top applied math phd programs

Overview of the PhD Program

For specific information on the Applied Mathematics PhD program, see the navigation links to the right. 

What follows on this page is an overview of all Ph.D. programs at the School; additional information and guidance can be found on the  Graduate Policies  pages. 

General Ph.D. Requirements

• 10 semester-long graduate courses, including at least 8 disciplinary.   At least 5 of the 10 should be graduate-level SEAS "technical" courses (or FAS graduate-level technical courses taught by SEAS faculty), not including seminar/reading/project courses.  Undergraduate-level courses cannot be used.  For details on course requirements, see the school's overall PhD course requirements  and the individual program pages linked therein.
• Program Plan (i.e., the set of courses to be used towards the degree) approval by the  Committee on Higher Degrees  (CHD).
• Minimum full-time academic residency of two years .
• Serve as a Teaching Fellow (TF) in one semester of the second year.
• Oral Qualifying Examination Preparation in the major field is evaluated in an oral examination by a qualifying committee. The examination has the dual purpose of verifying the adequacy of the student's preparation for undertaking research in a chosen field and of assessing the student's ability to synthesize knowledge already acquired. For details on arranging your Qualifying Exam, see the exam policies and the individual program pages linked therein.
• Committee Meetings : PhD students' research committees meet according to the guidelines in each area's "Committee Meetings" listing.  For details see the "G3+ Committee Meetings" section of the Policies of the CHD  and the individual program pages linked therein.
• Final Oral Examination (Defense) This public examination devoted to the field of the dissertation is conducted by the student's research committee. It includes, but is not restricted to, a defense of the dissertation itself.  For details of arranging your final oral exam see the  Ph.D. Timeline  page.
• Dissertation Upon successful completion of the qualifying examination, a committee chaired by the research supervisor is constituted to oversee the dissertation research. The dissertation must, in the judgment of the research committee, meet the standards of significant and original research.

Optional additions to the Ph.D. program

Harvard PhD students may choose to pursue these additional aspects:

• a Secondary Field (which is similar to a "minor" subject area).  SEAS offers PhD Secondary Field programs in  Data Science and in  Computational Science and Engineering .   GSAS  lists  secondary fields offered by other programs.
• a Master of Science (S.M.) degree conferred  en route to the Ph.D in one of several of SEAS's subject areas.  For details see here .
• a Teaching Certificate awarded by the Derek Bok Center for Teaching and Learning .

SEAS PhD students may apply to participate in the  Health Sciences and Technology graduate program  with Harvard Medical School and MIT.  Please check with the HST program for details on eligibility (e.g., only students in their G1 year may apply) and the application process.

In Applied Mathematics

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Ph.D. in Mathematics, Specializing in Applied Math

Table of contents, overview of applied mathematics at the courant institute.

• PhD Study in Applied Mathematics
• Applied math courses

Applied mathematics has long had a central role at the Courant Institute, and roughly half of all our PhD's in Mathematics are in some applied field. There are a large number of applied fields that are the subject of research. These include:

• Atmosphere and Ocean Science
• Biology, including biophysics, biological fluid dynamics, theoretical neuroscience, physiology, cellular biomechanics
• Computational Science, including computational fluid dynamics, adaptive mesh algorithms, analysis-based fast methods, computational electromagnetics, optimization, methods for stochastic systems.
• Data Science
• Financial Mathematics
• Fluid Dynamics, including geophysical flows, biophysical flows, fluid-structure interactions, complex fluids.
• Materials Science, including micromagnetics, surface growth, variational methods,
• Stochastic Processes, including statistical mechanics, Monte-Carlo methods, rare events, molecular dynamics

PhD study in Applied Mathematics

PhD training in applied mathematics at Courant focuses on a broad and deep mathematical background, techniques of applied mathematics, computational methods, and specific application areas. Descriptions of several applied-math graduate courses are given below.

Numerical analysis is the foundation of applied mathematics, and all PhD students in the field should take the Numerical Methods I and II classes in their first year, unless they have taken an equivalent two-semester PhD-level graduate course in numerical computing/analysis at another institution. Afterwards, students can take a number of more advanced and specialized courses, some of which are detailed below. Important theoretical foundations for applied math are covered in the following courses: (1) Linear Algebra I and II, (2) Intro to PDEs, (3) Methods of Applied Math, and (4) Applied Stochastic Analysis. It is advised that students take these courses in their first year or two.

A list of the current research interests of individual faculty is available on the Math research page.

Courses in Applied Mathematics

The following list is for AY 2023/2024:

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(MATH-GA.2701) Methods Of Applied Math

Fall 2023, Oliver Buhler

Description:  This is a first-year course for all incoming PhD and Masters students interested in pursuing research in applied mathematics. It provides a concise and self-contained introduction to advanced mathematical methods, especially in the asymptotic analysis of differential equations. Topics include scaling, perturbation methods, multi-scale asymptotics, transform methods, geometric wave theory, and calculus of variations.

Prerequisites : Elementary linear algebra, ordinary differential equations; at least an undergraduate course on partial differential equations is strongly recommended.

(MATH-GA.2704) Applied Stochastic Analysis

Spring 2024, Jonathan Weare

This is a graduate class that will introduce the major topics in stochastic analysis from an applied mathematics perspective.  Topics to be covered include Markov chains, stochastic processes, stochastic differential equations, numerical algorithms, and asymptotics. It will pay particular attention to the connection between stochastic processes and PDEs, as well as to physical principles and applications. The class will attempt to strike a balance between rigour and heuristic arguments: it will assume that students have some familiarity with measure theory and analysis and will make occasional reference to these, but many results will be derived through other arguments. The target audience is PhD students in applied mathematics, who need to become familiar with the tools or use them in their research.

Prerequisites: Basic Probability (or equivalent masters-level probability course), Linear Algebra (graduate course), and (beginning graduate-level) knowledge of ODEs, PDEs, and analysis.

(MATH-GA.2010/ CSCI-GA.2420) Numerical Methods I

• Fall 2023, Benjamin Peherstorfer

Description:   This course is part of a two-course series meant to introduce graduate students in mathematics to the fundamentals of numerical mathematics (but any Ph.D. student seriously interested in applied mathematics should take it). It will be a demanding course covering a broad range of topics. There will be extensive homework assignments involving a mix of theory and computational experiments, and an in-class final. Topics covered in the class include floating-point arithmetic, solving large linear systems, eigenvalue problems, interpolation and quadrature (approximation theory), nonlinear systems of equations, linear and nonlinear least squares, and nonlinear optimization, and iterative methods. This course will not cover differential equations, which form the core of the second part of this series, Numerical Methods II.

Prerequisites:   A good background in linear algebra, and some experience with writing computer programs (in MATLAB, Python or another language).

(MATH-GA.2020 / CSCI-GA.2421) Numerical Methods II

Spring 2024, Aleksandar Donev

This course (3pts) will cover fundamental methods that are essential for the numerical solution of differential equations. It is intended for students familiar with ODE and PDE and interested in numerical computing; computer programming assignments in MATLAB/Python will form an essential part of the course. The course will introduce students to numerical methods for (approximately in this order):

• The Fast Fourier Transform and pseudo-spectral methods for PDEs in periodic domains
• Ordinary differential equations, explicit and implicit Runge-Kutta and multistep methods, IMEX methods, exponential integrators, convergence and stability
• Finite difference/element, spectral, and integral equation methods for elliptic BVPs (Poisson)
• Finite difference/element methods for parabolic (diffusion/heat eq.) PDEs (diffusion/heat)
• Finite difference/volume methods for hyperbolic (advection and wave eqs.) PDEs (advection, wave if time permits).

Prerequisites

This course requires Numerical Methods I or equivalent graduate course in numerical analysis (as approved by instructor), preferably with a grade of B+ or higher.

( MATH-GA.2011 / CSCI-GA 2945) Computational Methods For PDE

Fall 2023, Aleksandar Donev & Georg Stadler

This course follows on Numerical Methods II and covers theoretical and practical aspects of advanced computational methods for the numerical solution of partial differential equations. The first part will focus on finite element methods (FEMs), and the second part on finite volume methods (FVMs) including discontinuous Galerkin (FE+FV) methods. In addition to setting up the numerical and functional analysis theory behind these methods, the course will also illustrate how these methods can be implemented and used in practice for solving partial differential equations in two and three dimensions. Example PDEs will include the Poisson equation, linear elasticity, advection-diffusion(-reaction) equations, the shallow-water equations, the incompressible Navier-Stokes equation, and others if time permits. Students will complete a final project that includes using, developing, and/or implementing state-of-the-art solvers.

In the Fall of 2023, Georg Stadler will teach the first half of this course and cover FEMs, and Aleks Donev will teach in the second half of the course and cover FVMs.

A graduate-level PDE course, Numerical Methods II (or equivalent, with approval of syllabus by instructor(s)), and programming experience.

• Elman, Silvester, and Wathen: Finite Elements and Fast Iterative Solvers , Oxford University Press, 2014.
• Farrell: Finite Element Methods for PDEs , lecture notes, 2021.
• Hundsdorfer & Verwer: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations , Springer-Verlag, 2003.
• Leveque: Finite Volume Methods for Hyperbolic Problems , Cambridge Press, 2002.

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( MATH-GA.2012 ) Immersed Boundary Method For Fluid-Structure Interaction

Not offered AY 23/24.

The immersed boundary (IB) method is a general framework for the computer simulation of flows with immersed elastic boundaries and/or complicated geometry.  It was originally developed to study the fluid dynamics of heart valves, and it has since been applied to a wide variety of problems in biofluid dynamics, such as wave propagation in the inner ear, blood clotting, swimming of creatures large and small, and the flight of insects.  Non-biological applications include sails, parachutes, flows of suspensions, and two-fluid or multifluid problems. Topics to be covered include: mathematical formulation of fluid-structure interaction in Eulerian and Lagrangian variables, with interaction equations involving the Dirac delta function; discretization of the structure, fluid, and interaction equations, including energy-based discretization of the structure equations, finite-difference discretization of the fluid equations, and IB delta functions with specified mathematical properties; a simple but effective method for adding mass to an immersed boundary; numerical simulation of rigid immersed structures or immersed structures with rigid parts; IB methods for immersed filaments with bend and twist; and a stochastic IB method for thermally fluctuating hydrodynamics within biological cells.  Some recent developments to be discussed include stability analysis of the IB method and a Fourier-Spectral IB method with improved boundary resolution.

Course requirements include homework assignments and a computing project, but no exam.  Students may collaborate on the homework and on the computing project, and are encouraged to present the results of their computing projects to the class.

Prerequisite:   Familiarity with numerical methods and fluid dynamics.

(MATH-GA.2012 / CSCI-GA.2945) :  High Performance Computing

Not offered AY 23/24

This class will be an introduction to the fundamentals of parallel scientific computing. We will establish a basic understanding of modern computer architectures (CPUs and accelerators, memory hierarchies, interconnects) and of parallel approaches to programming these machines (distributed vs. shared memory parallelism: MPI, OpenMP, OpenCL/CUDA). Issues such as load balancing, communication, and synchronization will be covered and illustrated in the context of parallel numerical algorithms. Since a prerequisite for good parallel performance is good serial performance, this aspect will also be addressed. Along the way you will be exposed to important tools for high performance computing such as debuggers, schedulers, visualization, and version control systems. This will be a hands-on class, with several parallel (and serial) computing assignments, in which you will explore material by yourself and try things out. There will be a larger final project at the end. You will learn some Unix in this course, if you don't know it already.

Prerequisites for the course are (serial) programming experience with C/C++ (I will use C in class) or Fortran, and some familiarity with numerical methods.

(MATH-GA.2011) Monte Carlo Methods

Fall 2023, Jonathan Weare and Jonathan Goodman

Topics : The theory and practice of Monte Carlo methods. Random number generators and direct sampling methods, visualization and error bars. Variance reduction methods, including multi-level methods and importance sampling. Markov chain Monte Carlo (MCMC), detailed balance, non-degeneracy and convergence theorems. Advanced MCMC, including Langevin and MALA, Hamiltonian, and affine invariant ensemble samplers. Theory and estimation of auto-correlation functions for MCMC error bars. Rare event methods including nested sampling, milestoning, and transition path sampling. Multi-step methods for integration including Wang Landau and related thermodynamic integration methods. Application to sampling problems in physical chemistry and statistical physics and to Bayesian statistics.

Required prerequisites:

• A good probability course at the level of Theory of Probability (undergrad) or Fundamentals of Probability (masters)
• Linear algebra: Factorizations (especially Cholesky), subspaces, solvability conditions, symmetric and non-symmetric eigenvalue problem and applications
• Working knowledge of a programming language such as Python, Matlab, C++, Fortran, etc.
• Familiarity with numerical computing at the level of Scientific Computing (masters)

Desirable/suggested prerequisites:

• Numerical methods for ODE
• Applied Stochastic Analysis
• Familiarity with an application area, either basic statistical mechanics (Gibbs Boltzmann distribution), or Bayesian statistics

(MATH-GA.2012 / CSCI-GA.2945) Convex & Non Smooth Optimization

Spring 2024, Michael Overton

Convex optimization problems have many important properties, including a powerful duality theory and the property that any local minimum is also a global minimum. Nonsmooth optimization refers to minimization of functions that are not necessarily convex, usually locally Lipschitz, and typically not differentiable at their minimizers. Topics in convex optimization that will be covered include duality, CVX ("disciplined convex programming"), gradient and Newton methods, Nesterov's optimal gradient method, the alternating direction method of multipliers, the primal barrier method, primal-dual interior-point methods for linear and semidefinite programs. Topics in nonsmooth optimization that will be covered include subgradients and subdifferentials, Clarke regularity, and algorithms, including gradient sampling and BFGS, for nonsmooth, nonconvex optimization. Homework will be assigned, both mathematical and computational. Students may submit a final project on a pre-approved topic or take a written final exam.

Prerequisites: Undergraduate linear algebra and multivariable calculus

Q1: What is the difference between the Scientific Computing class and the Numerical Methods two-semester sequence?

The Scientific Computing class (MATH-GA.2043, fall) is a one-semester masters-level graduate class meant for graduate or advanced undergraduate students that wish to learn the basics of computational mathematics. This class requires a working knowledge of (abstract) linear algebra (at least at the masters level), some prior programming experience in Matlab, python+numpy, Julia, or a compiled programming language such as C++ or Fortran, and working knowledge of ODEs (e.g., an undergrad class in ODEs). It only briefly mentions numerical methods for PDEs at the very end, if time allows.

The Numerical Methods I (fall) and Numerical Methods II (spring) two-semester sequence is a Ph.D.-level advanced class on numerical methods, meant for PhD students in the field of applied math, masters students in the SciComp program , or other masters or advanced undergraduate students that have already taken at least one class in numerical analysis/methods. It is intended that these two courses be taken one after the other, not in isolation . While it is possible to take just Numerical Methods I, it is instead strongly recommended to take the Scientific Computing class (fall) instead. Numerical Methods II requires part I, and at least an undergraduate class in ODEs, and also in PDEs. Students without a background in PDEs should not take Numerical Methods II; for exceptions contact Aleks Donev with a detailed justification.

The advanced topics class on Computational Methods for PDEs follows on and requires having taken NumMeth II or an equivalent graduate-level course at another institution (contact Aleks Donev with a syllabus from that course for an evaluation), and can be thought of as Numerical Methods III.

Q2: How should I choose a first graduate course in numerical analysis/methods?

• If you are an undergraduate student interested in applied math graduate classes, you should take the undergraduate Numerical Analysis course (MATH-UA.0252) first, or email the syllabus for the equivalent of a full-semester equivalent class taken elsewhere to Aleks Donev for an evaluation.
• Take the Scientific Computing class (fall), or
• Take both Numerical Methods I (fall) and II (spring), see Q1 for details. This is required of masters students in the SciComp program .

Applied Mathematics PhD

The Department of Mathematics offers both a PhD program in Mathematics and Applied Mathematics.

Students are admitted for specific degree programs: the PhD in Mathematics or PhD in Applied Mathematics. Requirements for the Mathematics and Applied Mathematics PhDs differ only in minor respects, and no distinction is made between the two in day-to-day matters. Graduate students typically take 5-6 years to complete the doctorate.

Continuing students wishing to transfer from one program to another should consult the graduate advisor in 910 Evans Hall. Transfers between the two PhD programs are fairly routine but must be done prior to taking the qualifying examination. It is a formal policy of the department that an applicant to the PhD program who has previous graduate work in mathematics must present very strong evidence of capability for mathematical research.

Students seeking to transfer to the department's PhD programs from other campus programs, including the Group in Logic and the Methodology of Science, must formally apply and should consult the Vice Chair for Graduate Studies.

Contact Info

[email protected]

Berkeley, CA 94720

At a Glance

Department(s)

Mathematics

Admit Term(s)

Application Deadline

December 17, 2024

Degree Type(s)

Doctoral / PhD

Degree Awarded

GRE Requirements

PhD Program

More information and a full list of requirements for the PhD program in Mathematics can be found in the University Bulletin .

During their first year in the program, students typically engage in coursework and seminars which prepare them for the  Qualifying Examinations .  Currently, these two exams test the student’s breadth of knowledge in algebra and real analysis. 

Starting in Autumn 2023, students will choose 2 out of 4 qualifying exam topics: 

• real analysis
• geometry and topology
• applied mathematics

Course Requirements for students starting prior to Autumn 2023

To qualify for candidacy, the student must have successfully completed 27 units of Math graduate courses numbered between 200 and 297.

Within the 27 units, students must satisfactorily complete a course sequence. This can be fulfilled in one of the following ways:

• Math 215A, B, & C: Algebraic Topology, Differential Topology, and Differential Geometry
• Math 216A, B, & C: Introduction to Algebraic Geometry
• Math 230A, B, & C: Theory of Probability
• 3 quarter course sequence in a single subject approved in advance by the Director of Graduate Studies.

Course Requirements for students starting in Autumn 2023 and later

To qualify for candidacy, the student must have successfully completed 27 units of Math graduate courses numbered between 200 and 297. The course sequence requirement is discontinued for students starting in Autumn 2023 and later.

By the end of Spring Quarter of their second year in the program, students must have a dissertation advisor and apply for Candidacy.

During their third year, students will take their Area Examination , which must be completed by the end of Winter Quarter. This exam assesses the student’s breadth of knowledge in their particular area of research. The Area Examination is also used as an opportunity for the student to present their committee with a summary of research conducted to date as well as a detailed plan for the remaining research.

Years 4&5

Typically during the latter part of the fourth or early part of the fifth year of study, students are expected to finish their dissertation research. At this time, students defend their dissertation as they sit for their University Oral Examination. Following the dissertation defense, students take a short time to make final revisions to their actual papers and submit the dissertation to their reading committee for final approval.

Throughout the PhD Program

All students continue through each year of the program serving some form of Assistantship: Course, Teaching or Research, unless they have funding from outside the department.

Our graduate students are very active as both leaders and participants in seminars and colloquia in their chosen areas of interest.

Applied Mathematics

Prospective ph.d. students, applied mathematics ph.d. program.

The Division of Applied Mathematics is devoted to research, education and scholarship. Our faculty engages in research in a range of areas from applied and algorithmic problems to the study of fundamental mathematical questions. By its nature, our work is and always has been inter– and multidisciplinary. Among the research areas represented in the division are dynamical systems and partial differential equations, control theory, probability and stochastic processes, numerical analysis and scientific computing, fluid mechanics, computational biology, statistics, and pattern theory. Our graduate program in applied mathematics includes around 50 Ph.D. students, with many of them working on interdisciplinary projects. Joint research projects exist with faculty in various biology and life sciences departments and the departments of Chemistry, Computer Science, Cognitive and Linguistic Sciences, Earth, Environmental, and Planetary Sciences, Engineering, Mathematics, Physics and Neuroscience, as well as with faculty in the Warren Alpert Medical School of Brown University.

Prospective PhD applicants who are interested in visiting the campus and meeting with a faculty member to discuss graduate and research programs are encouraged to contact Candida Hall , Student Affairs Manager (401.863.2463).

How to Apply

Please visit our webpage on the Graduate School for information and guidance on the application process, all relevant deadlines, and required materials. 

Inquire or Apply to our Ph.D. Program

• Applied Math code for GRE:  3094, GREs are required for the Academic Year,  2024-2025
• Applied Math code for TOEFL:  3094
• Brown University code for ETS:  3094

Ph.D. Program in Computational Biology

The Division of Applied Mathematics is one of four Brown academic units that contribute to the doctoral program administered by the Center for Computational Molecular Biology. Graduate students in this program who choose applied mathematics as their home department will receive a PhD in Applied Mathematics (with Computational Biology Annotation). For further information about this program, including the application process, please visit the  CCMB Graduate Program page .

Frequently Asked Questions

Applications which are missing materials will be considered, but may be at a disadvantage in regards to admission decisions.    

Admission to our programs depends on many factors. We cannot assess your chances of admission prior to reviewing your entire application. 

The Admission Committee reviews all aspects of your application, including personal statement, recommendation letters, grades, GRE scores, research experience and related original publications, etc. There is no precise formula followed to make an admission decision, but a strong showing in the above components is likely to increase your chances of admission. 

The Admission Committee reviews all aspects of your application when making decisions. The fact that a candidate attended a particular University X (whether X is Brown or any other institution) does not mean that an application will be treated any differently from other applications.

We expect to send the results to you before March 1. Our Student Affairs Manager, Candida Hall , can be contacted if you need information prior to that date.

Yes, we will organize a common Visiting Day for all admitted students sometime in March, and make arrangements for a visit on another day, if needed, to accommodate any schedule conflicts.  

Students are admitted to the Division of Applied Mathematics as a whole, and not to a particular professor or group. 

Over recent years, the incoming PhD class has averaged about 12-15 students per year. The target and actual enrollment for our program varies each year based on a number of factors.  For the academic year 2023-2024, the GRE scores are not required, and the deadline of application is December 9, 2023.

Each year, roughly half of the intake consists of international students. However, we do not have set quotas and decisions are made depending on the quality of the applicant. We are strongly committed to maintaining a fair and equitable admission process and to cultivating diversity in our student body. 

Your chances of admission depend on many factors including test scores (both the TOEFL score and the regular and subject GRE scores), transcripts, recommendation letters, research experience, statement of purpose and research interests, as well as the general background of the students. Improving any or all of these would improve your chances of admission.  For the academic year 2023-2024, the GRE scores are not required.  

Information about the research conducted in the Division can be found on the Division's webpages. If you have specific questions regarding a particular professor's research, you may e-mail that professor directly. 

If you have any technical difficulties with your applications or any other administrative questions related to your application, contact our Student Affairs Manager, [email protected] .

A Bachelors' degree is required, but the area does not have to be in mathematics.  Applicants are expected to have a strong background in mathematics. 

No, you only need to have a Bachelors' degree to apply for the PhD degree. However, you may also apply for a PhD degree after having completed the Masters' degree. 

A $75 application fee must be paid when an application is submitted. Applicants who want to be evaluated by more than one graduate program must submit a separate application and a separate fee for each additional program.

Applicants who are U.S. citizens or permanent residents may be eligible for fee waivers. (Please note that your completed application must be submitted 14 days in advance of the program’s application deadline in order to be considered for a fee waiver. Please choose the “Request a fee waiver” option as your method of payment on the payment information page.) Application fee waivers are not available for international applicants. 

Admission to our PhD program includes at least five years of guaranteed funding, including stipend, tuition, health services fee, and health insurance, for students who maintain good standing in the program.  

For the PhD program, the GRE general test is required and the GRE (mathematics) subject test is highly recommended.  Please note that although the subject test is not required, the absence of a subject score makes determining the quality of your application more challenging.  Nevertheless, it is possible that other portions of your application, such as general GRE scores, grades, letters of recommendation, etc. may provide enough information for a decision to be made.  For the academic year 2023-2024, the GRE scores are not required.

Yes, it is in your own interests to provide as much information as you can.  The more information we have, the more likely that we will be able to assess your application accurately.

Yes. The TOEFL cannot be waived unless you have completed an undergraduate or Masters degree at an accredited institution in which the medium of instruction is English in a predominantly English-speaking country (e.g., the United States, United Kingdom, Australia, New Zealand). The IELTS exam can be substituted for TOEFL. 

The minimum score for admission consideration is 577 on the paper-based test and 90 on the Internet-based test. For IELTS, the recommended minimum overall band score is 7. These exams should be taken early enough to allow the scores to reach the Graduate School by your program's deadline. Performance on the tests is one of many factors considered in making admission decisions. 

Admissions decisions are based on many factors of which test scores are just one (see Q11).  It is your overall performance which will be considered, so your performance in any particular area need not preclude your application being successful.  

We do not track and share average GRE or TOEFL scores. 

Brown University requires official and original test scores sent by ETS.  You may self-report your test scores and upload copies of your score report(s) into your application, prior to the reception of original test scores. 

All international applicants whose native language is not English must submit an official Test of English as a Foreign Language (TOEFL) or International English Language Testing System (IELTS) score.  Language proficiency exams are not required of those students who have earned a degree from a non-U.S. university where the primary language of instruction is English, or from a college or university in the United States, or in any of a number of countries.

We really cannot advise you on this or similar matters since we are not familiar with you or your history, and suggest that you contact an advisor at University X for advice about what choice of courses would be best for your specific circumstances. 

Transferring to the PhD program from the PhD program at another university happens only in very rare circumstances, and depends on many factors. It is unusual for a student's mathematical preparation to be sufficient to merit a transfer and in most cases, the student would need to start the program afresh as a new student. This is best accomplished by applying to the program as a regular applicant for admission in the following Fall.

Ph.D. Program

The degree of Doctor of Philosophy in Applied Mathematics and Computational Science is conferred in recognition of marked ability and high attainment in advanced applied and computational mathematics, including the successful completion of a significant original research project. The program typically takes four to five years to complete, although this length may vary depending on the student. Below, we describe the requirements and expectations of the program. All graduate students require a 3.0 GPA to graduate (no exceptions).

Written Preliminary Exam

Upon entry into the Ph.D. program, students are required to take the Written Preliminary Exam, typically scheduled the week before classes start in the Fall semester. The coverage of the exam is in Linear Algebra, Advanced Calculus, Complex Variables, and Probability at the undergraduate level. Details of the exam can be found here: Preliminary Exam Details

The student must pass the exam to continue as a Ph.D. student. The Written Exam is offered in April and August. If the student fails on the first attempt, two more attempts are granted (three attempts total).

Course Requirements

The student must take the following six core courses:

• Analysis: AMCS 6081/6091 (MATH 6080/6090)
• Numerical Analysis: AMCS 6025/6035
• Probability and Stochastic Processes: AMCS 6481/6491 (MATH 6480/6490)

These six core courses are to be completed in the first and second years of graduate studies.

Ten elective courses (a total of 14 courses) are required for graduation. These elective courses should be chosen according to the interests and/or research program of the student and must contain significant mathematical content. Whether a given course can be counted toward AMCS elective course credit will be decided in consultation with the Graduate Chair. Recent courses approved for elective credit can be discussed with your advisor and the Graduate Group Chair.

Deviations from the above may be necessary or recommended depending on the individual student; such decisions are made with the approval of the graduate chair.

Choosing an Advisor

In the first two years of graduate studies, students must choose their thesis advisor. Some students already have an advisor to whom they have committed upon entry to the program. Other students will typically start working with their prospective advisors in the latter half of the first year or the summer between the first and second year.

The purpose of the oral exam is to assess a student’s readiness to transition into full-time research and eventually write his or her dissertation. This exam will be taken by the end of the third year of graduate study.

First, an oral exam committee must be formed, consisting of three faculty members, two of whom must belong to the AMCS graduate faculty. The student must then produce a document of up to about 20 pages describing the research proposal and background material, which is then approved by the oral exam committee before the exam. In the exam, the student will give an oral presentation to the committee. A discussion with the committee follows this. In the oral exam, the committee may ask the student about the presentation as well as about necessary background material as seen fit by the committee. If the student fails this exam, the student will have one more attempt.

Dissertation and Defense

The dissertation must be a substantial original investigation in the field of applied mathematics and computational science, done under the supervision of a faculty advisor. A Ph.D. Thesis Committee consists of at least three faculty members, including the thesis advisor. When the dissertation is complete, it must be defended in a Dissertation Exam, at which the student will be expected to give a short public exposition of the results of the thesis and to satisfactorily answer questions about the thesis and related areas.

Teaching Assistant

Full-time students admitted to our Ph.D. program who are offered a financial support package for four years of study are required to be teaching assistants during the second year. Students for whom English is not their native language are required to pass a test the “Speak Test” (IELTS) demonstrating proficiency in English. More information can be found on the English Language Programs  web page.

https://www.elp.upenn.edu/institute-academic-studies/requirements

The department offers programs covering a broad range of topics leading to the Doctor of Philosophy and the Doctor of Science degrees (the student chooses which to receive; they are functionally equivalent). Candidates are admitted to either the Pure or Applied Mathematics programs but are free to pursue interests in both groups. Of the roughly 125 Ph.D. students, about 2/3 are in Pure Mathematics, 1/3 in Applied Mathematics.

The two programs in Pure and Applied Mathematics offer basic and advanced classes in analysis, algebra, geometry, Lie theory, logic, number theory, probability, statistics, topology, astrophysics, combinatorics, fluid dynamics, numerical analysis, mathematics of data, and the theory of computation. In addition, many mathematically-oriented courses are offered by other departments. Students in Applied Mathematics are especially encouraged to take courses in engineering and scientific subjects related to their research.

All students pursue research under the supervision of the faculty , and are encouraged to take advantage of the many seminars and colloquia at MIT and in the Boston area.

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Find Graduate Programs in the Mathematical Sciences offers comparative information on graduate programs in the mathematical sciences for prospective graduate students and their advisers. This web service provides only an overview of the programs offered; departments should be contacted directly for more detailed information. Currently 276 graduate programs are listed.

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In the wake of the pandemic, many mathematical sciences departments have made at least short-term changes in their requirements regarding the GRE for individuals applying for admission to graduate programs. The Google doc USA/Canada Math PhD Programs: GRE requirements and Qualifying Exams , curated by Brown University graduate student Emily Winn, provides a list of those current requirements.

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Academics   /   Graduate Study PhD in Applied Mathematics

The PhD program in applied mathematics is designed for students who want to pursue research in applied math at the highest level. The PhD program prepares students for independent research and emphasizes flexibility in adapting research areas to current opportunities. PhD students can obtain a master's degree after successful completion of the first year of the PhD program.

The applied mathematics faculty is involved in research that uses a variety of mathematical methods associated with asymptotic analysis, bifurcation theory, graph theory, scientific computing, ordinary and partial differential equations, probability and statistics, singular perturbations, stability theory, and stochastic processes.

These methods are employed in the investigation of problems arising in areas
 like biomechanics, combustion theory, complex networks, diffusion processes, fluid mechanics, geophysics, interfacial phenomena, materials science, molecular biology, neurobiology, reactor theory, solid mechanics, statistical mechanics, transport theory, and wave phenomena.

Learn more about the department's research areas

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Download a PDF program guide about your program of interest and get in contact with our graduate admissions staff.

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In early spring the department invites admitted PhD applicants to visit the department, usually for a departmental open house. For more information please see Addtional Information .

First-year PhD students are typically supported by university fellowships, which pay tuition costs and additionally provide a generous monthly stipend to cover living expenses. These fellowships carry no assigned duties, so students can direct all their efforts toward their studies.

After the first year, students are supported by research assistantships, teaching assistantships, or other special fellowships.

Career Paths

A degree in applied math opens the doors to many possibilities. Our graduates become professors at major research universities, teach at liberal arts colleges, get research positions in national labs and agencies, such as the NSA, and work in industry.

Graduates who want to get academic positions at major research universities typically have to do research as postdoctoral fellows first, usually for two or three years. Five graduates of our program have served as department chairs of their respective departments. These include the applied math department at the University of Washington (Nathan Kutz) and the math department at the University of British Columbia (Rachel Kuske). More detailed information about many PhD graduates can be found at Alumni .

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Department of Engineering Sciences and Applied Mathematics M c Cormick School of Engineering and Applied Science 2145 Sheridan Road, Room M426 Evanston, IL 60208 Phone: 847-491-3345 Fax: 847-491-2178 Email department

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2023 Virtual Information Session

We will hold a virtual information session on Zoom for prospective graduate students on Wednesday, November 8, from 3 to 4 PM (Central Time).

2024 Best Applied Mathematics Schools

Featured applied mathematics programs, choosing a great applied mathematics school, applied mathematics rankings by degree level, best schools for applied mathematics in the united states, 20 top schools in applied math.

There were roughly 262 applied mathematics students who graduated with this degree at UChicago in the most recent year we have data available. Degree recipients from the applied mathematics major at University of Chicago make $47,274 above the typical college grad with the same degree when they enter the workforce.

There were approximately 194 applied mathematics students who graduated with this degree at Brown in the most recent year we have data available. Applied Mathematics degree recipients from Brown University get an earnings boost of approximately $45,603 over the typical earnings of applied mathematics majors.

Put mathematical concepts to work to solve today's most complex real-world problems by studying applied mathematics with this specialized online bachelor's from Southern New Hampshire University.

Related Programs

There were about 364 applied mathematics students who graduated with this degree at WUSTL in the most recent data year.

There were about 152 applied mathematics students who graduated with this degree at Johns Hopkins in the most recent data year. Applied Mathematics degree recipients from Johns Hopkins University receive an earnings boost of about $45,889 above the average earnings of applied mathematics majors.

There were roughly 107 applied mathematics students who graduated with this degree at Harvard in the most recent data year. Those applied mathematics students who get their degree from Harvard University earn $42,778 more than the standard applied math grad.

There were about 144 applied mathematics students who graduated with this degree at MIT in the most recent data year.

There were roughly 101 applied mathematics students who graduated with this degree at Notre Dame in the most recent data year.

There were approximately 27 applied mathematics students who graduated with this degree at Carnegie Mellon in the most recent year we have data available.

There were approximately 295 applied mathematics students who graduated with this degree at USC in the most recent year we have data available. Students who graduate with their degree from the applied math program state that they receive average early career wages of $61,861.

There were approximately 50 applied mathematics students who graduated with this degree at Northwestern in the most recent year we have data available. Degree recipients from the applied mathematics program at Northwestern University make $20,852 more than the typical college graduate with the same degree shortly after graduation.

There were roughly 142 applied mathematics students who graduated with this degree at Vanderbilt in the most recent year we have data available. Degree recipients from the applied mathematics major at Vanderbilt University earn $29,324 above the standard college graduate with the same degree when they enter the workforce.

There were approximately 347 applied mathematics students who graduated with this degree at UCLA in the most recent data year. Students who receive their degree from the applied math program earn about $51,707 in their early career salary.

There were roughly 185 applied mathematics students who graduated with this degree at NYU in the most recent year we have data available.

There were approximately 85 applied mathematics students who graduated with this degree at SMU in the most recent year we have data available.

There were roughly 23 applied mathematics students who graduated with this degree at Cornell in the most recent year we have data available.

There were approximately 339 applied mathematics students who graduated with this degree at UC Berkeley in the most recent data year. Those applied mathematics students who get their degree from University of California - Berkeley receive $11,353 more than the standard applied math student.

There were approximately 90 applied mathematics students who graduated with this degree at Columbia in the most recent year we have data available. Applied Mathematics degree recipients from Columbia University in the City of New York get an earnings boost of about $11,798 over the typical income of applied mathematics graduates.

There were about 255 applied mathematics students who graduated with this degree at UW Seattle in the most recent data year. Applied Mathematics degree recipients from University of Washington - Seattle Campus get an earnings boost of around $23,954 over the average earnings of applied mathematics graduates.

Those applied mathematics students who get their degree from Virginia Tech make $16,484 more than the typical applied math student.

There were approximately 169 applied mathematics students who graduated with this degree at U Miami in the most recent year we have data available.

Rest of the Top Best Applied Mathematics Schools

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Applied Mathematics by Region

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Bachelor's degrees in applied math, doctor's degrees in applied math, highest paid grads in applied mathematics, best for veterans in applied mathematics, most popular in applied mathematics, most focused in applied mathematics, master's degrees in applied math, best value in applied mathematics, best for non-traditional students in applied mathematics, best online in applied mathematics, most popular online in applied mathematics, applied mathematics related rankings by major, applied math concentrations.

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Notes and References

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7 Best universities for Applied Mathematics in Saint Petersburg

Updated: February 29, 2024

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Below is a list of best universities in Saint Petersburg ranked based on their research performance in Applied Mathematics. A graph of 28.8K citations received by 4.12K academic papers made by 7 universities in Saint Petersburg was used to calculate publications' ratings, which then were adjusted for release dates and added to final scores.

We don't distinguish between undergraduate and graduate programs nor do we adjust for current majors offered. You can find information about granted degrees on a university page but always double-check with the university website.

1. St. Petersburg State University

For Applied Mathematics

2. Peter the Great St.Petersburg Polytechnic University

3. ITMO University

4. Saint Petersburg State Electrotechnical University

5. Leningrad State University

6. St. Petersburg State University of Architecture and Civil Engineering

7. St. Petersburg State University of Aerospace Instrumentation

Universities for Applied Mathematics near Saint Petersburg

Mathematics subfields in Saint Petersburg

Applied Mathematics

Undergraduate Program

The Applied Mathematics concentration consists of a broad undergraduate education in the mathematical sciences, especially in those subjects that have proved vital to an understanding of problems arising in other disciplines, and in some specific area where mathematical methods have been substantively applied. For concentrators, a core learning objective is building and demonstrating foundational knowledge in computation, probability, discrete, and continuous mathematics through the successful completion of the foundation and breadth courses. Students are also eligible to apply for an A.B./S.M. degree program.

Harvard School of Engineering offers a Doctor of Philosophy (Ph.D.) degree in Applied Mathematics. Doctoral students may earn the masters degree en route to the Ph.D. Students are drawn to Applied Mathematics by the flexibility it offers in learning about how to apply mathematical ideas to problems drawn from different fields, while remaining anchored to empirical data that drive these questions. Research and educational activities have particularly close links to Harvard’s efforts in Mathematics, Economics, Computer Science, and Statistics. Graduates go on to a range of careers in industry, academics, to professional schools in business, law, medicine, among others. All Ph.D.s are awarded through the Harvard Graduate School of Arts and Sciences.

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Learn about our programs.

Our graduate mathematics program is one of the largest in the country — and one of the best. The Department of Mathematics offers programs leading to the degrees of Master of Science and Doctor of Philosophy. Recent changes in the graduate program were aimed at improving opportunities for our students to develop in areas that suit their goals. We invite prospective students to visit to meet faculty and current students and to see first hand what Purdue has to offer.

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