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Ph.D. Program Overview

Description.

The graduate program in the field of mathematics at Cornell leads to the Ph.D. degree, which takes most students five to six years of graduate study to complete. One feature that makes the program at Cornell particularly attractive is the broad range of  interests of the faculty . The department has outstanding groups in the areas of algebra, algebraic geometry,  analysis, applied mathematics, combinatorics, dynamical systems, geometry, logic, Lie groups, number theory, probability, and topology. The field also maintains close ties with distinguished graduate programs in the fields of  applied mathematics ,  computer science ,  operations research , and  statistics .

Core Courses

A normal course load for a beginning graduate student is three courses per term. 

There are no qualifying exams, but the program requires that all students pass four courses to be selected from the six core courses. First-year students are allowed to place out of some (possibly, all) of the core courses. In order to place out of a course, students should contact the faculty member who is teaching the course during the current academic year, and that faculty member will make a decision. The minimum passing grade for the core courses is B-; no grade is assigned for placing out of a core course.

At least two core courses should be taken (or placed out) by the end of the first year. At least four core courses should be taken (or placed out) by the end of the second year (cumulative). These time requirements can be waived for students with health problems or other significant non-academic problems. They can be also waived for students who take time-consuming courses in another area (for example, CS) and who have strong support from a faculty; requests from such students should be made before the beginning of the spring semester. 

The core courses  are distributed among three main areas: analysis, algebra and topology/geometry. A student must pass at least one course from each group. All entering graduate students are encouraged to eventually take all six core courses with the option of an S/U grade for two of them. 

The six core courses are:

MATH 6110, Real Analysis

MATH 6120, Complex Analysis

MATH 6310, Algebra 1

MATH 6320, Algebra 2

MATH 6510, Introductory Algebraic Topology

MATH 6520, Differentiable Manifolds.

Students who are not ready to take some of the core courses may take MATH 4130-4140, Introduction to Analysis, and/or MATH 4330-4340, Introduction to Algebra, which are the honors versions of our core undergraduate courses.

"What is...?" Seminar

The "What Is...?" Seminar is a series of talks given by faculty in the graduate field of Mathematics. Speakers are selected by an organizing committee of graduate students. The goal of the seminar is to aid students in finding advisors.

Schedule for the "What Is...?" seminar

Special Committee

The Cornell Graduate School requires that every student selects a special committee (in particular, a thesis adviser, who is the chair or the committee) by the end of the third semester.

The emphasis in the Graduate School at Cornell is on individualized instruction and training for independent investigation. There are very few formal requirements and each student develops a program in conjunction with his or her special committee, which consists of three faculty members, some of which may be chosen from outside the field of mathematics. 

Entering students are not assigned special committees. Such students may contact any of the members on the Advising Committee if they have questions or need advice.

Current Advising Committee

Analysis / Probability / Dynamical Systems / Logic: Lionel Levine Geometry / Topology / Combinatorics: Kathryn Mann Probability / Statistics:  Philippe Sosoe Applied Mathematics Liaison: Richard Rand

Admission to Candidacy

To be admitted formally to candidacy for the Ph.D. degree, the student must pass the oral admission to candidacy examination or A exam. This must be completed before the beginning of the student's fourth year. Upon passing the A exam, the student will be awarded (at his/her request) an M.S. degree without thesis.

The admission to candidacy examination is given to determine if the student is “ready to begin work on a thesis.” The content and methods of examination are agreed on by the student and his/her special committee before the examination. The student must be prepared to answer questions on the proposed area of research, and to pass the exam, he/she must demonstrate expertise beyond just mastery of basic mathematics covered in the core graduate courses. 

To receive an advanced degree a student must fulfill the residence requirements of the Graduate School. One unit of residence is granted for successful completion of one semester of full-time study, as judged by the chair of the special committee. The Ph.D. program requires a minimum of six residence units. This is not a difficult requirement to satisfy since the program generally takes five to six years to complete. A student who has done graduate work at another institution may petition to transfer residence credit but may not receive more than two such credits.

The candidate must write a thesis that represents creative work and contains original results in that area. The research is carried on independently by the candidate under the supervision of the chairperson of the special committee. By the time of the oral admission to candidacy examination, the candidate should have selected as chairperson of the committee the faculty member who will supervise the research. When the thesis is completed, the student presents his/her results at the thesis defense or B Exam. All doctoral students take a Final Examination (the B Exam, which is the oral defense of the dissertation) upon completion of all requirements for the degree, no earlier than one month before completion of the minimum registration requirement.

Masters Degree in the Minor Field

Ph.D. students in the field of mathematics may earn a Special Master's of Science in Computer Science. Interested students must apply to the Graduate School using a form available for this purpose. To be eligible for this degree, the student must have a member representing the minor field on the special committee and pass the A-exam in the major field. The rules and the specific requirements for each master's program are explained on the referenced page.

Cornell will award at most one master's degree to any student. In particular, a student awarded a master's degree in a minor field will not be eligible for a master's degree in the major field.

Graduate Student Funding

Funding commitments made at the time of admission to the Ph.D. program are typically for a period of five years. Support in the sixth year is available by application, as needed. Support in the seventh year is only available by request from an advisor, and dependent on the availability of teaching lines. Following a policy from the Cornell Graduate School, students who require more than seven years to complete their degree shall not be funded as teaching assistants after the 14th semester.

Special Requests

Students who have special requests should first discuss them with their Ph.D. advisor (or with a field member with whom they work, if they don't have an advisor yet). If the advisor (or field faculty) supports the request, then it should be sent to the Director of Graduate Studies.  

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.

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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:

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

(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 .

Graduate Program

Our graduate program is unique from the other top mathematics institutions in the U.S. in that it emphasizes, from the start, independent research. Each year, we have extremely motivated and talented students among our new Ph.D. candidates who, we are proud to say, will become the next generation of leading researchers in their fields. While we urge independent work and research, there exists a real sense of camaraderie among our graduate students. As a result, the atmosphere created is one of excitement and stimulation as well as of mentoring and support. Furthermore, there exists a strong scholarly relationship between the Math Department and the Institute for Advanced Study, located just a short distance from campus, where students can make contact with members there as well as attend the IAS seminar series.  Our program has minimal requirements and maximal research and educational opportunities. We offer a broad variety of advanced research topics courses as well as more introductory level courses in algebra, analysis, and geometry, which help first-year students strengthen their mathematical background and get involved with faculty through basic course work. In addition to the courses, there are several informal seminars specifically geared toward graduate students: (1) Colloquium Lunch Talk, where experts who have been invited to present at the Department Colloquium give introductory talks, which allows graduate students to understand the afternoon colloquium more easily; (2) Graduate Student Seminar (GSS), which is organized and presented by graduate students for graduate students, creating a vibrant mathematical interaction among them; and, (3) What’s Happening in Fine Hall (WHIFH) seminar where faculty give talks in their own research areas specifically geared towards graduate students. Working or reading seminars in various research fields are also organized by graduate students each semester. First-year students are set on the fast track of research by choosing two advanced topics of research, beyond having a strong knowledge of three more general subjects: algebra, and real and complex analysis, as part of the required General Examination. It is the hope that one, or both, of the advanced topics will lead to the further discovery of a thesis problem. Students are expected to write a thesis in four years but will be provided an additional year to complete their work if deemed necessary. Most of our Ph.D.'s are successfully launched into academic positions at premier mathematical institutions as well as in industry .

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Overview of the PhD Program

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

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

General Ph.D. Requirements

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

Optional additions to the Ph.D. program

Harvard PhD students may choose to pursue these additional aspects:

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

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

In Applied Mathematics

• First-Year Exploration
• Areas of Application
• AM & Economics
• How to Declare
• Who are my Advisors?
• Secondary Field
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• Research for Course Credit (AM 91R & AM 99R)
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• Peer Concentration Advisors (PCA) Program
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• How to Apply
• PhD Timeline
• PhD Model Program (Course Guidelines)
• Oral Qualifying Examination
• Committee Meetings
• Committee on Higher Degrees
• Research Interest Comparison
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• Cross-Harvard Engagement
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Doctor of Philosophy in the Field of Mathematics (STEM)

Faculty expertise in the PhD program in mathematics covers a wide range of research fields, including analysis, ordinary and partial differential equations, dynamical systems, applied math (including numerical analysis), combinatorics, logic, topology and knot theory. With about 30 graduate students and 20 faculty members, there is lively interaction as well as extensive individual attention.

All graduate students have individual advisors throughout their enrollment, starting from the time of admission. New students also receive peer advisors. In addition, research seminars and the department colloquium series help students explore potential research areas. Teaching assistantships are available for full-time students. Assistants gain teaching experience with a moderate workload, leading recitations for one introductory undergraduate course per semester.

The graduate certificate in mathematics is offered for those who seek to strengthen their mathematical backgrounds at the advanced undergraduate and beginning graduate levels and better position themselves in their careers or prepare for graduate work in quantitative disciplines.

This is a STEM designated program.

Visit the program website for additional information.  

Supporting documents not submitted online should be mailed to:

Columbian College of Arts and Sciences, Office of Graduate Studies The George Washington University 801 22nd Street NW, Phillips Hall 107 Washington DC 20052

For additional information about the admissions process visit the Columbian College  of Arts and Sciences  Frequently Asked Questions  page.

[email protected] 202-994-6210 (phone)

Hours: 9:00 am to 5:00 pm, Monday through Friday

The following requirements must be fulfilled:

The general requirements stated under  Columbian College of Arts and Sciences, Graduate Programs .

The requirements for the  Doctor of Philosophy Program .

Pre-candidacy

Pre-candidacy requirements include satisfactory completion of 48 credits of coursework and achievement of a passing grade in the general examination.

After completing 36 credits of coursework, students may petition the graduate committee for approval to take MATH 6995 , but students may take no more than 12 credits in any combination of MATH 6995 and MATH 8999 in a single academic year.

Students wishing to take courses outside the department must petition and obtain the approval of the graduate committee.  The committee may limit the number of such courses that students take.

Subject to the approval of the graduate committee (requested via petition), students may take up to 12 credits of courses  offered by other institutions  in  the  Consortium of Universities of the Washington Metropolitan Area .   Students wishing to take such courses must petition and obtain the approval of the graduate committee.

Subject to the approval of the graduate committee (requested via petition) and the agreement of the instructor, students may take up to 12 credits from the following upper-level undergraduate courses for graduate credit, provided that additional graduate-level coursework is completed in these classes.

General examination

The general examination consists of two preliminary examinations. One examination is in two to four subjects selected from algebra, analysis, topology, and applied math, and the other is a specialty examination in a research area approved by the department.

Post-candidacy requirements

Post-candidacy requirements include the successful completion of an additional 24 credits of graduate coursework, including at least 6 credits of MATH 8999 ; the completion of the dissertation; and the successful defense of the dissertation in a final oral examination.

No more than 15 credits in any combination of MATH 6995 and MATH 8999 may be among the student's final 18 credits of required coursework.

Once a student successfully completes 24 post-candidacy credits, they must register for 1 credit of CCAS 0940  each subsequent fall and spring semester until they have successfully defended their dissertation, thereby completing the degree program.

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• Doing a PhD in Mathematics
• Doing a PhD

What Does a PhD in Maths Involve?

Maths is a vast subject, both in breadth and in depth. As such, there’s a significant number of different areas you can research as a math student. These areas usually fall into one of three categories: pure mathematics, applied mathematics or statistics. Some examples of topics you can research are:

• Number theory
• Numerical analysis
• String theory
• Random matrix theory
• Graph theory
• Quantum mechanics
• Statistical forecasting
• Matroid theory
• Control theory

Besides this, because maths focuses on addressing interdisciplinary real-world problems, you may work and collaborate with other STEM researchers. For example, your research topic may relate to:

• Biomechanics and transport processes
• Evidence-based medicine
• Fluid dynamics
• Financial mathematics
• Machine learning
• Theoretical and Computational Optimisation

What you do day-to-day will largely depend on your specific research topic. However, you’ll likely:

• Continually read literature – This will be to help develop your knowledge and identify current gaps in the overall body of knowledge surrounding your research topic.
• Undertake research specific to your topic – This can include defining ideas, proving theorems and identifying relationships between models.
• Collect and analyse data – This could comprise developing computational models, running simulations and interpreting forecasts etc.
• Liaise with others – This could take many forms. For example, you may work shoulder-to-shoulder with individuals from different disciplines supporting your research, e.g. Computer scientists for machine learning-based projects. Alternatively, you may need frequent input from those who supplied the data for your research, e.g. Financial institutions or biological research colleagues.
• Attend a wide range of lectures, seminars and events.

Browse PhD Opportunities in Mathematics

Application of artificial intelligence to multiphysics problems in materials design, study of the human-vehicle interactions by a high-end dynamic driving simulator, physical layer algorithm design in 6g non-terrestrial communications, machine learning for autonomous robot exploration, detecting subtle but clinically significant cognitive change in an ageing population, how long does it take to get a phd in maths.

The average programme duration for a mathematics PhD in the UK is 3 to 4 years for a full-time studying. Although not all universities offer part-time maths PhD programmes, those that do have a typical programme duration of 5 to 7 years.

Again, although the exact arrangement will depend on the university, most maths doctorates will require you to first register for an MPhil . At the end of your first year, your supervisor will assess your progress to decide whether you should be registered for a PhD.

Additional Learning Modules

Some Mathematics departments will require you to enrol on to taught modules as part of your programme. These are to help improve your knowledge and understanding of broader subjects within your field, for example, Fourier Analysis, Differential Geometry and Riemann Surfaces. Even if taught modules aren’t compulsory in several universities, your supervisor will still encourage you to attend them for your development.

Most UK universities will also have access to specialised mathematical training courses. The most common of these include Pure Mathematics courses hosted by Mathematics Access Grid Conferencing ( MAGIC ) and London Taught Course Centre ( LTCC ) and Statistics courses hosted by Academy for PhD Training in Statistics ( APTS ).

What Are the Typical Entry Requirements for A PhD in Maths?

In the UK, the typical entry requirements for a Maths PhD is an upper second-class (2:1) Master’s degree (or international equivalent) in Mathematics or Statistics [1] .

However, there is some variation on this. From writing, the lowest entry requirement is an upper second-class (2:1) Bachelor’s degree in any math-related subject. The highest entry requirement is a first-class (1st) honours Master’s degree in a Mathematics or Statistics degree only.

It’s worth noting if you’re applying to a position which comes with funding provided directly by the Department, the entry requirements will usually be on the higher side because of their competitiveness.

In terms of English Language requirements, most mathematics departments require at least an overall IELTS (International English Language Testing System) score of 6.5, with no less than 6.0 in each individual subtest.

Tips to Consider when Making Your Application

When applying to any mathematics PhD, you’ll be expected to have a good understanding of both your subject field and the specific research topic you are applying to. To help show this, it’s advisable that you demonstrate recent engagement in your research topic. This could be by describing the significance of a research paper you recently read and outlining which parts interested you the most, and why. Additionally, you can discuss a recent mathematics event you attended and suggest ways in how what you learnt might apply to your research topic.

As with most STEM PhDs, most maths PhD professors prefer you to discuss your application with them directly before putting in a formal application. The benefits of this is two folds. First, you’ll get more information on what their department has to offer. Second, the supervisor can better discover your interest in the project and gauge whether you’d be a suitable candidate. Therefore, we encourage you to contact potential supervisors for positions you’re interested in before making any formal applications.

How Much Does a Maths PhD Typically Cost?

The typical tuition fee for a PhD in Maths in the UK is £4,407 per year for UK/EU students and £20,230 per year for international students. This, alongside the range in tuition fees you can expect, is summarised below:

Note: The above tuition fees are based on 12 UK Universities [1]  for 2020/21 Mathematic PhD positions. The typical fee has been taken as the median value.

In addition to the above, it’s not unheard of for research students to be charged a bench fee. In case you’re unfamiliar with a bench fee, it’s an annual fee additional to your tuition, which covers the cost of specialist equipment or resources associated with your research. This can include the upkeep of supercomputers you may use, training in specialist analysis software, or travelling to conferences. The exact fee will depend on your specific research topic; however, it should be minimal for most mathematic projects.

What Specific Funding Opportunities Are There for A PhD in Mathematics?

Alongside the usual funding opportunities available to all PhD Research students such as doctoral loans, departmental scholarships, there are a few other sources of funding available to math PhD students. Examples of these include:

You can find more information on these funding sources here: DiscoverPhDs funding guide .

What Specific Skills Do You Gain from Doing a PhD in Mathematics?

A doctorate in Mathematics not only demonstrates your commitment to continuous learning, but it also provides you with highly marketable skills. Besides subject-specific skills, you’ll also gain many transferable skills which will prove useful in almost all industries. A sample of these skills is listed below.

• Logical ability to consider and analyse complex issues,
• Commitment and persistence towards reaching research goals,
• Outstanding verbal and written skills,
• Strong attention to detail,
• The ability to liaise with others from unique disciple backgrounds and work as part of a team
• Holistic deduction and reasoning skills,
• Forming and explaining mathematical and logical solutions to a wide range of real-world problems,
• Exceptional numeracy skills.

What Jobs Can You Get with A Maths PhD?

One of the greatest benefits maths PostDocs will have is the ability to pursue a wide range of career paths. This is because all sciences are built on core principles which, to varying extents, are supported by the core principles of mathematics. As a result, it’s not uncommon to ask students what path they intend to follow after completing their degree and receive entirely different answers. Although not extensive by any means, the most common career paths Math PostDocs take are listed below:

• Academia – Many individuals teach undergraduate students at the university they studied at or ones they gained ties to during their research. This path is usually the preferred among students who want to continue focusing on mathematical theories and concepts as part of their career.
• Postdoctoral Researcher – Others continue researching with their University or with an independent organisation. This can be a popular path because of the opportunities it provides in collaborative working, supervising others, undertaking research and attending conferences etc.
• Finance – Because of their deepened analytical skills, it’s no surprise that many PostDocs choose a career in finance. This involves working for some of the most significant players in the financial district in prime locations including London, Frankfurt and Hong Kong. Specific job titles can include Actuarial, Investment Analyst or Risk Modeller.
• Computer Programming – Some students whose research involves computational mathematics launch their career as a computer programmer. Due to their background, they’ll typically work on specialised projects which require high levels of understanding on the problem at hand. For example, they may work with physicists and biomedical engineers to develop a software package that supports their more complex research.
• Data Analyst – Those who enjoy number crunching and developing complex models often go into data analytics. This can involve various niches such as forecasting or optimisation, across various fields such as marketing and weather.

What Are Some of The Typical Employers Who Hire Maths PostDocs?

As mentioned above, there’s a high demand for skilled mathematicians and statisticians across a broad range of sectors. Some typical employers are:

• Education – All UK and international universities
• Governments – STFC and Department for Transport
• Healthcare & Pharmaceuticals – NHS, GSK, Pfizer
• Finance & Banking – e.g. Barclays Capital, PwC and J. P. Morgan
• Computing – IBM, Microsoft and Facebook
• Engineering – Boeing, Shell and Dyson

The above is only a small selection of employers. In reality, mathematic PostDocs can work in almost any industry, assuming the role is numerical-based or data-driven.

How Much Can You Earn with A PhD in Maths?

As a mathematics PhD PostDoc, your earning potential will mostly depend on your chosen career path. Due to the wide range of options, it’s impossible to provide an arbitrary value for the typical salary you can expect.

However, if you pursue one of the below paths or enter their respective industry, you can roughly expect to earn [3] :

Academic Lecturer

• Approximately £30,000 – £35,000 starting salary
• Approximately £40,000 with a few years experience
• Approximately £45,000 – £55,000 with 10 years experience
• Approximately £60,000 and over with significant experience and a leadership role. Certain academic positions can earn over £80,000 depending on the management duties.

Actuary or Finance

• Approximately £35,000 starting salary
• Approximately £45,000 – £55,000 with a few years experience
• Approximately £70,000 and over with 10 years experience
• Approximately £180,000 and above with significant experience and a leadership role.

Aerospace or Mechanical Engineering

• Approximately £28,000 starting salary
• Approximately £35,000 – £40,000 with a few years experience
• Approximately £60,000 and over with 10 years experience

Data Analyst

• Approximately £45,000 – £50,000 with a few years experience
• Approximately £90,000 and above with significant experience and a leadership role.

Again, we stress that the above are indicative values only. Actual salaries will depend on the specific organisation and position and responsibilities of the individual.

Facts and Statistics About Maths PhD Holders

The below chart provides useful insight into the destination of Math PostDocs after completing their PhD. The most popular career paths from other of highest to lowest is education, information and communication, finance and scientific research, manufacturing and government.

Note: The above chart is based on ‘UK Higher Education Leavers’ data [2] between 2012/13 and 2016/17 and contains a data size of 200 PostDocs. The data was obtained from the Higher Education Statistics Agency ( HESA ).

Which Noteworthy People Hold a PhD in Maths?

Alan turing.

Alan Turing was a British Mathematician, WW2 code-breaker and arguably the father of computer science. Alongside his lengthy list of achievements, Turning achieved a PhD in Mathematics at Princeton University, New Jersey. His thesis titled ‘Systems of Logic Based on Ordinals’ focused on the concepts of ordinal logic and relative computing; you can read it online here . To this day, Turning pioneering works continues to play a fundamental role in shaping the development of artificial intelligence (AI).

Ruth Lawrence

Ruth Lawrence is a famous British–Israeli Mathematician well known within the academic community. Lawrence earned her PhD in Mathematics from Oxford University at the young age of 17! Her work focused on algebraic topology and knot theory; you can read her interesting collection of research papers here . Among her many contributions to Maths, her most notable include the representation of the braid groups, more formally known as Lawrence–Krammer representations.

Emmy Noether

Emmy Noether was a German mathematician who received her PhD from the University of Erlangen, Germany. Her research has significantly contributed to both abstract algebra and theoretical physics. Additionally, she proved a groundbreaking theorem important to Albert Einstein’s general theory of relativity. In doing so, her theorem, Noether’s theorem , is regarded as one of the most influential developments in physics.

Other Useful Resources

Institute of Mathematics and its Applications (IMA) – IMA is the UK’s professional body for mathematicians. It contains a wide range of useful information, from the benefits of further education in Maths to details on grants and upcoming events.

Maths Careers – Math Careers is a site associated with IMA that provides a wide range of advice to mathematicians of all ages. It has a section dedicated to undergraduates and graduates and contains a handful of information about progressing into research.

Resources for Graduate Students – Produced by Dr Mak Tomford, this webpage contains an extensive collection of detailed advice for Mathematic PhD students. Although the site uses US terminology in places, don’t let that put you off as this resource will prove incredibly helpful in both applying to and undertaking your PhD.

Student Interviews – Still wondering whether a PhD is for you? If so, our collection of PhD interviews would be a great place to get an insider perspective. We’ve interviewed a wide range of PhD students across the UK to find out what doing a PhD is like, how it’s helped them and what advice they have for other prospective students who may be thinking of applying to one. You can read our insightful collection of interviews here .

[1] Universities used to determine the typical (median) and range of entry requirements and tuition fees for 2020/21 Mathematics PhD positions.

• http://www.lse.ac.uk/study-at-lse/Graduate/Degree-programmes-2020/MPhilPhD-Mathematics
• https://www.ox.ac.uk/admissions/graduate/courses/dphil-mathematics?wssl=1
• https://www.graduate.study.cam.ac.uk/courses/directory/mapmpdpms
• https://www.ucl.ac.uk/prospective-students/graduate/research-degrees/mathematics-mphil-phd
• http://www.bristol.ac.uk/study/postgraduate/2020/sci/phd-mathematics/
• https://www.surrey.ac.uk/postgraduate/mathematics-phd
• https://www.maths.ed.ac.uk/school-of-mathematics/studying-here/pgr/phd-application
• https://www.lancaster.ac.uk/study/postgraduate/postgraduate-courses/mathematics-phd/
• https://www.sussex.ac.uk/study/phd/degrees/mathematics-phd
• https://www.manchester.ac.uk/study/postgraduate-research/programmes/list/05325/phd-pure-mathematics/
• https://warwick.ac.uk/study/postgraduate/research/courses-2020/mathematicsphd/
• https://www.exeter.ac.uk/pg-research/degrees/mathematics/

[2] Higher Education Leavers Statistics: UK, 2016/17 – Outcomes by subject studied – https://www.hesa.ac.uk/news/28-06-2018/sfr250-higher-education-leaver-statistics-subjects

[3] Typical salaries have been extracted from a combination of the below resources. It should be noted that although every effort has been made to keep the reported salaries as relevant to Math PostDocs as possible (i.e. filtering for positions which specify a PhD qualification as one of their requirements/preferences), small inaccuracies may exist due to data availability.

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Pure Mathematics Research

Pure mathematics fields.

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

Pure Math Committee

Dissertations

Most Harvard PhD dissertations from 2012 forward are available online in DASH , Harvard’s central open-access repository and are linked below. Many older dissertations can be found on ProQuest Dissertation and Theses Search which many university libraries subscribe to.

Mathematics (PhD)

Program at a glance.

• In State Tuition
• Out of State Tuition

Learn more about the cost to attend UCF.

The Mathematics PhD program prepares students with a broad base in pure, applied and industrial mathematics.

The Doctor of Philosophy degree in Mathematics is intended to provide a broad base in applied and industrial mathematics. The goal of the program is to produce students who will attain distinction in their fields of research. In order to achieve this, the program has required core courses as well as a set of electives providing cross-disciplinary subjects.

Students in the program can specialize in one of many aspects of mathematics, including Approximation Theory, Applied and Computational Harmonic Analysis, Big Data and Mathematical Statistics, Combinatorics and Graph Theory, Commutative Algebra and Algebraic Geometry, Control and Optimization, Differential and Symplectic Geometry, Fluid and Plasma Dynamics, Functional Analysis, Inverse and Ill-posed Problems, Mathematical Biology, Mathematical Finance, Nonlinear Waves and Nonlinear Dynamics, Numerical Analysis, Orthogonal Polynomials, Partial Differential Equations, Probability and Stochastic Analysis, and Tomography and Medical Imaging. Responding to this wide variety of interests, the program offers flexibility in the composition of the core courses as well as the candidacy examination. The program is comprehensive with opportunities for students to pursue research in a variety of disciplines.

The Mathematics PhD program consists of at least 75 credit hours of course work beyond the bachelor's degree, of which a minimum of 39 hours of formal course work, exclusive of independent study, and 15 credit hours of dissertation research (7980) are required. The program requires 18 credit hours of core courses, and 6 to 12 credit hours in two 2-semester sequences.

Total Credit Hours Required: 75 Credit Hours Minimum beyond the Bachelor's Degree

Program Tracks/Options

• Mathematics (PHD) - Financial Mathematics Track

Application Deadlines

• International

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Program Prerequisites

Bachelor's degree in related field.

Students entering the graduate program with regular status are assumed to have a working knowledge of undergraduate calculus, differential equations, linear algebra (or matrix theory), and maturity in the language of advanced calculus (at the level of MAA 4226).

Degree Requirements

Required courses.

• All students are required to complete the following courses with grade of "B" or better.
• MAA5237 - Mathematical Analysis (3)
• MAT5712 - Scientific Computing (3)
• MAS5145 - Advanced Linear Algebra and Matrix Theory (3)
• MAP6385 - Applied Numerical Mathematics (3)
• MAA6306 - Real Analysis (3)
• MAA6405 - Complex Variables (3)
• MAP5336 - Ordinary Differential Equations and Applications (3)
• MAA6506 - Functional Analysis (3)
• MAP6111 - Mathematical Statistics (3)

Restricted Electives

• All students are required to complete two 2-semester sequences. Sequences are pairs of related courses that give advanced knowledge in an area of mathematics. Each sequence must be approved by the dissertation adviser, dissertation committee, and the graduate program director. The following shows examples of acceptable sequences using current courses. We expect that other sequences will be developed as our program grows. Note that some sequences consist of a core course plus one elective, while others consist of two electives. Thus, the credit hours in this requirement are variable (6 to 12 credit hours). A written examination on two such sequences will be required as part of the candidacy examination (see more details in Candidacy Examination section
• MAA6404 - Complex Analysis (3)
• MAD5205 - Graph Theory I (3)
• MAP6356 - Partial Differential Equations (3)
• MAA6238 - Measure and Probability I (3)
• MAA6245 - Measure and Probability II (3)
• MAA7239 - Asymptotic Methods in Mathematical Statistics (3)
• MAS5311 - Algebra I (3)
• MAS6312 - Algebra II (3)
• MAP6195 - Mathematical Foundations for Massive Data Modeling and Analysis (3)
• MAP6197 - Mathematical Introduction to Deep Learning (3)
• MTG5256 - Differential Geometry (3)
• MTG6345 - Algebraic Topology (3)

Unrestricted Electives

• Earn at least 30 credits from the following types of courses: Electives are chosen in consultation with the student's advisory committee and may be chosen from the suggested options: Discrete Mathematics, General Applied Mathematics, Mathematical Computer Tomography, Image Processing and Computer Graphics, Mathematical Finance, Mathematical Physics, Pure Mathematics, and Mathematical Statistics. A list of elective course options can be obtained from the graduate program director. Courses taken outside the Mathematics department must be approved by the adviser and graduate program director. These courses are selected in consultation with the student's advisory committee.

Dissertation

• Earn at least 15 credits from the following types of courses: XXXX 7980 Dissertation Research 15 Credit Hours (minimum)

Examinations

Qualifying examination.

• The qualifying/comprehensive examination is based on the core course work (MAA 5237 and MAS 5145 - Advanced Linear Algebra and Matrix Theory). To continue in the PhD program students must pass the examination at the PhD level. Two attempts are permitted. The examination will be administered twice a year: one in the Fall semester and the other in the Spring semester. To take the examination, students must have earned a "B" or better in each core course, must have a minimum grade point average of 3.0 (out of 4.0) in the program, or must obtain permission from the graduate program director. Students will normally take the examination after taking the core courses MAA 5237 and MAS 5145, and are expected to have passed it by the end of the second year of study unless a written request for a postponement has been approved by the Graduate Committee at least two months before the examination date. The student must pass the Qualifying Examination in at most two attempts. It is strongly recommended that the student select a dissertation adviser by the completion of 18 credit hours of course work, and it is strongly recommended that the student works with the dissertation adviser to form a dissertation committee within two semesters of passing the Qualifying Examination.

Candidacy Examination

• The Candidacy Examination consists of a written examination based on the materials from two of the selected two-semester sequence courses taken by the students beyond the core courses on Mathematical Analysis and Advanced Linear Algebra (MAA 5237, MAS 5145). A committee formed or selected by the Graduate Committee or the graduate program director is responsible for preparing and grading the written examinations. After passing the candidacy examination and meeting other requirements, the student can register for Doctoral Dissertation (MAP 7980 or MAA 7980). A minimum of 15 Doctoral Dissertation credit hours are required. The Candidacy Examination can be attempted after passing the qualifying examination. The Candidacy Examination must be completed within three years after passing the qualifying examination. A student must successfully pass the Candidacy Examination within at most two attempts.

Admission to Candidacy

• The following are required to be admitted to candidacy and enroll in dissertation hours: Completion of all course work, except for dissertation hours. Successful completion of the candidacy examination. The dissertation advisory committee is formed, consisting of approved graduate faculty and graduate faculty scholars. Submittal of an approved program of study.

Dissertation Proposal Examination

• After passing the candidacy examination, the student will prepare a dissertation proposal and orally present it to the dissertation advisory committee for approval. The proposal will include a description of the research performed to date and an agenda for the research planned to be completed for the dissertation. In addition to standards of correctness, indicating a suitable level of mastery of the material of the area of the dissertation, and suitability of the proposed dissertation topic, the presentation must meet current standards for professional presentations within the discipline of mathematics. For the successful completion of the Dissertation Proposal Examination the presentation must be judged as passing the requirements for the examination by the majority of the dissertation committee. This exam must be passed within 18 months of passing the candidacy examination and not later than the end of the sixth year of graduate study. A candidate must pass this examination within at most two attempts.

Dissertation Defense

• Upon completion of a student's research, the student's committee schedules an oral defense of the dissertation. Most students complete the program within five years after obtaining their bachelor's degree. Students are expected to complete the dissertation in no more than seven years from the date of admission to the program.

Independent Learning

• The required 15 credit hours of dissertation will provide ample opportunities for students to gain the independent learning experience through studying published research papers and deriving, on their own, new and meaningful research results.

Grand Total Credits: 75

Application requirements, financial information.

Graduate students may receive financial assistance through fellowships, assistantships, tuition support, or loans. For more information, see the College of Graduate Studies Funding website, which describes the types of financial assistance available at UCF and provides general guidance in planning your graduate finances. The Financial Information section of the Graduate Catalog is another key resource.

Fellowship Information

Fellowships are awarded based on academic merit to highly qualified students. They are paid to students through the Office of Student Financial Assistance, based on instructions provided by the College of Graduate Studies. Fellowships are given to support a student's graduate study and do not have a work obligation. For more information, see UCF Graduate Fellowships, which includes descriptions of university fellowships and what you should do to be considered for a fellowship.

The department offers over 20 Graduate Teaching Assistantships every year on a competitive basis. A few Graduate Research Assistantships are also available for qualified students.

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What Jobs Can You Get With A Maths Degree? - A New Scientist Careers Guide

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What Jobs Can You Get With A Maths Degree?

“What can I do with a maths degree?” you may wonder. How are algebra, imaginary numbers, calculus and other abstract mathematical concepts going to help you get a job tackling real-world problems? Indeed, many fail to see the relevance of mathematics in their day-to-day lives.

Nevertheless, mathematics is arguably the most fundamental scientific discipline, often referred to as the language of science. Mathematical principles are applied in virtually all aspects of modern civilisations, from advanced technologies such as artificial intelligence (AI) to everyday arithmetic. Maths graduates are highly employable due to their excellent numerical, analytical, critical thinking and problem-solving skills.

Popular areas in which you can apply different types of maths include statistics and data analysis , business and finance, IT and engineering , and teaching and research . This article outlines the top three highest paying graduate jobs in each of these four categories.

Statistics & Data Analysis

Statistics and d ata Analysis involve the collection, analysis, interpretation and visualisation of data, as well as prediction of outcomes. This discipline plays a vital role in decision-making, risk assessment and trend prediction in various sectors, such as healthcare, technology, business and research, with no shortage of jobs for mathematicians .

• Statistician

Job role: Statisticians work in various fields that generate large amounts of data. They may perform statistical tests on population datasets for a government to find trends and patterns; they may analyse results from clinical trials for pharmaceutical companies; or they may advise companies on new products based on consumer habits.

Route: Mathematics, economics, psychology or other degrees with high-level statistics will equip you with the necessary skills. Employers expect you to have gained exposure to the relevant industry through internships or postgraduate courses. With experience, you could manage statistical departments, become a consultant or learn advanced data science skills and become a data scientist.

Average salary (experienced): £62,000  

• Operational Researcher

Job role: Operational researchers collect and analyse data to evaluate how organisations function, identify areas for improvement and assess whether implemented changes have made a significant difference. Your job will also include observation of staff and interviewing them to obtain qualitative data.

Route: You usually need a degree in maths, computing or economics to apply for junior positions. Some employers may additionally require a postgraduate qualification in a subject such as management science. With experience, you could move into consultancy or become a project or operations manager.

Average salary (experienced): £60,000  

• Market Research Data Analyst

Job role: This is a more specialised version of a statistician and involves analysis of data obtained through surveys. You advise researchers on how to optimise their research design to gather high-quality data. High-level statistical tools will be at your disposal to perform data analysis and present your findings.

Route: A degree in maths and statistics, business and marketing, or data science is desirable. Postgraduate qualification will further enhance your employability, especially if you wish to work in sectors such as medicine or economic market research. With experience, you could become an independent market research consultant who advises industrial or research organisations.

Average salary (experienced): £60,000

Business and Finance

Some of the highest paying jobs with a maths degree can be found in the business and finance sector. Mathematical concepts such as algebra and statistics are used for various purposes, including risk assessment, quantitative analysis and financial modelling .  

• Chief Executive

Job role: Chief executive officers (CEOs) lead businesses and organisations, ensuring they run successfully. CEOs have several responsibilities, including policy implementation, setting the company’s agenda and devising strategies to meet it, and managing relationships with business partners.

Route: Although you don’t strictly require a degree to reach this role, rising competition means academic qualifications will put you in a strong position. Maths degrees provide you with the invaluable numerical and problem-solving skills required to be a successful executive or CEO. 

Additionally, you will have to acquire in-depth knowledge of business and the industry you wish to work in through years of industry experience, postgraduate courses also help

Average salary (experienced): £120,000  

Job role: Actuaries advise organisations on long-term financial costs and investment risks. This helps companies plan their business strategies and find solutions to economic issues. You may work in an office, at a client’s workplace or from home.

Route: Most junior jobs require a degree in maths, actuarial science, accounting or economics, along with work experience in industry. You usually train towards this role while working at an actuarial company; postgraduate courses may help you become an actuary faster. Specialising in fields such as data science, banking or insurance will enhance your job prospects.

Average salary (experienced): £70,000  

• Chartered Accountant

Job role: Accountants help individuals or businesses manage their money. They prepare financial statements, make cash-flow forecasts and advise on spending, costs, profits and taxes.

Route: You typically need a degree in accountancy, finance, business or maths followed by a graduate training scheme that provides you with a professional accountancy qualification. There are different professional bodies that you can train with, such as the Chartered Institute of Management Accountants. CIMA allows you to work privately once you are registered as a ‘member in practice’.

With experience, you could manage your own practice, start a company or become an auditor or a financial director.

Average salary (experienced): £65,000

IT & Engineering

IT and engineering are often viewed as applied mathematics and physics . Algebra, logic, arithmetic and geometry are essential for building machines, software and networks. As technologies are becoming increasingly complex, for example with the introduction of AI, individuals with advanced maths skills and knowledge are needed to solve modern engineering problems.  

• IT Systems Architect

Job role: IT architects are crucial to a business’ successful functioning. They plan and develop IT systems and software to meet their clients’ technical requirements​​. You may work at your own office, a client’s office or from home.

Route: You usually require a degree in computer science , software engineering or mathematics. With experience, you could take on managerial roles , become a strategy business planner or specialise in specific areas, such as finance or healthcare.

Average salary (experienced): £90,000  

• Machine Learning Engineer

Job role: Machine learning (ML) is a branch of data science and serves as a tool to automate tasks and algorithms, minimising constant human input. ML engineers develop algorithms and models that can be fed large datasets and, on their own, learn patterns and make predictions without explicit coding.

Route: A degree in maths, computer science, engineering or other numerate subject is usually required. As it is a highly advanced field of data science, many ML engineers will have a postgraduate qualification in ML and AI technologies. With experience, you could take on managerial roles in big companies, move into academia or start your own business.

Average salary (experienced): £82,500  

• Air Accident Investigator

Job role: Air accident investigators find the cause of major accidents involving civilian aircraft by applying mathematical and engineering principles. You may dismantle and reassemble wreckage, gather evidence and interview victims and witnesses. It is a demanding job requiring you to work in different settings, including remote areas, aircraft hangars and labs.

Route: You must have several years of experience in aerospace engineering before applying for this role. This entails a degree in aeronautical engineering, mechanical engineering , electrical engineering , physics or maths; sometimes you must also hold a pilot’s licence. A postgraduate qualification is highly desirable.

With substantial experience, you may be promoted to chief accident inspector, or move into consultancy and work with aerospace manufacturers, safety regulators or insurance companies.

Average salary (experienced): £82,000

Pure Mathematics & Science

Mathematical principles help us learn more about our world and new scientific observations, in turn, require maths to make sense of them. Traditional mathematicians are therefore some of the frontrunners in scientific innovation, undertaking cutting-edge research in fields such as cryptography, AI and quantum mechanics . Often researchers will also have a passion for teaching, which can be highly lucrative.  

• Headteacher

Job role: Headteachers manage and represent a school, ensuring it runs smoothly. They set the school’s values and vision, continuously improve the school environment and quality of teaching, maintain health and safety standards, control finances and engage with students, staff, parents and sometimes politicians. They may also continue to teach their usual subjects if they wish.

Route: With a maths degree, you could start working as a maths teacher once you have achieved qualified teacher status. After you have worked as a teacher, try to take on senior roles within your school; you may wish to complete the National Professional Qualification for Senior Leadership or Headship. Once you have several years of experience in a senior position, you could become a headteacher. 

Average salary (experienced): £131,000  

Job role: Astronomy can be classified as either observational or theoretical. Observational astronomers analyse images from satellites, spacecraft and telescopes, while theoretical astrophysicists test theories and make predictions using advanced computing methods. You could work at university, an observatory or in a lab.

Route: Start with completing a degree in physics, astrophysics or maths, and gain relevant work experience. To become a senior astrophysicist and lecturer, complete a PhD in astrophysics and aim to publish several research papers. You could also move into industrial sectors such as satellite or aerospace development and, after gaining experience, take on managerial roles.

• University Professor

Job role: Teaching maths or related scientific subjects is a highly prestigious career , especially if you work at the best universities for maths in the UK, such as Cambridge or Oxford. University professors are usually world-leading experts in their field.

Route: You will first need to study maths or other highly numerical disciplines followed by a master’s and a PhD. After working as a postdoc and publishing several research articles, you could get promoted to a professor. You will typically conduct research in and teach specific topics, such as calculus, cryptography or algebra.

Average salary (experienced): £55,000

Careers in maths are incredibly diverse and lucrative, thanks to the invaluable skills, aptitude and knowledge maths graduates gain. With expertise in problem-solving, data analysis and critical thinking, mathematicians are highly sought after professionals, commanding high-paying positions in finance, technology, research and beyond.

• Explore careers | National Careers Service [Internet]. Available from: https://nationalcareers.service.gov.uk/explore-careers
• Career development [Internet]. RSS. Available from: https://rss.org.uk/jobs-careers/career-development/
• About OR - the OR society [Internet]. Available from: https://www.theorsociety.com/about-or/
• Careers in Market Research | MRS Job Insights & Guide | Market Research Society [Internet]. Market Research Society. Available from: https://www.mrs.org.uk/resources/career-support
• Become an Actuary | Institute and Faculty of Actuaries [Internet]. Institute and Faculty of Actuaries. Available from: https://actuaries.org.uk/qualify/become-an-actuary
• Students [Internet]. Membership | AICPA & CIMA. Available from: https://www.aicpa-cima.com/resources/landing/students
• Get into tech: How to launch a career in IT | BCS [Internet]. Available from: https://www.bcs.org/it-careers/get-into-tech-how-to-build-a-career-in-it/
• Institute of Analytics - The Future is Here! [Internet]. IoA - Institute of Analytics. Available from: https://ioaglobal.org/
• Working for AAIB [Internet]. GOV.UK. 2014. Available from: https://www.gov.uk/government/organisations/air-accidents-investigation-branch/about/recruitment
• Get into teaching | Get into teaching GOV.UK [Internet]. Get Into Teaching. Available from: https://getintoteaching.education.gov.uk/
• Home | Advance HE [Internet]. Available from: https://www.advance-he.ac.uk/
• Careers | The Royal Astronomical Society [Internet]. Available from: https://ras.ac.uk/education-and-careers/careers

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Master of Science in Mathematics and Applied Statistics

At California State University Long Beach

The Department of Mathematics & Statistics at CSULB offers four Master of Science programs .

Teaching & Graduate Assistantships provide students with funding and with college teaching experience.

Graduates have found employment in both technical and academic workplaces. Many have obtained tenure-track community college professorships. Others have gone on to PhD programs.

MS in Mathematics, General Option

Study and explore concepts in areas including analysis, algebra, topology, and geometry, as well as the   deep connections between and among these subjects.

MS in Mathematics, Option in Applied Mathematics

Study applied math methods with an emphasis on computational skills.

MS in Mathematics, Option in Mathematics Education for Secondary School Teachers

A flexible program that includes coursework in mathematics and in mathematics education research & theory.

MS in Applied Statistics

Using conceptual foundations and statistical software packages ( SAS , R , and Python ), students are trained to analyze real world data appropriately and communicate their findings effectively. The tools learnt here will open the door for careers in data science and analytics, or prepare you for a PhD in a variety of related fields.

More Information & Application Instructions

Apply for Teaching and Graduate Assistantships

Contact Us with Questions

Dr. John Brevik, Pure Mathematics Graduate Advisor, [email protected]

Dr. Paul Sun, Applied Math Graduate Advisor, [email protected]

Dr. Xuhui Li, Mathematics Education Graduate Advisor, [email protected]

Dr. Kagba Suaray, Statistics Graduate Advisor, [email protected]

Department of Mathematics

Innovation in cancer treatment and mathematics: SciRIS awardees lead the way

SciRIS awards bolster essential research endeavors, such as the investigation of human cancer cells (pictured above).

Collaborative science has the power to change the world. The 2024 College of Science Research and Innovation Seed (SciRIS) award recipients aim to use that power to develop better treatments for cancer and unlock the mysteries of complex mathematical equations.

The SciRIS program funds projects based on collaborative research within the College of Science community and beyond. There are two tracks through the program: SciRIS (Stages 1-3) and the SciRIS individual investigator award (SciRIS-ii).

SciRIS Stages 1-3 funds teams in three stages of increasing funding to support training, research and capacity-building, accelerating work toward external funding opportunities. SciRIS-ii funds individual faculty to establish research relationships with external partners, enabling them to demonstrate the feasibility of their ideas and quickening the pace of scientific discovery.

SciRIS Stage 1

Professor Claudia Maier, alongside a multidisciplinary team including researchers from the Colleges of Engineering and Agricultural Science, received a SciRIS Stage 1 award to study on triple-negative breast cancer.

Maier’s team includes two other College of Science researchers, Yanming Di from the Department of Statistics and Chad Giusti from the Department of Mathematics.

In biology, cells exhibit a range of diverse characteristics known as cellular heterogeneity, regardless if the overall biology appears uniform. This diversity influences disease progression, treatment outcomes and the likelihood of disease recurrence. Single-cell proteomics is an emerging technique that allows researchers to study these differences at the individual cell level.

Collaborating with faculty from the College of Engineering and the College of Agricultural Science, the team aims to refine a single-cell mass spectrometry workflow focusing on triple-negative breast cancer and specifically targeting therapy-induced senescent cells. Senescent cells eventually stop multiplying but don’t die off, leading to the continued release of chemicals that can trigger inflammation and damage healthy cells. This research builds upon previous work and collaboration, moving from technology development to practical application in biomedicine.

By understanding the heterogeneity within breast cancer and the role of senescent cells in treatment resistance, the researchers aim to develop methods for detecting and characterizing TIS cells from tissue samples. This information will be crucial for developing treatments that target these cells, potentially improving outcomes for TNBC patients.

Kyriakos Stylianou

SciRIS-ii (Individual Investigator)

The following three scientists received SciRIS-ii awards: Kyriakos Stylianou, Christine Escher and Xueying Yu.

Materials scientist Kyriakos Stylianou will use his SciRISii award to study a new, more efficient way to diagnose and treat cancer using advanced technology that combines imaging and therapy in one tiny package.

Theranostics is a novel cancer approach that uses radiotracers, compounds made of radiation and chemicals that selectively bind to a specific target in the body. The tracers identify and then deliver radioactive drug therapy to the tumor, resulting in better outcomes and personalized treatments.

Stylianou will explore using metal-organic frameworks to build the nanoparticles. His research will also look at utilizing boron neutron capture therapy, a promising approach to cancer treatment that results in minimal consequences to normal cells.

By combining gadolinium for imagining and carborane-based ligands—which include boron—for therapy, the MOF would be able to diagnose and treat cancer after being activated specifically in tumor microenvironments.

The successful demonstration of the theranostic capabilities of the MOFs in lab settings will mark the initial phase towards more complex studies conducted in living organisms.

Christine Escher

Mathematics Professor Christine Escher will use her SciRISii award to delve into Global Riemannian geometry, a field studying the relationship between local and global geometric properties of space. Specifically, the focus is on understanding manifolds with lower curvature bounds by exploring symmetries.

Escher will be continuing to collaborate with Catherine Searle from Wichita State University, to achieve a comprehensive classification of such manifolds, contributing to a deeper understanding of Riemannian geometry.

Escher will be attending a semester-long program at the Mathematical Sciences Research Institute in Berkeley entitled, “New Frontiers in Curvature: Flows, General Relativity, Minimal Submanifolds and Symmetry.” This opportunity facilitates collaboration and provides access to specialized resources. One of Escher’s Ph.D. students, Augustin Bosgraaf, will also participate in the program, further enhancing the mentorship and educational aspects of this research endeavor.

Assistant Professor of Mathematics Xueying Yu received a SciRISii grant to understand the behavior of dispersive equations, which are fundamental in describing various natural phenomena such as light transmission, charge transport in DNA and particle interaction in atoms. While these equations are widely used across physics and biology, their long-term behavior remains largely unexplored.

Collaborating with researchers at the University of Bologna in Italy, the University of New York at Binghamton and Massachusetts Institute of Technology, Yu will focus on equations with variable coefficients which are more complex to analyze. The project aims to develop theories and tools to understand the long-term behavior of these variable coefficient dispersive equations, focusing on aspects like global well-posedness, scattering effects and unique continuation of solutions.

This project will not only contribute to advancing mathematical understanding but also have practical implications in various fields such as numerical simulations, optics, condensed matter, fluid mechanics and biology.

Read more stories about: news , faculty and staff , chemistry , mathematics , statistics , biomedical science , research , awards & recognition , interdisciplinary

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Reinke Is Appointed Chair of Genetics at Yale School of Medicine

Valerie Reinke, PhD, professor of genetics, has been appointed chair of the Department of Genetics at Yale School of Medicine (YSM), effective today. For the past six years, Reinke has served as vice-chair of the department, and since July, she has held the role of interim chair.

Reinke received an undergraduate degree in genetics from the University of Illinois, Champaign-Urbana, and a PhD in biomedical sciences from the University of Texas Health Science Center. After completing a postdoctoral fellowship at Stanford University, she joined the YSM Department of Genetics in 2000. Since then, her lab has advanced genomic technologies to discover novel and fundamental gene regulatory mechanisms that direct germ cell function, organization, and development to understand how genome organization and composition influence complex tissue-specific regulatory programs, using the model organism C. elegans . In addition, she has long been a leader in multiple genome consortia focused on comprehensive identification of functional sequence elements to define global gene regulatory networks in both C. elegans and Drosophila. Inside and beyond the lab, Reinke has a demonstrated commitment to training and mentorship, and as chair, she will continue to cultivate collaborations across YSM to facilitate research projects, funding, and training opportunities to foster outstanding science that leads to fundamental discoveries in basic and translational genetic research. She is committed to promoting the success of faculty in the department and to increasing scientific collaboration among sections within the department. Across these efforts, she is committed to integrating diversity, equity, and inclusion into departmental practices and to creating a culture of support, mentorship, and sponsorship.

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• Valerie Reinke, PhD Professor of Genetics; Chair, Genetics

Doctoral Candidate Presents Dissertation Findings at National Conference

Karmen Yu’s research addresses the question: How do undergraduate Calculus I students experience and navigate their learning of calculus in the parallel spaces of coursework and inquiry-oriented complementary instruction?

Posted in: Research Presentations

Doctoral candidate Karmen Yu recently presented findings from her dissertation study at the annual Research in Undergraduate Mathematics Education conference in Omaha, NE. Karmen’s talk, entitled Case Studies of Undergraduate Students’ Agentive Participation in the Parallel Spaces of Calculus I Coursework and Peer-Led, Inquiry-Oriented, Complementary Instruction.  She shared findings from one case study that included characterizations of the different forms of agentive participation afforded to students in each of the two spaces, as well as their complementary nature relative to learning calculus with understanding. It was a fantastic presentation. Karmen’s advisor, Dr. Steven Greenstein, was a contributor to the presentation and was there to support her. Great work, Karmen!

Best Global Universities for Engineering in Russia

These are the top universities in Russia for engineering, based on their reputation and research in the field. Read the methodology »

To unlock more data and access tools to help you get into your dream school, sign up for the  U.S. News College Compass !

Here are the best global universities for engineering in Russia

Itmo university, tomsk state university, tomsk polytechnic university, lomonosov moscow state university, novosibirsk state university, saint petersburg state university, peter the great st. petersburg polytechnic university, moscow institute of physics & technology, national research nuclear university mephi (moscow engineering physics institute).

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• # 307 in Best Universities for Engineering  (tie)
• # 696 in Best Global Universities  (tie)
• # 364 in Best Universities for Engineering  (tie)
• # 587 in Best Global Universities  (tie)
• # 396 in Best Universities for Engineering  (tie)
• # 879 in Best Global Universities  (tie)
• # 632 in Best Universities for Engineering  (tie)
• # 355 in Best Global Universities
• # 809 in Best Universities for Engineering  (tie)
• # 579 in Best Global Universities  (tie)
• # 847 in Best Universities for Engineering  (tie)
• # 652 in Best Global Universities
• # 896 in Best Universities for Engineering  (tie)
• # 679 in Best Global Universities  (tie)
• # 902 in Best Universities for Engineering  (tie)
• # 475 in Best Global Universities  (tie)
• # 915 in Best Universities for Engineering  (tie)
• # 483 in Best Global Universities  (tie)

100 Best universities for Mechanical Engineering in Russia

Updated: February 29, 2024

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Below is a list of best universities in Russia ranked based on their research performance in Mechanical Engineering. A graph of 714K citations received by 136K academic papers made by 158 universities in Russia was used to calculate publications' ratings, which then were adjusted for release dates and added to final scores.

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

1. Moscow State University

For Mechanical Engineering

2. Tomsk State University

3. St. Petersburg State University

4. Bauman Moscow State Technical University

5. Ufa State Aviation Technical University

6. Peter the Great St.Petersburg Polytechnic University

7. Tomsk Polytechnic University

8. Ural Federal University

9. South Ural State University

10. National Research University Higher School of Economics

11. Moscow Aviation Institute

12. Novosibirsk State University

13. ITMO University

14. N.R.U. Moscow Power Engineering Institute

15. National Research Nuclear University MEPI

16. Kazan Federal University

17. National University of Science and Technology "MISIS"

18. Moscow Institute of Physics and Technology

19. Samara National Research University

20. Moscow State Technological University "Stankin"

21. Novosibirsk State Technical University

22. RUDN University

23. Southern Federal University

24. Saratov State University

25. Ufa State Petroleum Technological University

26. Samara State Technical University

27. Siberian Federal University

28. Kazan National Research Technical University named after A.N. Tupolev - KAI

29. Perm State Technical University

30. Omsk State Technical University

31. Saint Petersburg State Electrotechnical University

32. Moscow Polytech

33. Saint-Petersburg Mining University

34. Magnitogorsk State Technical University

35. Saratov State Technical University

36. Moscow State University of Railway Engineering

37. Lobachevsky State University of Nizhni Novgorod

38. Nizhny Novgorod State Technical University

39. Tula State University

40. Belgorod State Technological University

41. Far Eastern Federal University

42. Novgorod State University

43. belgorod state university.

44. Finance Academy under the Government of the Russian Federation

45. Moscow Medical Academy

46. Kazan State Technological University

47. Russian State University of Oil and Gas

48. siberian state aerospace university.

49. Tambov State Technical University

50. Voronezh State University

51. Siberian State Industrial University

52. Saint Petersburg State Institute of Technology

53. Kalashnikov Izhevsk State Technical University

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

55. Mendeleev University of Chemical Technology of Russia

56. Murmansk State Technical University

57. South-Western State University

58. Ogarev Mordovia State University

59. Tomsk State University of Control Systems and Radioelectronics

60. south-russian state university of economics and service.

61. Perm State University

62. Kuzbass State Technical University

63. Russian National Research Medical University

64. Plekhanov Russian University of Economics

65. Ulyanovsk State Technical University

66. Ulyanovsk State University

67. Penza State University

68. Kuban State University of Technology

69. Polzunov Altai State Technical University

70. Chelyabinsk State University

71. Yaroslavl State University

72. University of Tyumen

73. National Research University of Electronic Technology

74. Leningrad State University

75. Moscow State Pedagogical University

76. Udmurt State University

77. Irkutsk State University

78. North-Eastern Federal University

79. Bashkir State University

80. Russian Presidential Academy of National Economy and Public Administration

81. Kuban State University

82. Kuban State Agricultural University

83. St. Petersburg State University of Aerospace Instrumentation

84. Kemerovo State University

85. Immanuel Kant Baltic Federal University

86. Orenburg State University

87. Baltic State Technical University "Voenmeh"

88. Tomsk State University of Architecture and Building

89. Chuvash State University

90. ivanovo state power university.

91. Irkutsk National Research Technical University

92. Orel State University

93. State University of Management

94. Tomsk State Pedagogical University

95. Volgograd State University

96. Petrozavodsk State University

97. Tver State University

98. Northern Arctic Federal University

99. Omsk State Transport University

100. Kaliningrad State Technical University

The best cities to study Mechanical Engineering in Russia based on the number of universities and their ranks are Moscow , Tomsk , Saint Petersburg , and Ufa .

Engineering subfields in Russia

/images/cornell/logo35pt_cornell_white.svg" alt="math phd fields"> Cornell University --> Graduate School

Mathematics ph.d. (ithaca), field of study.

Mathematics

Program Description

The graduate program in the field of mathematics at Cornell leads to the Ph.D. degree, which takes most students five to six years of graduate study to complete. One feature that makes the program at Cornell particularly attractive is the broad range of  interests of the faculty . The department has outstanding groups in the areas of algebra, algebraic geometry, analysis, applied mathematics, combinatorics, dynamical systems, geometry, logic, Lie groups, number theory, probability, and topology. The field also maintains close ties with distinguished graduate programs in the fields of  applied mathematics ,  computer science ,  operations research , and  statistics and data science.

Ph.D. students in the field of mathematics may earn a Special Master's of Science in Computer Science .

The field also offers a math minor and a math concentration to students in certain fields.

Contact Information

316 Malott Hall Cornell University Ithaca, NY  14853

Concentrations by Subject

• mathematics

Visit the Graduate School's Tuition Rates page.

Application Requirements and Deadlines

Fall, December 15; no spring admission

Note: The Graduate Field of Mathematics will not admit doctoral students for Fall 2024.  Our program is bigger than its typical steady-state size: strong recruiting has caused a modest increase, while COVID-impact has lengthened the typical time-to-PhD.  We are committed to providing enhanced funding to our current students.  We will resume regular admissions next year for students to matriculate in Fall 2025. 

Requirements Summary:

Applicants must demonstrate mastery of the material required for an undergraduate major in mathematics. The mathematics field welcomes applications from and admits students with various mathematics backgrounds. 

• GRE General and Subject Test scores are not required and will not be considered.
• Detailed academic requirements can be located in the graduate field handbook
• All  Graduate School Requirements , including the  English Language Proficiency Requirement for all applicants
• 3 Letters of recommendation (5 letters allowed)

Learning Outcomes

The Ph.D. program in mathematics teaches you to create and communicate mathematics. The chief requirement for the doctoral degree is to complete under the guidance of an advisor a dissertation that makes an original and substantial contribution to its subject matter. You will be expected to disseminate the main results of your dissertation in the form of journal articles and conference presentations.

We do not require you to choose a specific research area or a dissertation advisor at the outset of your graduate education. The best way to make an informed choice of a research area and to make headway in it is to gain knowledge in a number of areas of mathematics. As a beginning student, you will be taking various required core courses in basic subjects. Furthermore, aside from providing research training, we prepare our students for careers as professional mathematicians in a variety of settings including academia, business, and government. Our aim, therefore, is to educate flexible and broadly knowledgeable mathematicians, and to this end, we offer besides the core courses a wide selection of advanced courses and seminars.

We will help you develop the oral and written communication skills expected of a professional mathematician. You will acquire these skills in part through courses and dissertation work. Active participation in our many seminars, several of which are targeted at students, is another way to improve your presentation skills and will also ease your transition from a learner to a researcher of mathematics. In addition, many practicing mathematicians are involved in teaching at some level, and that is why we require every student to undergo our teaching assistant training program and to participate in the teaching mission of the mathematics department.

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