• AMATH 567, 568, 569
• AMATH 584, 585, 586
• AMATH 600: two, 2-credit readings, each with a different faculty member, to be completed prior to the start of the student's second year.
• Students must take a minimum of 15 numerically graded courses. At most two of these can be at the 400 level or be cross listed with courses at the 400 level. Graduate level courses previously taken at UW (e.g., during a Master's program) count toward this requirement. Graduate level courses taken outside of UW may count toward the requirement for 15 numerically graded courses with the approval of the Graduate Program Coordinator. The entire course of study of a student and all exceptions to this list must be approved by the Graduate Program Coordinator and the student’s advisor or faculty mentors.

For students who entered the doctoral program autumn 2017 or autumn 2018, please see these degree requirements. For students who entered the doctoral program prior to autumn 2017, please see these degree requirements.  

Faculty mentoring

Upon arrival, incoming students will be assigned two faculty mentors. Until a student settles on an advisor, the faculty mentors aid the student in selecting courses, and they each guide the student through a 2-credit independent reading course on material related to the student’s research interest. The faculty mentors are not necessarily faculty in the Department of Applied Mathematics.

Faculty advisor

By the end of a student’s first summer quarter, an advisor must be determined.  T he advisor provides guidance in designing a course of study appropriate for the student’s research interests, and in formulating a dissertation topic.

A full Supervisory Committee should be formed four months prior to the student’s General Exam. The full Supervisory Committee should have a minimum of three regular members plus the Graduate School Representative , and will consist of at least two faculty members from Applied Mathematics, one of whom is to be the Chair of the Committee. If the proposed dissertation advisor is a member of the Applied Mathematics faculty, then the advisor will be the Chair. The dissertation advisor may be from another department,  or may have an  affiliate  (assistant, associate, full) professor appointment with the Applied Mathematics department  and is then also a member of the Supervisory Committee.

The Dissertation Reading Committee , formed after the General Exam,  is a subset of  at least   three members from the Supervisory Committee   who are appointed to read and approve the dissertation.  Two members of the Dissertation Reading Committee must be from the Applied Mathematics faculty. At least one of the committee members must be a member of the core  Applied Mathematics faculty. It is required that this member is present for both the general and final examination, and is included on the reading committee.

While the principal source of guidance during the process of choosing specialization areas and a research topic is the thesis advisor, it is strongly advised that the student maintain contact with all members of the Supervisory Committee. It is suggested that the student meet with the Supervisory Committee at least once a year to discuss their progress until the doctoral thesis is completed.

Examination requirements for the Ph.D. program

Students in the Ph.D. program must pass the following exams:

• The  qualifying exam
• The  general exam
• The  final exam  (defense)

Satisfactory performance and progress

At all times, students need to make satisfactory progress towards finishing their degree. Satisfactory progress in course work is based on grades. Students are expected to maintain a grade point average of 3.4/4.0 or better. Satisfactory progress on the examination requirements consists of passing the different exams in a timely manner. Departmental funding is contingent on satisfactory progress.   The Graduate School rules regarding satisfactory progress are detailed in Policy 3.7: Academic Performance and Progress .   The Department of Applied Mathematics follows these recommended guidelines of the Graduate School including an initial warning, followed by a maximum of three quarters of probation and one quarter of final probation, then ultimately being dropped from the program.    We encourage all students to explore and utilize the many available  resources  across campus.

Expected academic workload

A first-year, full-time student is expected to register for a full course load, at least three numerically graded courses, typically totaling 12-18 credits. All other students are expected to consult with their advisor and register for at least 10-18 credits per quarter.  Students who do not intend to register for a quarter must seek approved  academic leave  in order to maintain a student status.   Students who do not maintain active student status through course registration or an approved leave request need to request reinstatement to rejoin the program. Reinstatement is at the discretion of the department. Students approved for reinstatement are required to follow degree requirements active at time of reinstatement. 

Annual Progress Report

Students are required to submit an Annual Progress Report to the Graduate Program Coordinator by the second week of Spring Quarter each year. The annual progress report should contain the professional information related to the student’s progress since the previous annual report. It should contain information on courses taken, presentations given, publications, thesis progress, etc., and should be discussed with the student's advisor prior to submission. Students should regard the Annual Progress Report as an opportunity to self-evaluate their progress towards completing the PhD. The content of the Annual Progress Report is used to ensure the student is making satisfactory progress towards the PhD degree.

Financial assistance

Financial support for Doctoral studies is limited to five years after admission to the Ph.D. program in the Department of Applied Mathematics. Support for an additional period may be granted upon approval of a petition, endorsed by the student’s thesis supervisor, to the Graduate Program Coordinator.

Master of Science program

Students in the Ph.D. program obtain an M.Sc. Degree while working towards their Ph.D. degree by satisfying the  requirements for the M.Sc. degree.  

Additional Ph.D. Degree Options and Certificates

Students in the Applied Mathematics Ph.D. program are eligible to pursue additional degree options or certificates, such as the  Advanced Data Science Option  or the  Computational Molecular Biology Certificate .  Students must be admitted and matriculated to the PhD program prior to applying for these options. Option or certificate requirements are in addition to the Applied Mathematics degree requirements. Successful completion of the requirements for the option or the certificate leads to official recognition of this fact on the UW transcript.

Career resources, as well as a look at student pathways after graduation, may be found   here.

FAQs |  Contact the Graduate Program  |  Apply Now

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Top 10 Best PhD in Mathematics Programs

A PhD in mathematics is an invaluable qualification in many sectors, but particularly academia. According to the Bureau of Labor Statistics, mathematicians and statisticians will see a 33% growth in jobs  from 2020 to 2030, especially with the growing need for data and cybersecurity  professionals.

Mathematics PhD programs equip you with the advanced skills and knowledge you need to progress to the highest levels of the field.

Table of Contents

Best PhD in Mathematics Programs

Stanford university, school of humanities & sciences.

Ph.D. in Mathematics

Stanford is a renowned Ivy League university known for its high-quality programs in several areas, including mathematics. Its faculty includes eminent figures in the field, and the school features a dedicated mathematics research center on campus. Students must pass two exams, one in algebra and the other in real analysis, by the second year to remain in the program. Progressing students are fully funded for five years.

• Courses : Multivariable calculus, linear algebra, and applied matrix theory.
• Duration : 5 years
• Delivery : On-campus
• Tuition : Refer tuition page
• Financial aid : Teaching/research assistantships and fellowships
• Acceptance rate: 5.2%
• Location : Stanford, California

Harvard University, Graduate School of Arts and Sciences

Harvard’s Department of Mathematics is one of the world’s most acclaimed centers for research and education in pure mathematics. You’ll need to complete four courses for each term of your PhD, though there are no compulsory courses. As one of the most impressive mathematics doctoral programs, this PhD emphasizes the love of the subject for its own sake, curiosity, creative ability, and the research process.

• Courses : Advanced real analysis, commutative algebra, and geometry & complexity theory.
• Credits : 16 half-courses
• Duration : 2-5 years
• Financial aid : Grants and fellowships
• Acceptance rate: 5%
• Location : Cambridge, Massachusetts

Brown University, Graduate School

Brown University is known for its flexible yet rigorous approach to education that gives students the intellectual freedom to apply creative thinking to their work. Its PhD in Math program encourages close student-faculty relationships and collaboration. This program doesn’t ask for GRE scores or foreign language courses for international students.

• Courses : Complex functions, topology, and algebraic geometry.
• Tuition : $7,835 per course
• Financial aid : Fellowships and teaching assistantships
• Acceptance rate:  7.7%
• Location : Providence, Rhode Island

The University of Pennsylvania, School of Arts & Sciences

UPenn is a world-renowned university that boasts the first medical school in the country. It has a very active mathematics department committed to serious research, and its faculty includes recognized figures from the mathematics community. In its doctorate in mathematics program, students can take courses from other departments in the university. Full-tuition scholarships for all five years of the program are available.

• Courses: Geometry-topology, mathematical physics, and combinatorics.
• Duration: 5 years
• Delivery: On-campus
• Tuition: Full funding
• Financial aid: Fellowships, teaching assistantships, and scholarships.
• Acceptance rate: 9%
• Location: Philadelphia, Pennsylvania

Rochester Institute of Technology, School of Mathematical Sciences

Ph.D. Mathematical Modeling

The School of Mathematical Sciences at RIT is world-renowned for mathematical and statistical research. It focuses on creative approaches to solving real-world problems using mathematical and statistical knowledge. Its doctorate degree in mathematics is the country’s first  mathematical modeling PhD  and offers five concentrations, including Biomedical mathematics, applied inverse problems, and dynamical systems & fluid dynamics.

• Courses : Numerical analysis, mathematical modeling, and high-performance computing for mathematical modeling.
• Credits : 60 minimum
• Duration : 7 years maximum
• Financial aid : Scholarships and graduate assistantships
• Acceptance rate : 73.9%
• Location : Rochester, New York

Princeton University, The Graduate School

Ph.D. Applied and Computational Math

Princeton University is one of the world’s most famous universities, scoring the top ranking in the US News & World Report’s Best Universities list for 2022 . Its PhD program in Applied and Computational Math is flexible with few central requirements: students can work with their advisers to build their own program.

• Courses : Differential equations, numerical algorithms, and applied dynamical systems.
• Credits : 12 hours/semester
• Duration : 4 years
• Tuition : $56,010 per year
• Financial aid : Fellowships, research/teaching assistantships, loans, grants, and federal work-study.
• Acceptance rate : 5.6%
• Location : Princeton, New Jersey

The University of Arizona, Department of Mathematics

Ph.D. in Applied Mathematics

The University of Arizona is well-known for its contributions to scientific research. Its PhD in Applied Mathematics is an interdisciplinary program that emphasizes mathematical modeling and computer simulation. The teaching staff is drawn from various disciplines, including engineering, computing, social, and biological sciences.

• Courses : Real analysis, algebra, and topology-geometry.
• Tuition : $13,400 per year
• Financial aid : Teaching assistantships, research assistantships, fellowships, and grants.
• Acceptance rate:  85%
• Location : Tucson, Arizona

Portland State University, Fariborz Maseeh Department of Mathematics &         Statistics

Ph.D. Mathematical Sciences

PSU’s innovative Mathematics Department has a joint research project  with Ohio University to make the theoretical science of math more relevant and practical. Its PhD program focuses on mathematics and statistics, with separate primary and secondary-level courses in both subjects.

• Courses : Mathematics, statistics, and mathematical/statistical literature problems.
• Credits : 81 minimum
• Tuition : $509 per credit
• Financial aid : Scholarships, fellowships, research assistantships, teaching assistantships, and veteran benefits.
• Acceptance rate : 95.2%
• Location : Portland, Oregon

Rice University, Wiess School of Natural Sciences

Ph.D. in Computational and Applied Mathematics

Rice University works closely with many prominent research, education, and science institutions in Houston, translating into exciting student opportunities. Its Computational and Applied Mathematics department only accepts 5-9 students into its PhD program, assuring a low student-faculty ratio. The program allows students to conduct high-impact research with the help of professors from diverse fields such as energy, mathematical biology, healthcare delivery, and sustainability.

• Courses : Advanced numerical analysis, computational science , and modeling mathematical physics.
• Duration : 4-6 years
• Tuition : $2,893 per credit
• Financial aid : Scholarships, fellowships, teaching assistantships, and loans.
• Acceptance rate : 10.9%
• Location : Houston, Texas

Florida Institute of Technology

Applied Mathematics, Ph.D.

Florida Institute of Technology is the only independent technological university in the Southeast. It is a private institution encouraging a hands-on approach to students’ learning. This PhD is one of the only 30 Applied Mathematics programs in the country and accommodates a wide range of specializations, including non-linear analysis, scientific computing, and optimization. The highly flexible program also allows students to choose their research areas.

• Courses : Partial differential equations, real analysis, and numerical methods for partial differential equations.
• Credits : 72 post-bachelor’s or 42 post-master’s
• Tuition : $1,241 per credit
• Financial aid: Scholarships, grants, awards, loans, graduate assistantships, and veteran benefits.
• Acceptance rate: 70.4%
• Location : Melbourne, Florida

What Do You Need to Get a PhD in Mathematics?

Exact entry requirements vary from program to program, but often you’ll need a master’s in a related field as a basic prerequisite. Some programs will accept graduates with a bachelor’s degree in mathematics or a related subject, though you’ll probably need to complete extra credit hours compared to students with a master’s.

You’ll need to submit an application, typically involving a research proposal and/or personal statement , academic resume , and letters of recommendation . You may need to present GRE  scores , but these aren’t required for every program.

Equally, the requirements to earn your doctorate will depend on the specific program, though there are several commonalities. Most PhD in mathematics programs include:

• Qualifying examinations and a final exam
• Teaching hours
• A research dissertation

If you want to prepare thoroughly for a math PhD program, it can be helpful to:

• Study qualifying exam subjects  such as real analysis and algebra
• Working with proof of math problems
• Participate in u ndergraduate research experience programs
• Build your network  and identify mentors

What to Consider When Choosing a Mathematics PhD Program

With so many strong math PhD programs on offer, it can be hard to know which is the right program for you.

While this is a very individual decision, here are some key factors to consider:

• Courses, specializations, and concentrations
• Funding options
• School and faculty reputation
• Mode of delivery: online, on-campus, or hybrid

Why Get a Doctorate in Mathematics?

Today’s world desperately needs data in every field to improve and grow. So, mathematics is now highly relevant across many fields. The Bureau of Labor Statistics estimates impressive job growth of 33%  for mathematicians and statisticians from 2020 to 2030.

A PhD in mathematics equips you with a range of skills, including critical and analytical evaluation, logical reasoning, mathematical solutions for real-life issues, verbal and written communication skills, and dedication.

According to the American Mathematical Society , the number of math PhD graduates has been increasing steadily over the last two decades, and is currently around 2,000 a year. In total, there have been more than 30,000 PhD graduates in mathematics over the last two decades.

Common Jobs for Doctor of Mathematics Grads

With a doctorate in mathematics, you may work in various roles, including:

• Assistant Professor ( $61,252  average annual salary)
• Professor ( $88,951 )
• Data Scientist ( $97,358 )
• Mathematician ( $70,920 )
• Postdoctoral Research Associate ( $51,968 )

Key Takeaways

The best PhD mathematics programs are offered by renowned universities across the country, including the most prestigious schools. While some are expensive, various funding options and even fully-funded Mathematics doctorate programs offer some relief.

If you have a real taste for the subject, a PhD in mathematics can take your career to unimaginable heights. It will also let you build high-level skills and in-depth knowledge that could make you a leader in your field.

If you’ve found this guide useful, take a look at our posts on the best online Master’s in Mathematics programs , the top online PhD programs , and the highest-paying PhDs .

PhD in Mathematics FAQs

How long is a phd in mathematics.

A PhD in Mathematics may take 3-7 years to complete, depending on program requirements. If you have a master’s degree , the program will likely be shorter in duration.

How Much Does a Math PhD Cost?

A math PhD costs anywhere from $13,000 to $45,000 per year, including tuition and other fees, based on various factors. You can also find fully funded PhD programs in Math, including from some of the most prestigious schools in the country.

Are Math PhDs in Demand?

A PhD in math is a qualification that has always had a demand in the academic world. With the increasing importance of mathematical principles and concepts in emerging technologies and disciplines, we’re also seeing a range of opportunities for mathematicians in other fields.

Is a PhD in Mathematics Difficult?

Mathematics is an abstract and mostly theoretical subject. Therefore, it’s best to have a certain degree of flair, or at least interest in the subject, otherwise, you may struggle. However, anyone can excel in a math PhD program with a little dedication and hard work.

What Do I Need to be Eligible for a PhD in Mathematics?

At a minimum, you’ll need a bachelor’s in math or another STEM field. However, some programs require a master’s or at least an honors undergraduate qualification.

Lisa Marlin

Lisa is a full-time writer specializing in career advice, further education, and personal development. She works from all over the world, and when not writing you'll find her hiking, practicing yoga, or enjoying a glass of Malbec.

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Mathematics at MIT is administratively divided into two categories: Pure Mathematics and Applied Mathematics. They comprise the following research areas:

Pure Mathematics

• Algebra & Algebraic Geometry
• Algebraic Topology
• Analysis & PDEs
• Mathematical Logic & Foundations
• Number Theory
• Probability & Statistics
• Representation Theory

Applied Mathematics

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.

• Combinatorics
• Computational Biology
• Physical Applied Mathematics
• Computational Science & Numerical Analysis
• Theoretical Computer Science
• Mathematics of Data

Why Study for a Mathematical Finance PhD?

I was emailed by a reader recently asking about mathematical finance PhD programs and the benefits of such a course. If you are considering gaining a PhD in mathematical finance, this article will be of interest to you.

If you are currently near the end of your undergraduate studies or are returning to study after some time in industry, you might consider starting a PhD in mathematical finance. This is an alternative to undertaking a Masters in Financial Engineering (MFE), which is another route into a quantitative role. This article will discuss exactly what you will be studying and what you are likely to get out of a PhD program. Clearly there will be differences between studying in the US, UK or elsewhere. I personally went to grad school in the UK, but I will discuss both UK and US programs.

Mathematical finance PhD programs exist because the techniques within the derivatives pricing industry are becoming more mathematical and rigourous with each passing year. In order to develop new exotic derivatives instruments, as well as price and hedge them, the financial industry has turned to academia. This has lead to the formation of mathematical finance research groups - academics who specialise in derivatives pricing models, risk analysis and quantitative trading.

Graduate school, for those unfamiliar with it, is a very different experience to undergraduate. The idea of grad school is to teach you how to effectively research a concept without any guidance and use that research as a basis for developing your own models. Grad school really consists of a transition from the "spoon fed" undergraduate lecture system to independent study and presentation of material. The taught component of grad school is smaller and the thesis component is far larger. In the US, it is not uncommon to have two years of taught courses before embarking on a thesis (and thus finding a supervisor). In the UK, a PhD program is generally 3-4 years long with either a year of taught courses, or none, and then 3 years of research.

A good mathematical finance PhD program will make extensive use of your undergraduate knowledge and put you through graduate level courses on stochastic analysis, statistical theory and financial engineering. It will also allow you to take courses on general finance, particularly on corporate finance and derivative securities. When you finish the program you will have gained a broad knowledge in most areas of mathematical finance, while specialising in one particular area for your thesis. This "broad and deep" level of knowledge is the hallmark of a good PhD program.

Mathematical Finance research groups study a wide variety of topics. Some of the more common areas include:

• Derivative Securities Pricing/Hedging: The technical term for this is "financial engineering", as "quantitative analysis" now encompasses a wide variety of financial areas. Some of the latest research topics include sophisticated models of options including stochastic volatility models, jump-diffusion models, asymptotic methods as well as investment strategies.
• Stochastic Calculus/Analysis: This is more of a theoretical area, where the basic motivation stems from the need to solve stochastic differential equations. Research groups may look at path-dependent PDEs, functional Ito calculus, measure theory and probability theory.
• Fixed Income Modeling: Research in this area centres on effectively modelling interest rates - such as multi-factor models, multi-curve term structure models as well as interest rate derivatives such as swaptions.
• Numerical Methods: Although not always strictly related to mathematical finance, there is a vast amount of university research carried out to try and develop more effective means of solving equations numerically (i.e. on the computer!). Recent developments include GPU-based Monte Carlo solvers, more efficient matrix solvers as well as Finite Differences on GPUs. These groups will almost certainly possess substantial programming expertise.
• Market Microstructure/High-Frequency Modeling: This type of research is extremely applied and highly valued by funds engaged in this activity. You will find many academics consulting, if not contracting, for specialised hedge funds. Research areas include creating limit order market models, high frequency data statistical modelling, market stability analysis and volatility analysis.
• Credit Risk: Credit risk was a huge concern in the 2007-2008 financial crisis and many research groups are engaged in determining such "counterparty risks". Credit derivatives are still a huge business and so a lot of research goes into collateralisation of securities as well as pricing of exotic credit derivatives.

These are only a fraction of the total areas that are studied within mathematical finance. The best place to find out more about research topics is to visit the websites of all the universities which have a mathematical finance research group, which is typically found within the mathematics, statistics or economics faculty.

The benefits of undertaking a PhD program are numerous:

• Employment Prospects: A PhD program sets you apart from candidates who only possess an undergraduate or Masters level ability. By successfully defending a thesis, you have shown independence in your research ability, a skill highly valued by numerate employers. Many funds (and to a lesser extent, banks) will only hire PhD level candidates for their mathematical finance positions, so in a pragmatic sense it is often a necessary "rubber stamp". In investment banks, this is not the case so much anymore, as programming ability is generally prized more. However, in funds, it is still often a requirement. Upon being hired you will likely be at "associate" level rather than "analyst" level, which is common of undergraduates. Your starting salary will reflect this too.
• Knowledge: You will spend a large amount of time becoming familiar with many aspects of mathematical finance and derivatives theory. This will give you a holistic view into the industry and a more transferable skill set than an undergraduate degree as you progress up the career ladder. In addition, you will have a great deal of time to learn how to program models effectively (without the day-to-day pressure to get something implemented any way possible!), so by the time you're employed, you will be "ahead of the game" and will know best practices. This aspect is down to you, however!
• Intellectual Prospects: You are far more likely to gain a position at a fund after completing a PhD than without one. Funds are often better environments to work in. There is usually less stress and a more relaxed "collegiate" environment. Compare this to working on a noisy trading floor, where research might be harder to carry out and be perceived as less important.

I would highly recommend a mathematical finance PhD, so long as you are extremely sure that a career in quantitative finance is for you. If you are still unsure of your potential career options, then a more general mathematics, physics or engineering PhD might be a better choice.

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