phd in applied mathematics

PhD in Applied Mathematics

Phd in applied mathematics degree.

Applied Mathematics at the Harvard John A. Paulson School of Engineering is an interdisciplinary field that focuses on the creation and imaginative use of mathematical concepts to pose and solve problems over the entire gamut of the physical and biomedical sciences and engineering, and increasingly, the social sciences and humanities. The program has focuses on understanding nature through the fusion of Artificial Intelligence, Computing (classical to quantum), and Mathematics. We value foundational contributions, societal impact, and ethics in our work. Our program uniquely interfaces with diverse fields, including physics, neuroscience, materials science, economics, biology and fluid mechanics, to tackle some of the most pressing challenges of our time, such as sustainability, responsible digital transformations, and health and well-being.

Working individually and as part of teams collaborating across the University and beyond, you will partner with faculty to quantitatively describe, predict, design and control phenomena in a range of fields. Projects current and past students have worked on include collaborations with mechanical engineers to uncover some of the fundamental properties of artificial muscle fibers for soft robotics and developing new ways to simulate tens of thousands of bubbles in foamy flows for industrial applications such as food and drug production.

Our core mission is to provide students with individualized programs tailored to their interests, needs, and background. We welcome students from diverse technical backgrounds. Our program is dedicated to the principles of diversity, equity, and inclusion. We celebrate and value differences among our members, and we strive to create an equitable and inclusive environment for people of all backgrounds.

APPLY NOW >

Applied Mathematics PhD Degree

Harvard School of Engineering offers a  Doctor of Philosophy (Ph.D.) degree in Applied Mathematics conferred through the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences . Doctoral students may earn the masters degree en route to the Ph.D. Prospective students apply through Harvard Griffin GSAS; in the online application, select  “Engineering and Applied Sciences” as your program choice and select “PhD Applied Math” in the Area of Study menu.

The Applied Mathematics program does not offer an independent Masters Degree.

Applied Mathematics PhD Career Paths

Our graduates have gone on to careers such as start-up pioneers, social innovators, and a range of careers in industry in organizations like the Kingdom of Morocco, Meta, and Bloomberg. Others have secured faculty positions at Dartmouth, Imperial College in London, and UCLA. More generally, students with a PhD in Applied Mathematics can go on to careers in academia, banking, data science, bioinformatics, management consulting, government/military research, and more. Also, r ead about some of our Applied Mathematics alumni .

Admissions & Academic Requirements

Please review the  admissions requirements and other information  before applying. Our website also provides  admissions guidance ,   program-specific requirements , and a PhD program academic timeline .

Academic Background

Applicants typically have bachelor’s degrees in the natural sciences, mathematics, computer science, or engineering. 

Standardized Tests

GRE General: Not Accepted

Applied Mathematics Faculty & Research Areas

View a list of our  Applied Mathematics faculty and applied mathematics  affiliated research areas , Please note that faculty members listed as “Affiliates" or "Lecturers" cannot serve as the primary research advisor.  

Applied Mathematics Centers & Initiatives

View a list of the research centers & initiatives at SEAS and the Applied Mathematics faculty engagement with these entities .

Graduate Student Clubs

Graduate student clubs and organizations bring students together to share topics of mutual interest. These clubs often serve as an important adjunct to course work by sponsoring social events and lectures. Graduate student clubs are supported by the Harvard Kenneth C. Griffin School of Arts and Sciences. Explore the list of active clubs and organizations .

Funding and Scholarship

Learn more about financial support for PhD students.

• How to Apply

Learn more about how to apply  or review frequently asked questions for prospective graduate students.

In Applied Mathematics

• First-Year Exploration
• Areas of Application
• AM & Economics
• How to Declare
• Who are my Advisors?
• Secondary Field
• Senior Thesis
• Research for Course Credit (AM 91R & AM 99R)
• AB/SM Information
• Peer Concentration Advisors (PCA) Program
• Student Organizations
• PhD Timeline
• PhD Model Program (Course Guidelines)
• Oral Qualifying Examination
• Committee Meetings
• Committee on Higher Degrees
• Research Interest Comparison
• Collaborations
• Cross-Harvard Engagement
• Clubs & Organizations
• Centers & Initiatives
• Alumni Stories

Graduate News

Master's student capstone spotlight: AI-Enabled Information Extraction for Investment Management

Extracting complicated data from long documents

Academics , AI / Machine Learning , Applied Computation , Computer Science , Industry

Master's student capstone spotlight: AI-Assisted Frontline Negotiation

Speeding up document analysis ahead of negotiations

Academics , AI / Machine Learning , Applied Computation , Computer Science

Master's student capstone spotlight: A Remote Sensing Framework for Rail Incident Situational Awareness Drones

Using drones to rapidly assess disaster sites

• Applied Mathematics (Ph.D.)

Applied mathematics addresses problems in science, engineering, and society. Find new ways to solve real-world problems through original, creative research in Illinois Tech’s applied mathematics Ph.D. program.

• Academic Programs

Illinois Tech’s Ph.D. program in Applied Mathematics is a flagship graduate program that prepares talented mathematicians and statisticians for careers in research or academia through a rigorous education, which includes advanced coursework, independent study, and original research. With almost 100 percent job placement at graduation, our alumni work at Goldman Sacks, UBS, Amazon, University of Michigan, DePaul University, United Airlines, Grant Thornton, as well as start-ups and early stage companies.

The Department of Applied Mathematics and the College of Computing offer generous scholarships in the form of teaching or research assistantships that cover tuition and provide a competitive monthly stipend. 

Courses cover a wide range of topics in applied mathematics and statistics including mathematical courses offered in popular graduate programs such as Data Science , Financial Technology , and Computational Decision Science and Operations Research . 

The Department of Applied Mathematics is a vibrant research hub with internationally recognized faculty working in a variety of applied research areas such as computational mathematics, stochastic analysis, statistics, data science, applied discrete mathematics, and optimal control. 

In addition to numerous academic activities, the Department of Applied Mathematics is home to several student organizations such as Illinois Tech SIAM Student Chapter, American Statistical Association, Association for Women in Mathematics, Machine Learning @IIT, and Fun Math Problems.

Program Overview

Prepare for a career in industrial research or academia through a rigorous education that includes advanced coursework, independent study, and original research to make a significant contribution to the field of applied mathematics.

Career Opportunities

Career opportunities exist across industries, as so many need mathematics experts.

• Operations researcher/analyst
• Mathematician/statistician
• Post-secondary mathematics/science teacher
• Post-secondary mathematics/science administrator

View Details

Admission Requirements

The program typically requires a bachelor’s degree in mathematics or applied mathematics. Candidates whose degree is in another field and whose background in mathematics is strong are also eligible for admission and are encouraged to apply.

Applicants should have a bachelor’s degree from an accredited university. A cumulative GPA of 3.5/4.0 is usually required.

TOEFL scores, if required, should be a minimum of 80/550 (internet-based/paper-based test scores).

A two-page professional statement of goals/objectives and a curriculum vitae must be submitted.

Three letters of recommendation are required.

All applications are automatically considered for full scholarship in from of Teaching or Research Assistantship, with no additional application process for such funds. The scholarships are awarded to top candidates based on the strength of the entire portfolio, the departmental needs and the availability of funds. Full consideration are given to applications for Fall semesters received before the priority deadline of January 31.

Ask a Professor

What do climate change, finance, data science, sports analytics, engineering, and software development all have in common? They all have foundations in mathematics. Discover how a degree in applied mathematics can open doors to these careers, and many more, by speaking with Professor Igor Cialenco, director of graduate studies at Illinois Tech’s Department of Applied Mathematics. These virtual visits occur on Wednesdays from 3 p.m. to 4 p.m. CST.

Featured Faculty

Igor Cialenco

Jinqiao (Jeffrey) Duan

Sonja Petrović

Fred J. Hickernell

Maggie Cheng

Our alumni hit the ground running

By planning together, the Department of Applied Mathematics and Jeffery Mudrock forged a plan allowing him to work full-time while pursuing a Ph.D. and his career goals.

Learning experiences from a network of collaborators helped Lluís Antoni Jiménez Rugama evaluate mathematics from a variety of perspectives.

After emigrating to the United States, Kabre ran her own business before pursuing a Ph.D. in applied mathematics and a career in academia.

Yicong Huang finds himself incorporating his research experience into his daily work.

With a combination of theoretical and practical research experience, Xiao Huang can’t find a work problem he is unable to solve.

Learn more...

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).

Get to know us: Virtual workshop

Attend talks by faculty and graduate students. Ask questions at the Q&A panel with faculty and students.

Saturday, December 2, 2023

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 not required for the Academic Year,  2023-2024
• 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.

Applied Mathematics

Share this page.

Applied Mathematics is an area of study within the Harvard John A. Paulson School of Engineering and Applied Sciences. Prospective students apply through Harvard Griffin GSAS; in the online application, select  “Engineering and Applied Sciences” as your program choice and select “PhD Applied Math” in the Area of Study menu.

Applied Mathematics at the Harvard John A. Paulson School of Engineering is an interdisciplinary field that focuses on the creation and imaginative use of mathematical concepts to pose and solve problems over the entire gamut of the physical and biomedical sciences and engineering, and increasingly, the social sciences and humanities

Working individually and as part of teams collaborating across the University and beyond, you will partner with faculty to quantitatively describe, predict, design and control phenomena in a range of fields. Projects current and past students have worked on include collaborations with mechanical engineers to uncover some of the fundamental properties of artificial muscle fibers for soft robotics and developing new ways to simulate tens of thousands of bubbles in foamy flows for industrial applications such as food and drug production.

Graduates of the program have gone on to a range of careers in industry in organizations like the Kingdom of Morocco, Meta, and Bloomberg. Others have secured faculty positions at Dartmouth, Imperial College in London, and UCLA.

Standardized Tests

GRE General:  Not accepted

APPLICATION DEADLINE

Questions about the program.

• Admission Policies
• Financial Support
• Ph.D. in Atmosphere Ocean Science
• M.S. at Graduate School of Arts & Science
• M.S. at Tandon School of Engineering
• Current Students

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:

--------------------------------------

(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.

-------------------------------------

( 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 .

Graduate Programs

Applied mathematics.

The graduate program provides training and research activities in a broad spectrum of applied mathematics. The breadth is one of the great strengths of the program and is further reflected in the courses we offer.

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.

Application Information

Application requirements, gre subject:.

Not Required

GRE General:

Dates/deadlines, application deadline, completion requirements.

Eight courses, of which at least six must be at the 2000 level, at least six must be applied mathematics courses, and at least six must be completed with a grade of B or better; preliminary oral examination; two semesters of teaching; dissertation and oral defense.

Alumni Careers

Contact and Location

Division of applied mathematics, mailing address.

• Program Faculty
• Program Handbook
• Graduate School Handbook

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.

Applied Mathematics, PHD

On this page:, at a glance: program details.

• Location: Tempe campus
• Second Language Requirement: No

Program Description

Degree Awarded: PHD Applied Mathematics

This PhD program in applied mathematics is intended for students with superior computational and mathematical modeling ability. It emphasizes a solid mathematical foundation and promotes creative scholarship in an application discipline.

The School of Mathematical and Statistical Sciences has faculty in applied mathematics with outstanding transdisciplinary research programs that have strong external funding. Current research interests include mathematical epidemiology and mathematical ecology, mathematical neuroscience, environmental fluid dynamics and high-performance computing, imaging and inverse problems, supply chain dynamics, control and optimization, computational methods for ordinary and partial differential equations, analysis of differential equations, and geophysical and environmental fluid dynamics.

Degree Requirements

84 credit hours, a written comprehensive exam, a prospectus and a dissertation

Required Core (3 credit hours) APM 505 Applied Linear Algebra (3)

Other Requirements (12 credit hours) APM 501 Differential Equations I (3) APM 502 Differential Equations II (3) APM 503 Applied Analysis (3) APM 504 Applied Probability and Stochastic Processes (3) APM 506 Computational Methods (3)

Electives and Research (57 credit hours)

Culminating Experience (12 credit hours) APM 799 Dissertation (12)

Additional Curriculum Information Students must pass:

• two qualifying examinations
• a written comprehensive examination
• an oral dissertation prospectus defense

Students should see the department website for examination information.

Each student must write a dissertation and defend it orally in front of five dissertation committee members.

Electives are chosen from math or related area courses approved by the student's supervisory committee.

Students choose four out of the five courses listed for other requirements.

Admission Requirements

Applicants must fulfill the requirements of both the Graduate College and The College of Liberal Arts and Sciences.

Applicants are eligible to apply to the program if they have earned a bachelor's or master's degree in mathematics, applied mathematics, economics, engineering or a natural science from a regionally accredited institution.

Applicants must have a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in the last 60 hours of their first bachelor's degree program or a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in an applicable master's degree program

All applicants must submit:

• graduate admission application and application fee
• official transcripts
• statement of education and career goals
• three letters of recommendation
• proof of English proficiency

Additional Application Information An applicant whose native language is not English must provide proof of English proficiency regardless of their current residency.

To demonstrate their competitiveness in an applicant pool, applicants must show evidence of coursework in linear algebra (equivalent to ASU course MAT 342 or MAT 343) and advanced calculus (equivalent to ASU course MAT 371). It is desirable that applicants have scientific programming skills.

Next Steps to attend ASU

Learn about our programs, apply to a program, visit our campus, application deadlines, learning outcomes.

• Able to complete original research in applied mathematics.
• Apply concepts and skills from applied mathematics to conduct original research.
• Apply advanced computational methods in their coursework and research.

Career Opportunities

Foundational knowledge in mathematics is required for building careers in science and technology. It can be applied in many different types of professions in fields that include engineering, life sciences, business, and economic and social sciences. These are just a few of the top careers possible with a doctorate in applied mathematics:

• biostatistician
• data scientist
• financial analyst
• government and military researcher
• industrial researcher
• mathematical modeling expert
• mathematician
• medical researcher
• operations research analyst
• university instructor and faculty member

Program Contact Information

If you have questions related to admission, please click here to request information and an admission specialist will reach out to you directly. For questions regarding faculty or courses, please use the contact information below.

• [email protected]
• 480/965-3951

Ph.D. Program

Degree requirements.

In outline, to earn the PhD in either Mathematics or Applied Mathematics, the candidate must meet the following requirements.

• Take at least 4 courses, 2 or more of which are graduate courses offered by the Department of Mathematics
• Pass the six-hour written Preliminary Examination covering calculus, real analysis, complex analysis, linear algebra, and abstract algebra; students must pass the prelim before the start of their second year in the program (within three semesters of starting the program)
• Pass a three-hour, oral Qualifying Examination emphasizing, but not exclusively restricted to, the area of specialization. The Qualifying Examination must be attempted within two years of entering the program
• Complete a seminar, giving a talk of at least one-hour duration
• Write a dissertation embodying the results of original research and acceptable to a properly constituted dissertation committee
• Meet the University residence requirement of two years or four semesters

Detailed Regulations

The detailed regulations of the Ph.D. program are the following:

Course Requirements

During the first year of the Ph.D. program, the student must enroll in at least 4 courses. At least 2 of these must be graduate courses offered by the Department of Mathematics. Exceptions can be granted by the Vice-Chair for Graduate Studies.

Preliminary Examination

The Preliminary Examination consists of 6 hours (total) of written work given over a two-day period (3 hours/day). Exam questions are given in calculus, real analysis, complex analysis, linear algebra, and abstract algebra. The Preliminary Examination is offered twice a year during the first week of the fall and spring semesters.

Qualifying Examination

To arrange the Qualifying Examination, a student must first settle on an area of concentration, and a prospective Dissertation Advisor (Dissertation Chair), someone who agrees to supervise the dissertation if the examination is passed. With the aid of the prospective advisor, the student forms an examination committee of 4 members.  All committee members can be faculty in the Mathematics Department and the chair must be in the Mathematics Department. The QE chair and Dissertation Chair cannot be the same person; therefore, t he Math member least likely to serve as the dissertation advisor should be selected as chair of the qualifying exam committee . The syllabus of the examination is to be worked out jointly by the committee and the student, but before final approval, it is to be circulated to all faculty members of the appropriate research sections. The Qualifying Examination must cover material falling in at least 3 subject areas and these must be listed on the application to take the examination. Moreover, the material covered must fall within more than one section of the department. Sample syllabi can be reviewed online or in 910 Evans Hall. The student must attempt the Qualifying Examination within twenty-five months of entering the PhD program. If a student does not pass on the first attempt, then, on the recommendation of the student's examining committee, and subject to the approval of the Graduate Division, the student may repeat the examination once. The examining committee must be the same, and the re-examination must be held within thirty months of the student's entrance into the PhD program. For a student to pass the Qualifying Examination, at least one identified member of the subject area group must be willing to accept the candidate as a dissertation student.

College of Arts & Sciences

PhD Program in Applied Mathematics

Phd program.

Applying to PhD Program

PhD Degree Requirements

PhD Coursework

PhD Institutional Funding

Applied Mathematics Graduate Seminar

Program Contact

The PhD program in Applied Mathematics prepares students for careers in academia, industry, or government. Students will receive expert guidance in research and a foundation in teaching pedagogy.

• Menu  Close 
• Search 

PhD in Applied Mathematics and Statistics

Create knowledge at the nation’s leading research institution. Our doctoral program in applied mathematics and statistics prepares you for leadership, no matter what professional path you choose.

With an emphasis on mathematical reasoning, mathematical modeling and computation, interdisciplinary, and the development of new methodology, the department’s doctoral program produces graduates with broad expertise, who have worked in cutting-edge advances in applied mathematics and statistics, and who are prepared to pursue careers in academia, industry, the public sector, and more.

Research and academic opportunities in the doctoral program cover a wide range of interconnected areas within applied mathematics that reflect the department faculty’s diverse scientific interests. These include:

• Data Science with a focus on theoretical foundations, modeling techniques, and algorithm development.
• Probability and statistics with a focus both on theory and applications, particularly related to machine learning.
• Financial mathematics , as well as a concentration in optimization/operations research, which address discrete and continuous optimization problems.
• Discrete mathematics , with a focus on combinatorics and graph theory.
• Computational and applied mathematics , with a focus on the development of mathematical and numerical models for various applications in fluid dynamics, shape analysis and biomedical engineering.

Ph.D. Student Handbook

Explore your options

Pursue a PhD with a focus on computational medicine

Doctoral candidates can focus their studies on computational medicine, through a partnership with JHU’s Institute for Computational Medicine, a collaboration between JHU’s School of Engineering and the renowned School of Medicine.

A commitment to discovery

Connecting the Dots - Carey Priebe has worked with hospitals hoping to reduce stress on surgical nurses, shown how satellites can spot Martian volcanoes, and given law enforcement new tools against human traffickers, shadowy financial networks, and corporate fraud.

NSF Program Support Career-readiness for Graduate Students - To help prepare graduate students in the mathematical sciences for careers in business, industry, and government, the Johns Hopkins Engineering and the University of Illinois at Urbana-Champaign are partnering to create the Internship Network in the Mathematical Sciences (INMAS).

Skip to Content

University of Colorado Denver

• Campus Directory
• Events Calendar
• Human Resources
• Student Services
• Auraria Library
• CU Denver Police
• University Policies

Schools and Colleges

• College of Architecture and Planning
• College of Arts & Media
• Business School
• School of Education & Human Development
• College of Engineering, Design and Computing
• Graduate School
• College of Liberal Arts and Sciences
• School of Public Affairs

Campus Affiliates

• CU Anschutz Medical Campus
• CU Colorado Springs

Other ways to search:

• University Directory

PhD in Applied Mathematics

Our PhD in Applied Mathematics program provides comprehensive training in applied mathematics and/or statistics and opportunities for cutting-edge research in close collaboration with internationally recognized scholars in the fields of

• Computational Mathematics
• Discrete Mathematics
• Optimization and Operations Research
• Probability

Some highlights of our exciting research projects include evolutionary dynamics, climate modeling, wildfire simulations, machine learning, genetic inheritance and association, optimization in data analysis, and more. Current research funding includes grants from NSF, NIH, DoD, and NASA.

The degree is designed to give students a contemporary, comprehensive education in subjects such as high-performance computing, numerical analysis, optimization, statistical methods, and operations research. In all of its activities, the department embodies the outlook that mathematics, statistics, computing, and data science are powerful tools that can be used to solve problems of immediate and practical importance. Our program emphasizes the training of skills valued by many employers. These skills include problem solving, critical thinking, analysis, facility with data, the ability to process quantitative information, and most important of all, the ability to learn and master new skills and concepts quickly. These strengths make our students highly marketable for careers in industry as well as in academia.  Scholarships and assistantships​  for graduate students are available and awarded competitively.​​

Ph.D. Program Quick Links

• Program Goals & Objectives
• Admissions for Ph.D. in Applied Mathematics
• Graduate Student Financial Resources
• Degree Requirements
• ​Linear Algebra Syllabus
• Previous Linear Algebra Exams
• Applied Analysis Syllabus
• Previous Applied Analysis Exams
• Current Student Resources
• Request Info/Contact

Graduate Program Director

Florian Pfender [email protected]

Director of Statistical Programs

Yaning Liu [email protected]

General Inquiries

Miriam Venzor Program Assistant [email protected] Phone: 303-315-1702

• Website Feedback
• Privacy Policy
• Legal Notices
• Accreditation

© 2021  The Regents of the University of Colorado , a body corporate. All rights reserved.

Accredited by the Higher Learning Commission . All trademarks are registered property of the University. Used by permission only.

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.

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

Applied Mathematics, PhD

• Program description
• At a glance
• Degree requirements
• Admission requirements
• Tuition information
• Application deadlines
• Program learning outcomes
• Career opportunities
• Contact information

Computational, Computational Mathematics, Mathematical Models, Mathematical modeling, Mathematician, Mathematics, Mathematics Science, Performance, applied, approved for STEM-OPT extension, computing, dynamics, math, systems

Are you passionate about making a difference in the world of climate or environmental issues, or advancing the fight against cancer or infectious diseases? You can apply your enhanced mathematics expertise to devising solutions to these types of current problems.

This PhD program in applied mathematics is intended for students with superior computational and mathematical modeling ability. It emphasizes a solid mathematical foundation and promotes creative scholarship in an application discipline.

The School of Mathematical and Statistical Sciences has faculty in applied mathematics with outstanding transdisciplinary research programs that have strong external funding. Current research interests include mathematical epidemiology and mathematical ecology, mathematical neuroscience, environmental fluid dynamics and high-performance computing, imaging and inverse problems, supply chain dynamics, control and optimization, computational methods for ordinary and partial differential equations, analysis of differential equations, and geophysical and environmental fluid dynamics.

This program may be eligible for an Optional Practical Training extension for up to 24 months. This OPT work authorization term may help international students gain skills and experience in the U.S. Those interested in an OPT extension should review ASU degrees that qualify for the STEM-OPT extension at ASU's International Students and Scholars Center website.

The OPT extension only applies to students on an F-1 visa and does not apply to students completing a degree through ASU Online.

• College/school: The College of Liberal Arts and Sciences
• Location: Tempe

84 credit hours, a written comprehensive exam, a prospectus and a dissertation

Required Core (3 credit hours) APM 505 Applied Linear Algebra (3)

Other Requirements (12 credit hours) APM 501 Differential Equations I (3) APM 502 Differential Equations II (3) APM 503 Applied Analysis (3) APM 504 Applied Probability and Stochastic Processes (3) APM 506 Computational Methods (3)

Electives and Research (57 credit hours)

Culminating Experience (12 credit hours) APM 799 Dissertation (12)

Additional Curriculum Information Students must pass:

• two qualifying examinations
• a written comprehensive examination
• an oral dissertation prospectus defense

Students should see the department website for examination information.

Each student must write a dissertation and defend it orally in front of five dissertation committee members.

Electives are chosen from math or related area courses approved by the student's supervisory committee.

Students choose four out of the five courses listed for other requirements.

Applicants must fulfill the requirements of both the Graduate College and The College of Liberal Arts and Sciences.

Applicants are eligible to apply to the program if they have earned a bachelor's or master's degree in mathematics, applied mathematics, economics, engineering or a natural science from a regionally accredited institution.

Applicants must have a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in the last 60 hours of their first bachelor's degree program or a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in an applicable master's degree program

All applicants must submit:

• graduate admission application and application fee
• official transcripts
• statement of education and career goals
• three letters of recommendation
• proof of English proficiency

Additional Application Information An applicant whose native language is not English must provide proof of English proficiency regardless of their current residency.

To demonstrate their competitiveness in an applicant pool, applicants must show evidence of coursework in linear algebra (equivalent to ASU course MAT 342 or MAT 343) and advanced calculus (equivalent to ASU course MAT 371). It is desirable that applicants have scientific programming skills.

Program learning outcomes identify what a student will learn or be able to do upon completion of their program. This program has the following program outcomes:

• Apply concepts and skills from applied mathematics to conduct original research.
• Able to complete original research in applied mathematics.
• Apply advanced computational methods in their coursework and research.

Foundational knowledge in mathematics is required for building careers in science and technology. It can be applied in many different types of professions in fields that include engineering, life sciences, business, and economic and social sciences. These are just a few of the top careers possible with a doctorate in applied mathematics:

• biostatistician
• data scientist
• financial analyst
• government and military researcher
• industrial researcher
• mathematical modeling expert
• mathematician
• medical researcher
• operations research analyst
• university instructor and faculty member

School of Mathematical and Statistical Sciences | WXLR A213 [email protected] 480-965-3951

PhD in Mathematics

The PhD in Mathematics consists of preliminary coursework and study, qualifying exams, a candidacy exam with an adviser, and creative research culminating in a written dissertation and defense. All doctoral students must also do some teaching on the way to the PhD. There are minimal course requirements, and detailed requirements and procedures for the PhD program are outlined in the  PhD Handbook .

Please note that our department alternates recruiting in-coming classes that are focused on either applied or pure mathematics. For the Fall 2024 admissions (matriculation in September 2024), we are focusing on students interested in areas of applied mathematics.

All our professors are active in research, and are devoted to teaching and mentoring of students. Thus, there are many opportunities to be involved in cutting-edge research in pure and applied mathematics. Moreover, the seven other research universities in the Boston area are all within easy reach, providing access to many more classes, seminars and colloquia in diverse areas of mathematical research.

Teaching assistantships are available for incoming PhD students, as well as a limited number of University-wide fellowships. Tufts has on-campus housing for graduate students, but many choose to live off-campus instead.

In addition to the above, PhD students often:

• Mentor undergraduates as teaching assistants and course instructors, and through graduate-student run programs like the Directed Reading Program.
• Meet with advisors and fellow students to share research and collaborate with scholars across disciplines
• Attend professional development workshops and present research at conferences

Department of Mathematics

Mathematics phd program.

The Ph.D. program in the Department of Mathematics provides students with in-depth knowledge and rigorous training in all the subject areas of mathematics. A core feature is the first-year program, which helps bring students to the forefront of modern mathematics. Students work closely with faculty and each other and participate fully in both research and student-run seminars.

Questions? Email [email protected]

• The firm deadline for applications for Autumn 2025, is December 5, 2024.
• The (general and advanced) GRE tests are no longer accepted. Please do not submit these scores.

Math Graduate Program Admissions

The application process... .

is  entirely electronic  through the UA Graduate College. Completion of the application requires payment of the application fee to the Graduate College. If you are affiliated with a federally sponsored program serving under-represented students, you are eligible for a  Graduate College Application Fee Waiver . 

Application Deadline: January 1, 2024

*NOTE*  Applications can be accepted after the deadline. Please contact the graduate office to inquire after the deadline.

Admission Requirements

To be admitted, applicants are expected to have completed the equivalent of an undergraduate major in Mathematics, with a minimum of one semester each of upper-division or higher level work, including one semester each of advanced analysis at the level of  MATH 425A , abstract algebra at the level of  MATH 415A , and linear algebra at the level of  MATH 413 . A full year of Abstract Algebra, Real Analysis, and a semester of point set topology are preferred, and graduate coursework is valued.

PhD applicants should also have completed a semester of complex variables at the level of  MATH 424 . GRE scores are not required, but applicants are welcome to include them if they believe they add to their application.

International Students

To be considered for admisson....

International applicants whose native language is not English must submit their scores from either the TOEFL (Test of English as a Foreign Language) or IELTS (International English Language Testing System) exams. For the TOEFL exam the minimum score to be considered for admission is 550 for the paper based (PB) TOEFL or 79 for the internet based (iBT) TOEFL. For the IELTS exam the minimum score to be considered for admission is a composite score of 7 with no subject area below a 6.

To be considered for financial support...

To be considered for financial support, international applicants whose native language is not English must submit scores from either the iBT TOEFL exam or the IELTS exam. (Applicants cannot be considered for financial support on the basis of the paper based TOEFL exam since it does not test speaking ability.) In addition to the requirements above, international students must have a minimum score of 24 on the speaking section of the iBT TOEFL or a minimum score of 7.5 on the speaking component of the IELTS. In addition to these minimum scores, phone or Skype interviews of applicants are usually required as well as further evaluation of the applicants English once he or she has arrived on campus.

Application Procedure

On the graduate online application, you will be required to submit online the following items. Prior research experience is also valuable, but not required.

Three (3) Letters of Recommendation

You should arrange for at least three letters of recommendation. The letters must be  dated  and may not be more than a year old.

Statement of Purpose

Your statement of purpose should describe your interests in mathematics generally, and in The University of Arizona in particular. If you have done a research project of some kind, you should include some description of it in your statement.

Transcripts & Degree Certificates

Both domestic and international applicants must submit transcripts from all previous colleges and universities attended. Unofficial transcripts are accepted for the initial application. Official transcripts will be required upon admittance and the applicant will be notified by the Graduate College.

Description of Math-Related Coursework

This list should include all your mathematics coursework at the junior level or higher. For each course please include the course number as it appears on your transcript, the topics covered and the textbook used. If the course is a graduate course, you should indicate this.

TOEFL or IELTS Score Report (International Applications)

International applicants whose native language is not English must also submit their scores from either the TOEFL (Test of English as a Foreign Language) or IELTS (International English Language Testing System) exams. The minimum requirements for the scores can be found under the "International Students" section of this page. The test date must be within two years of the intended term of enrollment to be considered valid.  You will be required to either upload a copy of your TOEFL (or IELTS) score report during the online application process or arrange to have ETS (Educational Testing Service) send your scores to us electronically.  (School code: 4832; Department code: 0703)

GRE Score Report

You may take both the  GRE  and the GRE Subject Test in Mathematics, however they are not required.   If you have chosen to take the exam you may either upload a copy of your GRE score report during the online application process or arrange to have  ETS  (Educational Testing Service) send your scores to us electronically.  (School code: 4832; Department code: 0703)

Our Support Center is a hub of quick information for everyone. Whether you're a current student, future student, Mathematics Department member, donor, or a curious member of UArizona, you can find the information you're looking for below!

Support Center

• Colleges & Degrees
• Academic Calendar
• International Education
• Graduate Studies
• Accreditation
• Tuition and Fees
• Parking & Maps
• Careers with CSULB
• Alumni Home
• Alumni Volunteering
• Alumni Giving

Campus Life

• Centers & Organizations
• Commencement
• Student Life
• Office of the President
• Office of the Provost
• Administration & Finance
• Student Affairs
• University Relations & Development
• Information Technology
• Beach Shops
• Campus Directory
• Enrollment Services
• Financial Aid
• Schedule of Classes
• Student Records
• 49er Foundation
• Research Foundation

1250 BELLFLOWER BOULEVARD LONG BEACH, CALIFORNIA 90840 562.985.4111

Computational and Applied Math Graduate Program

The Master of Science in Computational and Applied Mathematics provides a thorough program in the theory and use of advanced computational and applied mathematics that will prepare students to successfully model and solve mathematical and computational problems arising in the real world.

More information can be found in M.S. in Computational and Applied Mathematics - CSULB Catalog (2024-2025) .

There are ten full-time faculty members who teach courses in this program. The course titles are:

• Introduction to Numerical Analysis
• Intermediate Numerical Analysis
• Introduction to Partial Differential Equations
• Fourier Analysis
• Mathematics of Financial Derivatives
• Mathematical Modeling
• Mathematical Optimization
• Finite Element Method
• Matrix Method in Data Analysis and Pattern Recognition
• Applied Analysis
• Applied Nonlinear Ordinary Differential Equations
• Partial Differential Equations
• Advanced Scientific Computing
• Stochastic Calculus and Applications
• Calculus of Variations
• Numerical Analysis
• Numerical Solution of Partial Differential Equations
• Numerical Linear Algebra
• Advanced Mathematical Modeling

Degree Plan

Graduate students have the following options:

Thesis Option

Comprehensive exams option.

For the Thesis Option you need your thesis proposal to be approved by the Computational and Applied Mathematics Committee using the following procedure:

• Find your thesis advisor amongst Applied Math faculty
• Committee (three faculty members including your thesis advisor and at least one additional applied math faculty member)
• Objective: formulate your thesis problem in rigorous mathematical terms and state your goals.
• Methodologies: state mathematical methods that you will be implementing/analyzing in order to achieve your objective.
• Submit your approved thesis proposal along with your graduate progress report and file your advancement with grad advisor.

For the Comprehensive Exams Option you need to decide which (Numerical Analysis, PDEs, and ODEs) of two subjects you will be taking and take those courses prior to taking the exams. See Comprehensive Exam Preparation for more information.