online phd in applied mathematics

Applied Mathematics

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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

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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 (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

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...

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 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 11, 2023

Degree Type(s)

Doctoral / PhD

Degree Awarded

GRE Requirements

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.

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.

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

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

Committee on Computational and Applied Mathematics

Phd program, academic progress and milestones.

Course Requirements

First-year students are required to register for three courses per quarter for a total of nine (9) graduate courses during their first year in the program. They are required to follow the analytical sequence: 

        Applied Dynamical Systems ( CAAM 31410 ),

                 Applied Functional Analysis ( CAAM 31440 ),

        Partial Differential Equations ( CAAM 31220 ),

as well as the computational sequence:

                    Mathematical Computation I: Matrix Computation ( CAAM 30900 ),

                    Mathematical Computation II: Optimization ( CAAM   31020  or  CAAM 31015 ),

                     Applied Approximation Theory ( CAAM 31050 ).

They should receive a grade of B or above in each course and have an average of B+ or above in each sequence.  The remaining three courses may be chosen freely from CAM-related graduate programs at The University of Chicago. Approval for the three electives is required from the first-year PhD student advisor.

Graduate students are required to complete a third sequence during their first three years in the program. The sequence is composed of

                       Machine Learning ( CAAM 37710 ) 

or equivalent and two classes from the following list:

                       Measure Theoretic Probability (STAT 381, STAT 383, STAT 385),

                       Data Assimilation and Inverse Problems,

                       Variational Methods in Image Processing,

                       Monte Carlo Simulations,

                       Numerical PDE,

                       Fast Algorithms,

                       Algorithms for Massive Datasets,

                       Computational Neuroscience (CAAM 42610),

                       Stochastic processes in gene regulation (CAAM 35420).

Students who do not complete these requirements as noted above may be placed on academic probation.

Graduate students need to complete at least twelve (12) CAM-related graduate courses to graduate.

Qualifying Exams

Graduate students take written qualifying exams at the end of their first year, typically in the month of June. They need to take two out of three exams from the analytical sequence and two out of three exams from the computational sequence.

Students failing all four exams may be dismissed from the program at the end of their first year.

Students failing between one and three exams will need to retake those same exams the following year unless CCAM provides an alternative path. Students failing any exam for a second time may be dismissed from the program at the end of their second year.

Thesis Advisor and Thesis Committee

Students are encouraged to identify a potential PhD advisor at the end of their first year to engage/enroll in a summer Reading & Research course. If such an advisor cannot be identified, students are required to present a plan for their first-year summer quarter that needs approval from CCAM.

PhD students are free to change PhD advisors during their enrollment in the program. PhD advisors are free to discontinue working with a PhD student and then cease to be the student’s advisor should the collaboration not meet expectations.

Students are required to form a thesis committee once they have a designated advisor and no later than the Spring quarter of their third year. The committee will first meet (in person) at the end of the student’s second year. If such a thesis committee cannot be formed at that time, students need to present a plan for their future in the program that needs approval from CCAM.

The thesis committee is composed of a minimum of three researchers physically present in Chicago. At least two members need to be affiliated with CCAM. Thesis committees report to CCAM on the student progress at the end of every academic year.

Students are expected to present progress in their PhD work to the thesis committee once during year 3, and again once during year 4. One of these meetings may be used for advancement to candidacy (see below).

Our expectation is for students to graduate at the end of their fifth year in the program. Staying in the program for a sixth year requires approval by CCAM. Students would need to petition by the end of Winter quarter of their fifth year, provide a research plan for completing their degree in a timely manner, and receive approval from their PhD advisor.

Proposal Presentation and Admission to Candidacy

No later than the end of Spring Quarter of the fourth year, students should have scheduled and completed a proposal presentation to their committee, in order to be advanced to candidacy. The proposal presentation is typically an hourlong meeting that begins with a 30-minute presentation by the student, followed by a question and discussion period with the committee. The proposal meeting will be scheduled by the student and their committee and reported to the CAM student affairs administrator. Acceptance of the proposal by the Dissertation Committee is a formal requirement of CAM’s Ph.D. program; all committee members must sign the form approving the proposal. After a successful proposal presentation, the student will be formally admitted to candidacy for the Ph.D. degree. By University rules, the dissertation defense cannot occur earlier than 8 months after admission to candidacy, and the student should keep this in mind when scheduling both the proposal presentation and the defense.

Following advancement to candidacy, during each year that the student remains, the student is required to have a yearly meeting with the dissertation committee leading up to the final thesis defense.

Dissertation Defense

The Ph.D. degree will be awarded following a successful defense and the electronic submission of the final version of the dissertation to the University's Dissertation Office. In this process, a number of University and Department deadlines have to be obeyed. Listed in reverse order, the steps are:

a) Submission of Final Version of Dissertation: The deadline is set by the University and is generally on a Friday in the 6th or 7th week of the quarter when the degree will be awarded. See:

• Information for Ph.D. Students:  https://www.lib.uchicago.edu/research/scholar/phd/students/
• Dissertation Deadlines:   https://www.lib.uchicago.edu/research/scholar/phd/students/dissertation-deadlines/
• Dissertation Templates (LaTeX and Lyx):   https://wiki.uchicago.edu/display/DissertationTemplate/Home

for this deadline as well as guidelines for the formatting of dissertations. b) Dissertation Defense : The thesis defense will be an open seminar announced to the department. Following the regular question-and-answer session, the committee will remain, together with any interested faculty, and continue questioning the candidate. The decision on the thesis will then be reached in a closed meeting of the faculty present. The defense is to be scheduled at least two weeks before the University deadline indicated in point (a). A final draft of the dissertation must be made available to the entire faculty 8 days before the dissertation presentation. c) Committee Approval of Scheduled Defense: A draft of the dissertation should be distributed to the members of the dissertation committee no later than five weeks before the dissertation defense. At least four weeks before the defense, the student must file a departmental form in the Department office, signed by all members of the dissertation committee, indicating that the student can reasonably expect to defend the thesis within four weeks. These rules delineate the minimum level of involvement of the dissertation committee. We strongly recommend that students set up their committees early and that they interact regularly with the members of their committees once they are established. In particular, we strongly recommend that those students wishing to complete the degree before September schedule their defense before the Summer Quarter, else unanticipated committee requirements may lead to the degree being delayed to the Winter Quarter.

Students with questions may contact Jonathan Rodriguez (Student Affairs Administrator), Bahareh Lampert (Dean of Students in the Physical Sciences Division), or Amanda Young (Associate Director, Graduate Student Affairs) in UChicagoGRAD.

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

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Best Online Doctorates in Mathematics: Top PhD Programs, Career Paths, and Salary

An online PhD in Mathematics can land you a job as a math education specialist, data scientist, or information technology professional. If you prefer theoretical, dissertation, and course-based doctoral programs, then our list of the best online PhD in Mathematics is for you.

This list of PhD in Mathematics degree programs includes information technology, education, statistics, and data science postgraduate degrees with advanced mathematics components. Keep reading to find out what schools offer these programs, the course curriculum, acceptance rates, highest-paying mathematics jobs for PhD grads, and tuition rates.

Find your bootcamp match

Can you get a phd in mathematics online.

Yes, you can get a PhD in Mathematics online. However, finding online doctoral programs primarily focused on statistics, mathematics, or applied mathematics is rare. An online PhD route is apt for students looking to pursue information technology, education, or analytics-based mathematics fields.

Earning an online PhD in Information Technology allows you to advance your education in topics covering discrete mathematics, algorithms, and quantitative methods. An online PhD in Mathematics focuses on core graduate courses with a theoretical research and dissertation process. Some academic programs have a residency component required either on-site or online.

Is an Online PhD Respected?

Yes, an online PhD is respected. Online PhD programs are often considered untraditional, course-based doctoral programs, but they are currently seeing rising popularity and are on par with the curriculum of a campus-based program. Your opportunities for career advancement coincide with an on-campus PhD.

However, an online PhD in Mathematics offers limited research, topics, and dissertation scopes and can limit your career possibilities. Breaking into a mathematics academia or research profession with an online PhD can be difficult.

If you want to pursue a traditional mathematics PhD where subjects include metric space, differential geometry, algebra, and calculus, then an on-campus program is for you.

What Is the Best Online PhD Program in Mathematics?

The best online PhD program in mathematics is a PhD in Mathematics Education. This program focuses on the mathematics curriculum and educational management to help you enter the academic field. You will learn mathematical thinking , qualitative and quantitative research, geometry, and mathematical modeling.

The online math PhD programs discussed in this article cover mathematical subject areas and are more apt for those looking to enter the business, data, or tech fields. You can apply to Stanford University, Columbia University, or Harvard University to find the best in-person mathematics PhDs.

Why the University of Wyoming Has the Best Online PhD Program in Mathematics

University of Wyoming has the best mathematics PhD program because it offers an affordable and reputable online mathematics education doctorate program. This program provides in-depth coverage of mathematics and qualifies you for research or post-secondary positions in the field.

Best Online Master’s Degrees

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Online PhD in Mathematics Admission Requirements

The admission requirements of an online PhD in Mathematics include educational qualifications, GPA, and professional experience components. Universities require doctoral students to earn a 3.0 or higher GPA and have a master’s or bachelor’s degree from an accredited institution.

PhD programs from the accredited institutions in this article also require two to three letters of recommendation, a resume, and graduate courses covering math education.

Depending on the school, you might also need to submit GRE and GMAT scores and proof of three to five years of relevant professional experience. Some schools also require candidates to complete a doctoral interview that discusses their interests in the program.

• 3.0 or higher GPA and a relevant undergraduate degree or master’s degree
• Official transcripts from all attended universities
• GRE and GMAT scores
• Resume, personal statement, and letters of recommendation
• Financial assistance application
• Doctoral interview with program faculty
• Prerequisites for graduate courses providing a math education
• Proof of three to five years of professional experience in math, statistics, or a relevant field

Best Online PhDs in Mathematics: Top Degree Program Details

Best online phds in mathematics: top university programs to get a phd in mathematics online.

The top university programs to get a PhD in Mathematics are offered at prestigious institutions like Texas A&M, Northcentral University, and the University of Wyoming. Below is a list of the best online PhDs in Mathematics, along with their program descriptions, tuition costs, and admission requirements.

This list of online PhDs in Mathematics covers a wide range of subject areas, including information technology, data science, education, and business analytics, all of which encompass mathematics topics to help you further your career in math and tech-based professions.

Capella University is a private online university founded in 1993. Its online academic degrees are catered toward adult learners and busy professionals looking to get a higher education. It offers programs covering business, information technology, psychology, education, nursing, and health science fields. 

PhD in Information Technology

If you’re interested in furthering your education in information technology with an integrated capstone project and in-field curriculum, then this degree is for you. The program includes 70 online courses covering tech research and quantitative analysis topics and allows the transfer of 12 credits. 

This program has courses on assurance controls, tech consulting, quantitative design, and complex adaptive systems. This doctoral degree program can help you enter mathematical and tech research fields. You must also complete two virtual residencies and one dissertation focusing on risk management, data science processes , or advanced computing systems. 

PhD in Information Technology Overview

• Accreditation: Higher Learning Commission
• Program Length: Maximum 4 years 9 months
• Acceptance Rate: N/A
• Tuition and Fees: $750/credit per quarter; $175 resource kit fee; $3,000/capstone per quarter

PhD in Information Technology Admission Requirements

• $50 application fee
• Transcripts from all attended universities
• Master’s degree from an accredited university
• Minimum 3.0 GPA

Capitol Technology University was established in 1927 and is a private accredited institution offering both on-campus and online degrees. It is regionally recognized as a top STEM field university and offers online bachelor's, master's, and doctoral programs. Prospective students can apply for online programs in engineering management, cyber security, computer science, product management, or artificial intelligence.

PhD in Business Analytics and Data Science

This business analytics and data science doctorate degree is apt for statisticians wanting to venture into leadership positions in data or business. Although this is an online-based program, it still requires campus visits for its residency course, oral defense, and dissertation presentation. 

The curriculum includes applied statistics, quantitative methods, big data warehousing, applied research, and economic management. You can land lucrative statistician jobs at top companies or pursue a career in data research. 

PhD in Business Analytics and Data Science Overview

• Accreditation: Middle States Commission on Higher Education
• Program Length: 3 years
• Tuition and Fees: $933 plus fees/credit for 2022 to 2023 academic year

PhD in Business Analytics and Data Science Admission Requirements

• Essay, relevant experience, and skills covering business analytics and data science 
• Two letters of recommendation
• Resume showcasing a minimum of five years of industry-relevant work experience
• Undergraduate or master’s degree
• Official college transcripts

City University of Seattle specializes in providing flexible degree programs apt for adult education. It was founded in 1973 and offers more than 65 degree and certification programs. Prospective students can enroll in its advanced online degrees covering business, project management, computer systems, or education administration. 

Several industry professionals in information technology or computer science have extensive math education. This PhD is an excellent choice for graduate students wanting to build a career in instructional technology, computer science research, or another STEM field. 

It offers cyber security, data science, artificial intelligence, or cloud computing concentrations. The data science specialization is best suited for students wanting to get theoretical and hands-on experience across math subjects. 

Some of its core courses in advanced math education cover discrete math, evidence-based practices, quantitative research, computing algorithms, and differential equation topics. The degree also requires research, a dissertation, and a residency in computer science and research. 

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• Accreditation : Northwest Commission on Colleges and Universities
• Program Length: 4 to 5 years
• Tuition and Fees: $765/credit
• Background in programming, database management, operating systems, or networking
• Resume showcasing more than two years of tech-based experience
• 400 to 600-word goal statement
• Three references and their contact information
• Admissions statement questions form
• Interview with the program faculty

Colorado Technical University offers online and on-campus programs in business, mechanical engineering, criminal justice, computer science, educational administration, and nursing education. It was founded in 1965 and currently has a robust online advanced degree program. It offers more than 80 programs to its online students. 

PhD in Computer Science with a Concentration in Big Data Analytics

According to PayScale reports, the average salary of a lead data scientist is $135,887 per year. This doctoral degree provides you with the leadership skills and hands-on experience needed to land lead data scientists’ or other senior tech positions. The degree requires you to fulfill its curriculum, residency, dissertation process, and required credit hours. 

Some of its core courses include advanced quantitative analysis, research methods, big data analysis, information systems, and business intelligence. 

PhD in Computer Science with a Concentration in Big Data Analytics Overview

• Tuition and Fees: $598/credit

PhD in Computer Science with a Concentration in Big Data Analytics Admission Requirements

• Contact an admissions officer
• Online application
• Background in research, theoretical, or other relevant fields
• Contact program faculty to find other requirements
• Doctoral interview

Grand Canyon University is an accredited school founded in 1949. It is home to over 23,500 on-campus students and provides education to more than 90,000 working adult students through its online schools as of 2021. 

Some of its popular online programs include graphic design, communication, business administration, information systems, education administration, health sciences, and data science topics. Its online options offer both undergraduate and postgraduate degrees. 

DBA in Data Analytics 

If your career goals involve business or data analytics, then this DBA will help you get your dream job in data analytics . It focuses on quantitative research and math education and includes data complexity, business administration, financial management, and research design courses. 

Its residency and dissertation process requires you to pursue research in statistical mechanics, quantitative analytics, and data science topics. You can also opt to attend its on-campus evening classes instead of its online option. 

DBA in Data Analytics Overview

• Accreditation: Higher Learning Commission 
• Program Length: 150 weeks 
• Acceptance Rate: 73%
• Tuition and Fees: $715/credit

DBA in Data Analytics Admission Requirements

• 3.4 or above GPA is preferred, with 3.0 as the minimum GPA
• MBA or business-related graduate degree
• Official transcripts of all universities attended

Iowa State University is recognized as one of the flagship public schools with a total undergraduate enrollment of 26,846 students for its fall 2020 cohort. According to US News & World Report, Iowa State University ranks 122 among the best national universities and 45 in the best undergraduate engineering programs.

PhD in Information Systems and Business Analytics 

This tech and business doctoral degree is apt for students looking for career advancement across organizational leadership, computer science academia, or business intelligence management fields. The degree map incorporates dissertation research, oral defense, and advanced degrees. 

You will learn financial management , information technology research, business analytics, and organizational leadership. Some of its math-based courses include applied statistics, logistics, differential equations, and economics. 

PhD in Information Systems and Business Analytics Overview

• Program Length: 5 years
• Tuition and Fees: $6,491/semester (in state); $14,490/semester (out of state) for fall 2021 and spring 2022

PhD in Information Systems and Business Analytics Admission Requirements

• Master’s degree preferred but not required
• Statement of purpose, CV, and college transcripts
• GMAT scores preferred, 600 minimum or equivalent GRE Scores
• Three letters of recommendation
• A writing sample relevant to the doctorate subject

Northcentral University is a private institution founded in 1996 and offers undergraduate, master's, and doctoral-level online programs. Its doctoral programs are fully online and do not include any physical residency requirements. You can choose to enroll in business administration, health sciences, education law, elementary education, technology, or public administration online programs.  

PhD in Data Science

According to the US Bureau of Labor Statistics, the job outlook for data scientists is 22 percent between 2020 to 2030 . The high job outlook makes a data science doctoral degree investment highly profitable. It’s a research-based program that includes 20 courses in subjects like data warehousing, information systems, and data science theories. 

PhD in Data Science Overview

• Program Length: 3 years 4 months
• Tuition and Fees: $68,560/program

PhD in Data Science Admission Requirements

• Academic evaluation with an enrollment advisor

Texas A&M University is a public school that was established in 1876. It has a large student population with a fall 2021 enrollment of 73,284 students. Its distance learning school offers 104 undergraduate degrees, certificates, and graduate programs. Distance learners can enroll in public administration, education law, public health, criminal justice, or engineering programs.

PhD in Curriculum and Instruction, Emphasis in Mathematics Education 

If you want to become a professor or enter the elementary education sector, then this online doctoral degree in mathematics is for you. It helps you become an education specialist in mathematics. The degree plan requires you to complete a dissertation, curriculum development, research, and core credits. 

Its courses include statistical analysis, qualitative and quantitative research, theories of education, categorical data analysis, and linear models. Some other career prospects include math education administration, data science, or statistical analytics fields. 

PhD in Curriculum and Instruction, Emphasis in Mathematics Education Overview

• Accreditation: Southern Association of Colleges and Schools, Commission on Colleges
• Program Length: 4 years 
• Tuition and Fees: $4,254/semester (in state); $9,019/semester (out of state) fall 2022 or spring 2023 for 9 credit hours per semester

PhD in Curriculum and Instruction, Emphasis in Mathematics Education Admission Requirements

• $89 application fee for online application
• Departmental essays 
• Three letters of reference

University of Central Florida is a public research institution founded in 1963. The school is best known for its engineering, computer science, public administration, and programs in education. According to US News & World Report, the University of Central Florida is among the top public schools and ranks 15 among the most innovative schools .

PhD in Education

This online PhD in Education offers course tracks across educational psychology, English, instructional design, empirical research, social science, and K-8 mathematics. If your career goals involve being in fields like education policy, education administration, or math education, then this degree is for you. 

It requires you to complete research electives, a dissertation, and 18 credit hours of core courses. It helps you master evidence-based decisions, teacher education administrative tasks, and curriculum development skills. 

PhD in Education Overview

• Tuition and Fees: $327/credit (in state); $1,151/credit (out of state)

PhD in Education Admission Requirements

• Speak with a graduate program success coach
• GMAT or GRE scores
• Transcripts from all universities attended
• Immunization forms
• Residency class form
• Contact program faculty for other requirements

University of Wyoming is among the reputable national universities and offers both on-campus and online bachelor's, master's, and doctoral programs. According to US News & World Report, the University of Wyoming ranks 196 among the best national universities and number 99 in the top public schools. 

Its online students can choose from its accounting, education policy, public administration, education law, finance, or nurse practitioner certificate or degree programs.

PhD in Mathematics Education

This math education doctoral degree focuses on qualitative research, mathematical modeling, quantitative reasoning, differential equations, and associated geometry topics. It also requires you to complete dissertation research and a preliminary exam. 

You can use this degree to venture into mathematics professor, math doctorate degree advisor, or elementary education specialist professions. 

PhD in Mathematics Education Overview

• Tuition and Fees: $7,182/year (in state); $18,324/year (out of state) for 18 credits per academic year

PhD in Mathematics Education Admission Requirements

• GRE Scores: 144 quantitative, 151 verbal, 4.0 analytical writing
• Master’s degree
• Three years of P-12 teaching or other relevant professional experience
• Meet residency and academic status requirements
• Letter of intent, three letters of recommendation, an academic writing sample, and a resume

Online Mathematics PhD Graduation Rates: How Hard Is It to Complete an Online PhD Program in Mathematics?

It is extremely hard to complete an online PhD program in mathematics. This is due to the advanced core courses, extensive research, and dissertation process requirements. Several mathematics online PhDs tend to branch into difficult subjects covering information technology, data science, or business analytics .

This requires doctoral candidates to have an extensive background in technical and advanced mathematics subjects. The timeline and approval aspects of a PhD are also demanding, leading to high attrition rates.

How Long Does It Take to Get a PhD in Mathematics Online?

It takes about four years to get a PhD in Mathematics online. However, the exact time duration can range anywhere from three to six years, depending on the candidate and program requirements. Doctoral students with transfer credit hours can complete the degree in a shorter time frame.

Students who fail to meet their dissertation process requirements or get early approval for their research topics will finish their degrees in a longer time frame. Online PhDs are catered toward full-time professionals or adult learners by offering a flexible schedule. Those who take fewer credit hours and opt for a flexible schedule will increase their degree timeline.

How Hard Is an Online Doctorate in Mathematics?

An online Doctorate in Mathematics is extremely hard. It incorporates an array of advanced subject areas and dissertation topic possibilities. The best online mathematics PhDs are offered in information technology, data science, or business analytics fields. All of these subject areas require technical knowledge and extensive research commitment and self-discipline.

Best PhD Programs

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What Courses Are in an Online Mathematics PhD Program?

The courses in an online mathematics PhD program include mathematical modeling, statistics, quantitative reasoning, and discrete mathematics. Depending on your choice of a doctoral program, your core courses might also include curriculum development, data analytics, programming languages , or financial analysis.

Regardless of the primary PhD concentration, the mathematics components of the degree will help you master advanced applied math topics.

Main Areas of Study in a Mathematics PhD Program

• Data science
• Differential equations
• Mathematics curriculum development
• Business analytics
• Quantitative and qualitative research
• Discrete mathematics

How Much Does Getting an Online Mathematics PhD Cost?

It costs $19,314 per year to get a PhD in Mathematics, according to the National Center of Education Statistics (NCES). Your online PhD tuition rates will typically range in this area but will vary from one university program to another.

NCES further reports the graduate tuition of private institutions from 2018 to 2019 was $25,929 and for public institutions was $12,171. This disparity in tuition costs is also present in online PhDs in Mathematics.

How to Pay for an Online PhD Program in Mathematics

You can pay for an online PhD program in mathematics by applying for doctoral funding programs, scholarship opportunities, or education loan financing. You can also apply for an employee sponsorship program at your workplace.

The online aspect of the PhD limits student exposure to research opportunities and graduate assistantships. You can still get in touch with a university representative to find potential internal funding opportunities, though. Doctoral students can also apply for external funding available for math programs.

How to Get an Online PhD for Free

You cannot get an online PhD for free. However, PhDs and other advanced degrees may be eligible for fully-funded scholarships, depending on the student applying for them. You can apply for university or external scholarships to fully fund your PhD, but the chances of getting a full-tuition grant are low for online programs.

What Is the Most Affordable Online PhD in Mathematics Degree Program?

The most affordable online PhD in Mathematics degree program is offered by the University of Central Florida for $18,675. The school’s PhD in Education charges $327.32 per credit hour and consists of 51 to 57 credits for the program.

Most Affordable Online PhD Programs in Mathematics: In Brief

Why you should get an online phd in mathematics.

You should get an online PhD in Mathematics because it will help you land lucrative positions in post-secondary education, management, or research. A PhD is highly respected and showcases your expertise in advanced applied mathematics topics.

Top Reasons for Getting a PhD in Mathematics

• Multifaceted career outcome opportunities. An online PhD in Mathematics covers technical and non-technical subjects, including computer science, data analytics, education management, and applied mathematics. This opens up career possibilities across several in-demand industries.
• Specialized skills. Mathematics PhDs help you acquire specialized skills in quantitative research, statistical analytics, and discrete mathematics. These skills are highly demanded in the data, research, artificial intelligence, and business industries.
• Higher salaries. These doctoral programs also qualify you for senior roles. Senior-level roles offer higher salaries and an increased earning potential.
• High job security. The US Bureau of Labor Statistics notes that the job outlook for computer and information research scientists is 22 percent between 2020 to 2030. Several online mathematics PhDs will qualify you for jobs under these tech occupations and provide you with a high job security rate.

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What Is the Difference Between an On-Campus Mathematics PhD and an Online PhD in Mathematics?

The difference between an on-campus mathematics PhD and an online PhD in Mathematics the core course topics and research potential. The career outcomes of these types of degrees also vary depending on your choice of program. Below are the key differences between an on-campus vs an online mathematics PhD program.

Online PhD vs On-Campus PhD: Key Differences

• Tuition affordability. An online PhD in Mathematics is typically less expensive compared to an on-campus PhD. However, although an on-campus degree is more costly, there are more funding opportunities.
• More math concentrations. There aren’t many mathematics concentrations in an online program compared to in-person degrees. Most online programs offer information technology, data science, or education focused on advanced math subjects.
• Research scopes. The research topic scopes in an on-campus program are more practical, unlike dissertation-based online PhDs.
• Socialization opportunities. An on-campus PhD offers ample socialization opportunities and easy access to the program faculty and financial aid offices. This is more difficult for online doctoral students.

How to Get a PhD in Mathematics Online: A Step-by-Step Guide

To get a PhD in Mathematics online, you need to fulfill the school’s admissions prerequisites and program requirements. Below are the five main steps required to complete an online PhD in Mathematics.

You must pay the application fee, upload official transcripts, and fill out the application questionnaire. You will also submit all proof of professional experience, letters of recommendation, resumes, and personal statements. 

The admissions requirements section is where you complete a doctoral interview with the program faculty. The requirements for this interview will vary from school to school. You will discuss degree outcomes, passions, and financial payment plans during the interview.

The third step is to contact your program advisor and discuss the degree timeline and requirements. You will discuss potential research and dissertation topics.

Next, you must complete all core courses. The courses will vary depending on your major, but it is crucial to complete them to graduate both successfully and on time.

The last step is to complete your research, residency, and dissertation process. This step requires prior approval from the program’s doctoral faculty. 

Online PhD in Mathematics Salary and Job Outlook

The salary and job outlook for online mathematics PhDs will vary by industry, profession, and degree concentration. The salary can range from $79,640 to $110,000 per year, and the job growth rate can go as high as 33 percent.

What Can You Do With an Online Doctorate in Mathematics?

With an online Doctorate in Mathematics, you can become a research analyst , mathematics professor, statistician, or data scientist. You can also enter higher education management, data analytics, quantitative research, or finance analytics professions. The career possibilities with an online Doctorate in Mathematics are countless and cover many industries.

Best Jobs with a PhD in Mathematics

• Mathematics Professor
• Statistician
• Operations Research Analyst
• Mathematician
• Data Scientist

Potential Careers With a Mathematics Degree

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What Is the Average Salary for an Online PhD Holder in Mathematics?

The average salary for PhD in Mathematics holders is $110,000 per yea r, according to PayScale. However, your salary range will vary depending on your choice of profession. Education sector professions often offer lower average salaries compared to information technology or data science sectors.

Highest-Paying Mathematics Jobs for PhD Grads

Best mathematics jobs for online phd holders.

The best mathematics jobs for online PhD holders opens up opportunities across the research, data, analytics, or information technology fields. Below are the highest-paying jobs you can apply for with your online PhD in Mathematics.

A data scientist is responsible for analyzing, visualizing, and sorting raw data into useful information. They can work in a wide range of industries and work to extract useful data for optimal business operations or scientific results. 

• Salary with a Mathematics PhD: $100,910
• Job Outlook: 22% job growth from 2020 to 2030
• Number of Jobs: 33,000
• Highest-Paying States: Oregon, Arizona, Texas

A statistician is a mathematics and data science professional who uses quantitative and qualitative research in surveys to find valuable information. They are responsible for designing, conducting, and analyzing each quantitative survey. 

• Salary with a Mathematics PhD: $96,280
• Job Outlook: 33% job growth from 2020 to 2030
• Number of Jobs: 44,800
• Highest-Paying States: New York, Connecticut, Massachusetts 

A mathematician conducts research on theoretical mathematical principles to advance the mathematics, physics, engineering, data science, or economics fields. They work with mathematical modeling, statistical analysis, and quantitative reasoning to study mathematics. 

• Highest-Paying States: District of Columbia, New York, New Jersey

An operations research analyst is responsible for evaluating an organization’s operations and production plan and must suggest further productivity solutions. They use data analytics, mathematics, quantitative and qualitative research, and statistics for their jobs. 

• Salary with a Mathematics PhD: $82,360
• Job Outlook: 25% job growth from 2020 to 2030
• Number of Jobs: 104,100
• Highest-Paying States: Virginia, Alabama, Maryland

A mathematics professor works at post-secondary educational institutions and develops the course curriculum, assignments, and exams to test students in their knowledge and comprehension. They teach introductory and advanced math classes to college students and can also lead research departments. 

• Salary with a Mathematics PhD: $79,640
• Job Outlook: 12% job growth from 2020 to 2030
• Number of Jobs: 1,276,900
• Highest-Paying States: California, Rhode Island, Oregon

Is It Worth It to Do a PhD in Mathematics Online?

Yes, it is worth it to do a PhD in Mathematics online. Mathematics online PhDs cover subject areas that open up career opportunities in lucrative sectors of academics, business, data science, and tech.

PhD programs are worth it for doctorate students looking to earn a higher salary and achieve increased job security. Mathematics doctoral programs qualify you for many high-paying jobs, including university professor, research analyst, and mathematician.

Additional Reading About Mathematics

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Online PhD in Mathematics FAQ

Online mathematics PhDs cover courses in applied mathematics, discrete mathematics, statistics, quantitative research, and operations analytics topics. The core subject areas will vary depending on the focus subject of the online degree.

No, it is not easy to get an online mathematics PhD. A doctoral degree comprises research, dissertation, and advanced courses covering highly technical topics.

It will take around three to five years to complete an online mathematics PhD. Your degree timeline depends on your course schedule, dissertation process, and transfer credits.

A data science concentration is best for an online mathematics PhD. Data science is a rapidly growing field that encompasses tons of in-demand professions with high salaries. Moreover, with a data science focus, you will also get to work on a wide range of real-world problems instead of just theoretical scenarios.

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AM PhD Model Program

The overall set of courses must constitute a coherent, rigorous program appropriate for a Ph.D. specifically in the field of Applied Mathematics, and the faculty recommend that students take Applied Math graduate courses to the greatest extent possible and relevant.

Listed here are examples of courses the Applied Math faculty have identified as appropriate for Ph.D. Program Plans in Applied Math.  Note that the list is not exclusive, and each student’s individual plan requires review and approval by the CHD.   Students should also note the school's overall PhD Program Plan requirements .

Examples of courses for students studying machine learning and artificial intelligence

• AM 216 Inverse Problems in Science and Engineering
• AM 221 Advanced Optimization
• CS 234r Topics on Computation in Networks and Crowds
• CS 280r Advanced Topics in Artificial Intelligence
• or CS 181 Machine Learning if more appropriate given the student’s background
• ES 250 Information Theory
• Other Computer Science or Engineering Sciences courses relevant to the student’s research

Examples of courses for students studying computational math, inference, and prediction

• AM 207 Advanced Scientific Computing: Stochastic Methods for Data Analysis, Inference and Optimization
• AM 231 Decision Theory
• AC 209a/b Data Science 1/2
• CS 205 Computing Foundations for Computational Science
• ES 255 Statistical Inference with Engineering Applications
• 200-level Statistics courses appropriate for the student’s area of research

Examples of courses for students with an interest in physical modelling and applications

• AM 201/202 Physical Mathematics I/II
• AM 203 Introduction to Disordered Systems and Stochastic Processes
• AM 205 Advanced Scientific Computing: Numerical Methods
• AM 225 Advanced Scientific Computing: Numerical Methods for Partial Differential Equations
• AP 225 Introduction to Soft Matter, or other Applied Physics courses
• ES 220 Fluid Dynamics
• ES 240 Solid Mechanics
• Examples of Statistical Mechanics courses: AP 284, Physics 262
• Examples of Electromagnetism courses: AP 216, Physics 232
• Examples of Solid State Physics courses: AP 295a/b

Examples of courses for students with an interest in biological modelling and applications

• AM 217 Instabilities and Patterns in Soft Matter and Biophysics
• CS 289 Biologically-inspired Multi-agent Systems
• Math 243 Evolutionary Dynamics
• MCB 199 Statistical Thermodynamics and Quantitative Biology

Examples of courses for students with an interest in economics

• CS 236r Topics at the Interface between Computer Science and Economics
• Econ 2020a/b Microeconomic Theory I/II

Note that taking “G-level” courses at MIT is certainly an option, as MIT offers a different course selection than is available at SEAS and Harvard.  Examples of MIT courses taken by Applied Math PhD students include 2.29, 6.252J, 6.851, 8.334, 16.920, 18.1021,18.335J, 18.336.

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What are the deadlines for applications? December 1st is the priority deadline for applicants;  we strongly encourage applicants to submit their applications by the priority deadline.  Please see the admissions page for the  final deadline for remaining applicants; a pplicants who apply by the final deadline will still have their files formally reviewed by the admissions committee for full consideration, but not necessarily on a priority basis. 

When are applications reviewed? The admissions committee will begin reviewing applications as early as December 1st, but everyone who applies by the deadline receives full consideration.  The admissions page shows when applicants can expect a response.  We kindly request that you not email us to ask about the admissions decision unless you have not heard from us and February 15th has passed.

How can I get feedback on my application? PhD applicants who identify as members of underrepresented or marginalized groups may receive feedback from the volunteers in the  PAR program . 

Can I apply for any quarter? We only offer autumn admission.

How can I request an application fee waiver?  Step 1: See if you qualify for a UW Grad School application fee  waiver  and follow the instructions provided if so.  Step 2: If ineligible for the above waiver, the department offers a limited number of waivers. Waivers may be granted to applicants who identify as members of underrepresented or marginalized groups. Waivers may also be granted on a need basis. If requesting a waiver for these reasons, please  email  [email protected]  to explain why you are seeking a waiver  (please include your citizenship).

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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.​​

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Applied Mathematics, Ph.D.

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The Ph.D. in Applied Mathematics

Graduates with a master's in applied mathematics can expand their subject matter expertise by choosing a PhD in applied mathematics at Florida Tech. As one of only 30 applied mathematics programs in the United States, Florida Tech's doctoral program offers several specializations in the field, including nonlinear analysis, stochastic analysis, optimization, numerical analysis, scientific computing, and statistics.

A Degree with Real Flexibility

In addition to the areas of specialization, Florida Tech provides additional flexibility in its PhD in applied mathematics program, allowing doctoral students to design a curriculum that fits their specific research interests and career goals. As a national research university, Florida Tech is committed to providing students with a variety of applied mathematics research experiences, opening up careers in a wide range of industries.

Small Classes–World Renowned Faculty

Students in the PhD in applied mathematics program at Florida Tech work closely with professors and fellow students. A small faculty-to-student ratio creates a close-knit academic community that is often impossible at larger universities. Professors in the math department have doctoral degrees in applied and computational mathematics and statistics. Professors—not graduate students—teach all courses, supervise student research projects, and conduct their own meaningful research studies that are often open for student collaboration.

Advanced Research Opportunities

As in any doctoral program, research is the core of the academic program. The PhD in applied mathematics program explores many applied mathematics topics. Research is conducted in areas of science, engineering, medicine, and business through interdisciplinary teams, as well as in the areas of concentration needed for the doctoral degree program. Students take part in research projects such as dynamical systems and chaos theory, stem cell research, computational number theory, optimal control and inverse problems, and antagonistic stochastic games, to name a few.

Full-pay tuition scholarships are available for full-time doctoral graduate research assistants.

High-Tech Laboratory Facilities

The facilities and resources available for doctoral students at Florida Tech include access to the engineering and science labs, four mathematics labs that feature advanced software such as Wolfram Mathematica, MATLAB, the R Project, Sage, and IBM SPSS. Additionally, the new computational mathematics and statistics research lab includes a 55-inch touchscreen Mondopad.

Great Location

Many doctoral students in the PhD in applied mathematics program are working professionals living in close proximity to the campus in Melbourne, Florida. The university is also a top pick among students around the world for its location within the Florida High Tech Corridor–home to more than 5,000 high-tech companies and the fifth largest high-tech workforce in the nation.

Graduates with a PhD in applied mathematics work in a variety of fields ranging from engineering and science to medicine and economics. Some examples of the organizations, corporations, and research institutes that hire mathematicians include government labs, electronics and computer manufacturers, medical device companies, and financial services firms.

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Computational and Applied Mathematics

Computational & Applied Mathematics   began in 1981, as the main scientific publication of the Brazilian Society of Computational and Applied Mathematics (SBMAC). The objective of the journal is the publication of original research in Applied and Computational Mathematics, with interfaces in Physics, Engineering, Chemistry, Biology, Operations Research, Statistics, Social Sciences and Economy. The journal has the usual quality standards of scientific international journals, and we aim high level of contributions in terms of originality, depth and relevance. CAM is currently reviewed in Mathematical Reviews and Institute of Scientific Information (Webofscience).

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