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math phd curriculum

PhD Program

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

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

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

  • real analysis
  • geometry and topology
  • applied mathematics

Course Requirements for students starting prior to Autumn 2023

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

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

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

Course Requirements for students starting in Autumn 2023 and later

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

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

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

Years 4&5

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

Throughout the PhD Program

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

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

Department of Mathematics

Mathematics phd program.

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

Questions? Email [email protected]

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

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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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.

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.

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» » Graduate PhD Program

The Department of Mathematics offers a program leading to the degree of Doctor of Philosophy.

The PhD program is an intensive course of study designed for the full-time student planning a career in research and teaching at the university level or in quantitative research and development in industry or government. Admission is limited and highly selective. Successful applicants have typically pursued an undergraduate major in mathematics.

In the first year of PhD studies, students must pass written examinations in the areas of the basic . In the second year an oral examination on two selected topics must be passed. Subsequent years are devoted to seminars, research, and the preparation of a dissertation. Students are required to serve as a teaching assistant or instructor for four years beginning with the second year of study. All students must serve as a primary instructor for at least one semester; all others semesters students will serve as a teaching assistant. En route to the Ph.D., students will earn three degrees: a Master of Arts (after year one), a Master of Philosophy (after year four), and the Doctorate of Philosophy (after a successful thesis defense).

There are also allied doctoral programs in , , and .

The Mathematics Department is housed in a comfortable building containing an excellent , computing and printing facilities, faculty and graduate student offices, a lounge for tea and conversation, and numerous seminar and lecture rooms.

The department has a broad fellowship program designed to help qualified students achieve the PhD degree in the shortest practicable time. Each student admitted to the PhD program is appointed a fellow in the Department of Mathematics for a period of five years, contingent on good progress. A fellow receives a stipend for the nine-month academic year and is exempt from payment of tuition.

A fellow in the Department of Mathematics may hold a fellowship from a source outside Columbia University. When not prohibited by the terms of the outside fellowship, the University supplements the outside stipend to bring it up to the level of the University fellowship. Candidates for admission are urged to apply for fellowships for which they are eligible (e.g., National Science Foundation, Ford and Hertz Foundations).

All students admitted to the PhD program become fellows in the Department and are exempt from tuition. Students may be responsible for certain : a student activity fee and transcript fee.

Students in the PhD program are entitled to affordable University housing near the Department in Morningside Heights. This makes it possible to live comfortably in the University neighborhood on the fellowship stipend.

The PhD program in mathematics has an enrollment of approximately 60 students. Typically, 10-12 students enter each year. While students come from all over the world, they form an intellectually cohesive and socially supportive group.

New York City is America’s major center of culture. Columbia University’s remarkably pleasant and sheltered , near the Hudson River and Riverside Park, is situated within 20-30 minutes of Lincoln Center, Broadway theaters, Greenwich Village, and major museums. Most department members live within a short walk of the University.

Since receiving its charter from King George II in 1754, Columbia University has played an eminent role in American education. In addition to its various faculties and professional schools (such as Engineering, Law, and Medicine), the University has close ties with nearby museums, schools of music and theology, the United Nations, and the city government.

The application deadline is typically early December for admission the following September. Precise details on requirements and deadlines can be found . Applicants must submit all required documents by the posted deadline. Students whose undergraduate degree was not from an English-speaking country must also submit scores from the TOEFL or IELTS.  Applications must be filed .

 

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Graduate School of Arts and Sciences
Columbia University
Office of Student Affairs
107 Low Library, MC 4304
New York, NY 10027
212-854-6729

Michael Harris
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Department of Mathematics
Columbia University
2990 Broadway
509 Mathematics, MC 4406
New York, NY 10027


Welcome to the Math PhD program at Harvard University and the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences.

Learn more about Harvard’s Math community and our statement on diversity and inclusion.

The Harvard Griffin GSAS Office of Equity, Diversity, Inclusion & Belonging offers student affinity groups for graduate students and many other resources.

The Harvard University Office for Gender Equity has dedicated GSAS Title IX resource coordinators who work with and support graduate students.

open. The application deadline is December 15, 2021. -->

The application deadline for fall 2024 admission has passed. Applications for fall 2025 admission will open in September 2024.

For information on admissions and financial support, please visit the Harvard Harvard Kenneth C. Griffin Graduate School of Arts and Sciences.

Harvard Griffin GSAS is committed to ensuring that our application fee does not create a financial obstacle. Applicants can determine eligibility for a fee waiver by completing a series of questions in the Application Fee section of the application. Once these questions have been answered, the application system will provide an immediate response regarding fee waiver eligibility.

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Doctor of Philosophy (PhD)

Program synopsis and training.

Math Graduate Program

 The Doctor of Philosophy (PhD) in mathematics is the highest degree offered by our program. Graduates will have demonstrated their ability to conduct independent scientific research and contribute new mathematical knowledge and scholarship in their area of specialization. They will be well-supported and well prepared for research and faculty positions at academic institutions anywhere in the world. Owing to their independence, analytic abilities, and proven tenacity, our PhD graduates are also sought after by private and government employers.

Our PhD program offers two tracks, one for Theoretical Mathematics and one for Applied Mathematics . The tracks differ only in the course and  qualifying requirements during the first two years. Applicants are required to decide on one of the tracks and applications will be evaluated subject to respective criteria described below.

Once students have passed their Qualifying Requirements, the two tracks merge and there is no distinction in later examinations and research opportunities. In particular, the candidacy exam for both tracks consists of a research proposal, the graduate faculty available for advising is the same, and the final degree and thesis defense are independent of the initially chosen track.

Expected Preparations for Admission

Competitive applicants to the theoretical track are expected to have strong foundations in Real Analysis and Abstract Algebra, equivalent to our Math 5201 - 5202  and Math 5111 - 5112 sequences.

Expected preparations for the applied track include the equivalents of a rigorous Real Analysis course (such as Math 5201 ), a strong background in Linear Algebra, as well as an introductory course in Scientific Computing.

Besides these basic requirements, competitive applicants in either track submit evidence for a broad formation in mathematics at the upper-division or beginning graduate level. Relevant coursework in other mathematical or quantitative sciences may also be considered, especially for the applied track.

Prior research experiences are not required for either track, and we routinely admit students without significant research background. Nevertheless, applicants are encouraged to include accounts of research and independent project endeavors as well as letters of supervising mentors in order to be more competitive for fellowship considerations. The research component is likely to have greater weight in applications to the applied track.

These prepared documents serve to provide our admission committee with a narrative overview of the applicant's mathematical trajectory. Their primary focus should, therefore, be to enumerate and describe any evidence of mathematical ability and mathematical promise. The information included in the documents should be well-organized, comprehensive, informative, specific, and relevant. This will help our committee to properly and efficiently evaluate the high number of applications we receive each year.

Our Graduate Recruitment Committee will generally  not  consider GRE test scores for this Autumn 2024 admissions. If you have already taken the test, please do not self-report the scores to us. In exceptional circumstances students may have the option to report unofficially. 

International students whose native language is not English and are not exempt should score at least a 20 on the Speaking portion of the TOEFL or at least 6.5 on the IELTS Speaking portion.  We also recommend an overall score of at least 95 on TOEFL or at least 7.0 on IELTS.  For a list of exempt countries, please see  https://gpadmissions.osu.edu/intl/additional-requirements-to-apply.html

Qualifying Requirements by Track

The qualifying requirements for the theoretical track are fulfilled by passing our Abstract Algebra course sequence  ( Math 6111 , Math 6112 ) and  our Real Analysis course sequence ( Math 6211 , Math 6212 ), each with at least an A-, or  by passing a respective examination.

The qualifying requirements for the applied track combine a mandatory Scientific Computing course ( Math 6601 ), one of the algebra or analysis courses, and three additional courses chosen from  Math 6602 , Math 6411 ,  Math 6451 , and the courses comprising the algebra and analysis sequences.

The breadth requirements in the applied track are more flexible than in the theoretical track, but also include a mandatory graduate course in a non-math STEM department from an approved list. 

You can find more information about our PhD program requirement here .

Opportunities & Outcomes

The research opportunities and academic outcomes of our doctoral program are described in detail in the Graduate Program Prospectus  [pdf]. 

Our department has about 80 active graduate faculty on the Columbus and regional campuses. Virtually every area of mathematics is represented in our program, with a sampling displayed below.

  • Commutative, Non-commutative, & Quantum Algebra
  • Analytic, Algebraic, Computational Number Theory
  • Algebraic Geometry, Tropical Geometry
  • Applied Mathematics, Mathematical Physics
  • Real and Complex Analysis
  • Functional Analysis, Operator Algebras
  • Combinatorics and Graph Theory
  • Differential Geometry
  • Dynamical Systems and Ergodic Theory
  • Financial and Actuarial Mathematics
  • Logic and Foundations
  • Probability Theory, Statistical Mechanics
  • Mathematical Biology
  • Ordinary and Partial Differential Equations
  • Representation theory
  • Scientific Computing
  • Topology, Topological Data Analysis

See also our  Applied Mathematics Topics List  [pdf].

Our program offers many support opportunities without teaching duties as well, to allow more time for scientific endeavors. These opportunities include university fellowships, external funding, and departmental fellowships and special assignments. See the  Financial Support  page for more details.

The median time to degree completion in our program is below six years but also varies significantly among our students, with as little as four years for students entering with substantial prior preparations. Funding is guaranteed for six years and can be extended to seven years with advisor support and the permission of the Graduate Studies Committee.  

Most of our graduates continue their careers in academia. Post-doctoral placements in the last two years include, for example, UCLA, Stanford, ETH-Zürich, Brown University, University of Michigan, Northwestern University, University of Vienna, EPF Lausanne, Free University at Berlin, Purdue University, and University of Utah. In recent years our graduates also went to Princeton University, IAS, University of Chicago, Yale University, University of Michigan, Cal-Tech, Northwestern University, University of Texas, Duke University, SUNY Stony Brook, Purdue University, University of North Carolina - Chapel Hill, and Indiana University. Recent non-academic placements include Google, Facebook, Amazon, NSA, and prestigious financial institutions.

Students also have access to training and networking opportunities that prepare them better for careers in private industry and teaching - for example, through the Erdős Institute  - and are regularly offered highly competitive positions in the industry. 

Nearly half of the graduate population consists of domestic students coming from both larger universities and smaller liberal arts colleges with a solid math curriculum. And as a program group member of the National Math Alliance , we are dedicated to enhancing diversity in our program and the scientific community. The International students in our program come from all parts of the world with a wide variety of educational backgrounds.

Prospective students:  [email protected]

Graduate Office Department of Mathematics The Ohio State University 231 W 18th Avenue ( MA 102 ) Columbus, Ohio 43210 United States of America                

Phone: (614) 292-6274 Fax: (614) 292-1479

[pdf] - Some links on this page are to .pdf files. If you need these files in a more accessible format, please email  [email protected] . PDF files require the use of Adobe Acrobat Reader software to open them. If you do not have Reader, you may use the following link to Adobe to download it for free at:  Adobe Acrobat Reader .

Mathematics

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Undergraduate Program

Mathematics is the science of order, and mathematicians seek to identify instances of order and to formulate and understand concepts that enable us to perceive order in complicated situations. The concentration in Mathematics is designed to acquaint the student with the most important general concepts underlying the three branches of modern mathematics. Concentration in mathematics will provide an adequate basis for further study in either pure or applied mathematics. Students are also eligible to apply for an A.B./A.M. degree program.

The Bachelor of Liberal Arts degree is designed for industry professionals with years of work experience who wish to complete their degrees part time, both on campus and online, without disruption to their employment. Our typical student is over 30, has previously completed one or two years of college, and works full time.

The graduate Mathematics Program at Harvard is designed for students who hope to become research mathematicians and show definite promise in this direction. Once the student has demonstrated a command of basic mathematical concepts by passing the qualifying examination, the emphasis is on getting to the frontiers of some field by independent reading, advanced courses, and seminars. The Cambridge area is one of the most active centers of mathematics in the world. Harvard, Brandeis, and Northeastern Universities and the Massachusetts Institute of Technology have an especially close association in mathematics, and the Harvard University Center of Mathematical Sciences and Applications hosts conferences, seminars, and workshops.

Mathematical Sciences

Mellon college of science, ph.d. programs, doctor of philosophy in mathematical sciences.

Students seeking a Ph.D. in Mathematical Sciences are expected to show a broad grasp of mathematics and demonstrate a genuine ability to do mathematical research. The Doctor of Philosophy in Mathematical Sciences is a traditional research degree, and its requirements are representative of all doctoral programs.

After being admitted to graduate status by the Department, a student seeking a Ph.D. must be admitted to candidacy for this degree by fulfilling the appropriate program requirements.

The most important requirement for the Ph.D. degree is timely completion and public defense of an original Ph.D. thesis. The Ph.D. thesis is expected to display depth and originality and be publishable by a refereed journal.

Doctor of Arts in Mathematical Sciences

The Doctor of Arts degree shares all requirements and standards with the Ph.D., except with regard to the thesis. The D.A. thesis is not expected to display the sort of original research required for a Ph.D. thesis, but rather to demonstrate an ability to organize, understand, and present mathematical ideas in a scholarly way, usually with sufficient innovation and worth to produce a publishable work. Whenever practical, the department provides D.A. candidates with the opportunity to use materials developed to teach a course. While a typical Ph.D. recipient will seek a position that has a substantial research component, the D.A. recipient will usually seek a position where research is not central.

Doctor of Philosophy in Algorithms, Combinatorics, and Optimization (ACO)

This program is administered jointly by the Department of Mathematical Sciences, the Department of Computer Science, and the Tepper School of Business. It focuses on discrete mathematics and algorithmic issues arising in computer science and operations research, particularly the mathematical analysis of these issues. The participating units evaluate applicants separately. The requirements for this degree and information on participating faculty are available at the ACO page .

Doctor of Philosophy in Pure and Applied Logic (PAL)

This is an interdisciplinary program with faculty from the Department of Mathematical Sciences, the Department of Philosophy, and the School of Computer Science. The participating units evaluate applicants separately and set their own program requirements. Students who have been admitted to the PAL program, and who complete the requirements for the Ph.D. in Mathematical Sciences with a thesis in the area of logic, can choose to receive either a Ph.D. in Pure and Applied Logic or a Ph.D. in Mathematical Sciences. The choice of which degree to receive is usually based on the intended career path.

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

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

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

Degree Requirements

Degree requirements consist of:

  • Oral qualifying exam
  • Classroom teaching
  • Original thesis and defense

Prospective students are invited to consult the graduate career timeline for more information, and to read about the application procedure .

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Home / Graduate / Ph.D. Program

Ph.D. Degree Program Overview

The objectives of the Mathematics Ph.D. program are to prepare students for a career in academia, industry, or teaching. At the end of their studies, students will possess the ability to solve problems and communicate solutions in rigorous mathematical language, to communicate mathematical concepts effectively, and to conduct independent research. Ph.D. students are expected to obtain their Ph.D. degree within six years. Students admitted to the Ph.D. program may receive a master's degree en route to the Ph.D.

Ph.D. Degree Requirements

Students are required to complete all of the following:

  • Obtain a first-level pass on at least one of the three written preliminary examinations , and a second-level pass on at least one other. Students must complete the full sequence in the track associated with the preliminary examination they did not achieve a first-level pass*;
  • Satisfy the foreign language requirement;
  • Pass the oral qualifying examination ;
  • Complete three quarters as a Teaching Assistant ;
  • Complete the required coursework : Math 288A** plus six graduate courses in mathematics other than Math 200, 201, 202, 204, 205, 206, 208, 209 and 210. No more than three courses may be independent study or thesis research courses.
  • Write a Ph.D. thesis  (dissertation).

Degree Timetable

Preliminary Exams End of second year
Language Exam End of third year
Oral Qualifying Exam
(Advancement to Candidacy)		 Between seventh and twelfth quarters
Dissertation Defense Four to six years

Graduate Advising

Entering graduate students are advised initially by the Graduate Vice Chair, and then assigned a faculty mentor who will be an ongoing advising resource for the student. Within the first two years, and typically after passing the preliminary examinations, the student selects a faculty advisor in the area of the student's research interest. This is done in consultation with the Graduate Vice Chair.

Each graduate student is expected to consult with his or her advisor to formulate a plan of student and research. The student's advisor ultimately will be the student's thesis advisor.

Annual meetings with the Graduate Vice Chair and the Graduate Advisor and Program Coordinator are conducted with each student on a one-on-one basis. These meetings serve to notify the student of their current progress within the program and outline expectations for the continuation of normative progress toward the Ph.D. degree.

Graduate students are encouraged to review the  Graduate Student Mentoring Packet for resources on how to get the most out of the graduate student-mentor relationship.

  • Preliminary Exams

Ph.D. students should complete the preliminary exams and introductory sequence requirements by the end of the second year in order to make satisfactory progress.  If a graduate student does not fulfill the above requirement by the end of the second year, depending on the student's progress, they can be placed on academic probation.  If a graduate student has not fulfilled the above requirements by the end of the third year, they are subject to dismissal from the program.

Advancing to Candidacy

To make satisfactory progress, a Ph.D. student should advance to candidacy by the end of their fourth year. A Ph.D. student who has not advanced to candidacy by the end of the fourth year will be placed on academic probation or be subject to dismissal from the program.

  • Dissertation

Each graduate student in the Ph.D. program is required to write a Ph.D. dissertation or thesis on a research topic in mathematics. Ph.D. students are expected to obtain their Ph.D. degree in four to six years.

Applying for Graduation

Ph.D. students must complete the Application for the Doctor of Philosophy Degree form by the appropriate quarter’s deadline listed in the current Academic calendar . The form can be found on the Graduate Division website or can be provided by the Mathematics Department. The form should be turned in to the Graduate Advisor and Program Coordinator for review and submission to the Graduate Division.

  • Graduate Program
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Mathematics, PhD

The Department of Mathematics of the University of Pennsylvania offers a full Graduate Program in Mathematics, conferring the degrees of Master of Arts (A.M.), Master of Philosophy (M.Phil.), and Doctor of Philosophy (Ph.D.). The educational aim of this program is to provide well-rounded mathematical training for a career of research and teaching in mathematics, as well as for other careers which use advanced mathematics in a substantial way. Students are offered the possibility of a dual Ph.D. degree in Mathematics and a Masters degree in other subjects, including physics and computer science.

This program covers a variety of areas, including analysis, geometry-topology, algebra, mathematical physics, combinatorics, logic, and probability. See our Graduate Mathematics Page and our Math Department home page for detailed information about other aspects of this program and of the Penn Math Department as a whole, including its colloquia, seminars and lecture series, and the research interests of the faculty.

Full time Ph.D. students receive financial support for five years, at least two of which are in the form of a fellowship.

In addition, Penn offers an interdisciplinary graduate program in Applied Mathematics and Computational Science , for which there is separate application process.

For more information: https://www.math.upenn.edu/graduate/

View the University’s Academic Rules for PhD Programs .

Sample Plan of Study

A total of 20 course units are required for graduation.

Course List
Code Title Course Units
Year 1
Topology and Geometric Analysis
Topology and Geometric Analysis
Algebra
Algebra
Analysis
Analysis
Year 2
Year 3
Year 4
Year 5

The degree and major requirements displayed are intended as a guide for students entering in the Fall of 2024 and later. Students should consult with their academic program regarding final certifications and requirements for graduation.

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Mathematics

General information, program offerings:, director of graduate studies:, graduate program administrator:.

The Department of Mathematics graduate program has minimal requirements and maximal research and educational opportunities. It differentiates itself from other top mathematics institutions in the U.S. in that the curriculum emphasizes, from the start, independent research. Our students are extremely motivated and come from a wide variety of backgrounds. While we urge independent work and research, a real sense of camaraderie exists among our graduate students. As a result, the atmosphere created is one of excitement and stimulation and mentoring and support. There also exists a strong scholarly relationship between the department and the Institute for Advanced Study (IAS), located a short distance from campus. Students can contact IAS members as well as attend the IAS seminar series.

Students are expected to write a dissertation in four years but may be provided an additional year to complete their work if deemed necessary. Each year, our graduates are successfully launched into academic positions at premier mathematical institutions and industry.

Program Offerings

Program offering: ph.d..

The department offers a broad variety of research-related courses as well as introductory (or “bridge”) courses in several areas, which help first-year students strengthen their mathematical background. Students also acquire standard beginning graduate material primarily through independent study and consultations with the faculty and fellow students.

Language(s)

Students must satisfy the language requirement by demonstrating to a member of the mathematics faculty a reasonable ability to read ordinary mathematical texts in one of the following three languages: French, German, or Russian. Students must pass the language test by the end of the first year and before standing for the general exam.

Additional pre-generals requirements

Seminars The department offers numerous seminars on diverse topics in mathematics. Some seminars consist of systematic lectures in a specialized topic; others present reports by students or faculty on recent developments within broader areas. There are regular seminars on topics in algebra, algebraic geometry, analysis, combinatorial group theory, dynamical systems, fluid mechanics, logic, mathematical physics, number theory, topology, and other applied and computational mathematics. Without fees or formalities, students may also attend seminars in the School of Mathematics at the IAS.

The department also facilitates several informal seminars specifically geared toward graduate students: (1) Colloquium Lunch Talk, where experts who have been invited to present at the department colloquium will 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 and helps in creating a vibrant mathematical interaction among the graduate students; and, (3) What’s Happening in Fine Hall (WHIFH) seminar, where faculty members present talks in their own research areas specifically geared towards graduate students. Reading seminars are also organized and run by graduate students.

General exam

Beyond needing a strong knowledge of three more general subjects (algebra, and real and complex analysis), first-year students are set on the fast track of research by choosing two advanced research topics as part of their general exam. The two advanced topics are expected to come from distinct major areas of mathematics, and the student’s choice is subject to the approval of the department. Usually, by the second year, students will begin investigations of their own that lead to the doctoral dissertation.

General Exam in Mathematical Physics For a mathematics student interested in mathematical physics, the general exam is adjusted to include mathematical physics as one of the two special topics.

Qualifying for the M.A.

The Master of Arts (M.A.) degree is considered an incidental degree on the way to full Ph.D. candidacy. It is earned once a student successfully passes the language requirement and the general exam, and the faculty recommends it. It may also be awarded to students who, for various reasons, may leave the Ph.D. program, provided that the following requirements are met: passing the language requirement as well as the three general subjects (algebra, and real and complex analysis) of the general exam, and receiving department approval.

During the second, third, and fourth years, graduate students are expected to either grade or teach two sections of an undergraduate course, or the equivalent, each semester. Although students are not required to teach to fulfill department Ph.D. requirements, they are strongly encouraged to do so at least once before graduating. Teaching letters of recommendation are necessary for most postdoctoral applications.

Post-Generals requirements

Selection of a Research Adviser Upon completion of the general exam, the student is expected to choose a thesis adviser.

Dissertation and FPO

Two to three years is usually necessary for the completion of a suitable dissertation. Upon completion and acceptance of the dissertation by the department and Graduate School, the candidate is admitted to the final public oral examination. The dissertation is presented and defended by the candidate.

The Ph.D. is awarded after the candidate’s doctoral dissertation has been accepted and the final public oral examination sustained.

  • Igor Rodnianski

Associate Chair

  • János Kollár

Director of Graduate Studies

  • Lue Pan (associate)
  • Chenyang Xu

Director of Undergraduate Studies

  • Jennifer M. Johnson (associate)
  • Michael Aizenman
  • Noga M. Alon
  • Manjul Bhargava
  • Sun-Yung A. Chang
  • Maria Chudnovsky
  • Fernando Codá Marques
  • Peter Constantin
  • Mihalis Dafermos
  • Charles L. Fefferman
  • David Gabai
  • June E. Huh
  • Alexandru D. Ionescu
  • Nicholas M. Katz
  • Sergiu Klainerman
  • Peter Steven Ozsváth
  • Peter C. Sarnak
  • Paul Seymour
  • Amit Singer
  • Christopher M. Skinner
  • Allan M. Sly
  • Zoltán Szabó
  • Paul C. Yang
  • Shou-Wu Zhang

Assistant Professor

  • Bjoern Bringmann
  • Matija Bucic
  • Marc Aurèle Tiberius Gilles
  • Jonathan Hanselman
  • Susanna Haziot
  • Ana Menezes
  • Ravi Shankar
  • Jacob Shapiro
  • Jakub Witaszek
  • Ruobing Zhang

Associated Faculty

  • John P. Burgess, Philosophy
  • René A. Carmona, Oper Res and Financial Eng
  • Bernard Chazelle, Computer Science
  • Hans P. Halvorson, Philosophy
  • William A. Massey, Oper Res and Financial Eng
  • Frans Pretorius, Physics
  • Robert E. Tarjan, Computer Science
  • Ramon van Handel, Oper Res and Financial Eng
  • Louis Esser
  • Sepehr Hajebi
  • Kimoi Kemboi
  • Dmitry Krachun
  • Anubhav Mukherjee
  • Sung Gi Park
  • Semon Rezchikov
  • Joshua X. Wang
  • Mingjia Zhang

University Lecturer

  • Jennifer M. Johnson

Senior Lecturer

  • Jonathan M. Fickenscher
  • Mark W. McConnell
  • Tatyana Chmutova
  • Tatiana K. Howard
  • John T. Sheridan
  • David Villalobos

Visiting Professor

  • Bhargav B. Bhatt

Visiting Lecturer with Rank of Professor

  • Camillo De Lellis
  • Helmut H. Hofer
  • Akshay Venkatesh

For a full list of faculty members and fellows please visit the department or program website.

Permanent Courses

Courses listed below are graduate-level courses that have been approved by the program’s faculty as well as the Curriculum Subcommittee of the Faculty Committee on the Graduate School as permanent course offerings. Permanent courses may be offered by the department or program on an ongoing basis, depending on curricular needs, scheduling requirements, and student interest. Not listed below are undergraduate courses and one-time-only graduate courses, which may be found for a specific term through the Registrar’s website. Also not listed are graduate-level independent reading and research courses, which may be approved by the Graduate School for individual students.

COS 522 - Computational Complexity (also MAT 578)

Mat 500 - effective mathematical communication, mat 515 - topics in number theory and related analysis, mat 516 - topics in algebraic number theory, mat 517 - topics in arithmetic geometry, mat 518 - topics in automorphic forms, mat 519 - topics in number theory, mat 520 - functional analysis, mat 522 - introduction to pde (also apc 522), mat 525 - topics in harmonic analysis, mat 526 - topics in geometric analysis, mat 527 - topics in differential equations, mat 528 - topics in nonlinear analysis, mat 529 - topics in analysis, mat 531 - introduction to riemann surfaces, mat 547 - topics in algebraic geometry, mat 549 - topics in algebra, mat 550 - differential geometry, mat 555 - topics in differential geometry, mat 558 - topics in conformal and cauchy-rieman (cr) geometry, mat 559 - topics in geometry, mat 560 - algebraic topology, mat 566 - topics in differential topology, mat 567 - topics in low dimensional topology, mat 568 - topics in knot theory, mat 569 - topics in topology, mat 572 - topics in combinatorial optimization (also apc 572), mat 577 - topics in combinatorics, mat 579 - topics in discrete mathematics, mat 585 - mathematical analysis of massive data sets (also apc 520), mat 586 - computational methods in cryo-electron microscopy (also apc 511/mol 511/qcb 513), mat 587 - topics in ergodic theory, mat 589 - topics in probability, statistics and dynamics, mat 595 - topics in mathematical physics (also phy 508), mat 599 - extramural summer research project, phy 521 - introduction to mathematical physics (also mat 597).

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Department of Mathematics

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Mathematics Graduate Program

Thinking of applying to graduate school in mathematics.

math phd curriculum

Penn was ranked 8th among all US universities in a leading national study , and our mathematics graduate program was recently highest in a study of graduate programs in arts and sciences at Penn. We have a very active and involved mathematics faculty , diverse course offerings and a broad seminar schedule , with a variety of research projects and strengths in algebra, analysis, geometry-topology, combinatorics, logic, probability, and mathematical physics. We have a supportive atmosphere, with personal attention from the faculty and extensive interaction among graduate students. Our grad students can take courses not only in the Mathematics Department but also elsewhere at Penn, and the wide resources of the university are available. Our former graduate students have gone on to mathematical careers both in academia and in industry.

Our full-time Ph.D. students receive a generous and competitive support package including

  • five years of funding with a combination of  fellowships and teaching assistantships;
  • a stipend and a full tuition scholarship;
  • no teaching responsibilities for at least two years (generally including the first and fourth year);
  • health insurance coverage provided at no cost to the student.

We invite you to learn about our graduate program, our math department, our university and living in Philadelphia, a cosmopolitan city and a true mathematical hub, with easy access to nearby mathematics departments and research institutes.

We are looking for interested, mathematically talented and dedicated students to be a part of our group of excellent future mathematicians. Consider applying to Penn for your graduate education. Questions?

PhD in Mathematical Sciences

The Doctor of Philosophy (PhD) program in Mathematical Sciences represents achievement of a broad knowledge of the various branches of mathematics, of the ability to communicate mathematics in both written and oral form, and of a demonstrated creative ability in a particular branch of mathematics.

The program prepares students for careers in academia and as professional mathematicians. Students have the opportunity to work with some of the world's leading experts in a variety of research areas. The Department has particular strength in the interrelated fields of analysis , partial differential equations , and probability . Students interested in more applied directions can work with experts in  mathematical bioscience , computational finance , and  cryptography ..

Admission Requirements

Students applying for the program should have or be expecting to obtain a bachelor’s degree either in mathematics or a related field.

The following background is required:

  • Multivariable calculus at the level of MATH 2063
  • Ordinary differential equations at the level of MATH 2073 or MATH 2074
  • Linear algebra at the level of MATH 2076
  • Experience with reading and constructing mathematical proofs at the level of MATH 3001

An official GRE score is not required for admission, but a quantitative score of 160 or higher is recommended.

Proficiency in English is required of international students whose native language is not English.  The minimum scores required for admission and to be considered for an assistantship are as follows:
  Admission Assistantship
TOEFL 80 93
IELTS (overall band) 6.5 7
PEARSON (PTE) 54 64
Duolingo 110 115

The English proficiency requirement is met for applicants with degrees earned in English from accredited universities and colleges in  the US or other English-speaking countries .

Financial Support

Most of our PhD students receive full financial support via a teaching or research assistantship, and most are supported through their entire UC career.  Travel support is available for students to attend or present their work at conferences.  

All applicants for the PhD program are automatically reviewed for graduate assistantship eligibility at the time of application. 

  • Financial aid opportunities for students in the Mathematical Sciences Department
  • Tuition and fees for graduate and professional students

Application Instructions

Applicants will need to meet the minimum requirements to be considered for the program. Completed applications will be reviewed beginning February 1 . We will continue to receive applications until all positions are filled.

All application materials from international students requiring a US visa must be received prior to April 1 (but sooner is better) in order to allow time for the necessary paperwork to be processed. The visa application process can often take 90 days or more to complete.

How to apply: 

1. Create  an online application

2. Include these documents in your application:

  • Three letters of recommendation. The application system will automatically send an email to each of the recommenders with a link to submit their letters. 
  • Unofficial copy of transcript (official transcript will be required if you are admitted to the program). 
  • Statement of purpose/cover letter 
  • English Proficiency for international students

3. Pay the application fee

UC’s CEEB college code is 1833, as established by The College Board . CEEB codes are used to ensure that test scores are sent to the correct institution. 

  • More information about submitting your application materials
  • FAQs for the admission process

For additional guidance, watch the video below for step-by-step application instructions:

To access the video description, click the "i" icon on the top right of the video.

Program Description

The credit-hour requirement includes a minimum of 90 graduate credits beyond the bachelor's degree or a minimum of 60 credits beyond a master's degree, including 7 hours in dissertation research, with a GPA of 3.3 or higher. 

All incoming PhD students are required to take the  qualifying exam before the beginning of their first semester. Students who do not pass this exam at the PhD level are placed in the appropriate 6000 - level courses. The Mathematics Qualifying Exam is based on the two-semester sequence Advanced Calculus MATH6001-6002 and the one semester course Abstract Linear Algebra MATH6003. 

All PhD students must pass four preliminary examinations . Each Preliminary Exam is offered twice a year. Examinations based on a course given during Fall Semester are offered after the end of Spring Semester and at the beginning of the following Fall Semester. Examinations based on a course given during the Spring Semester are offered at the beginning of the Fall Semester and at the beginning of the following Spring Semester.

After the preliminary examinations, an advanced examination in the area of examination of the student is required. An advanced exam may either be a written exam, a presentation or a series of presentations. The exam will be administered by a committee. Generally, this committee will form the students’ dissertation committee.

Visit the curriculum guide to learn about the required courses. More details concerning the requirements of the program are explained in the Mathematical Sciences Department’s Graduate Handbook . See the course descriptions for information on the content.

About Cincinnati

Cincinnati is a big city with a small-town feel. The cost of living is low, but the quality of life is high.  Forbes named Cincinnati the #5 most affordable city and the #9 best city for raising a family. Cincinnati has ranked  the best place to live in Ohio by U.S. News & World Report, also the fourth-best city in the country for parks . UC is home to over 10,500 graduate students, 20% of which are international students.

  • Why Cincinnati  
  • Estimated living expenses  (for international students)

For further information, please contact the Graduate Program Director, Dr. Robert Buckingham:

See the full list of our graduate programs

Mathematics Education PhD

Doctor of philosophy.

The Doctor of Philosophy (Ph.D.) degree emphasizes research competencies. The degree requires a scholarly dissertation of intellectual merit and sound research methodology. Dissertation research may include analytical studies of the process of teaching or experimental studies of the teaching-learning process, including studies of verbal learning and laboratory practice or historical studies.

A graduate student studies in the TC library using a book and her laptop.

Admissions Information

Displaying requirements for the Spring 2025, Summer 2025, and Fall 2025 terms.

  • Points/Credits: 75
  • Entry Terms: Spring, Summer, Fall
  • Enrollment Formats: Full-Time, Part-Time

Application Deadlines

Entry Term AvailablePriority DeadlinesFinal DeadlinesExtended Deadlines
SpringNovember 15, 2024November 15, 2024N/A
SummerDecember 1, 2024June 1, 2025N/A
FallDecember 1, 2024July 1, 2025N/A

Select programs remain open beyond our standard application deadlines , such as those with an extended deadline or those that are rolling (open until June or July). If your program is rolling or has an extended deadline indicated above, applications are reviewed as they are received and on a space-available basis. We recommend you complete your application as soon as possible as these programs can close earlier if full capacity has been met.

Application Requirements

 Requirement
  , including Statement of Purpose and Resume
 
 Results from an accepted (if applicable)
 $75 Application Fee
 Two (2) Letters of Recommendation

Requirements from the TC Catalog (AY 2023-2024)

Displaying catalog information for the Fall 2023, Spring 2024 and Summer 2024 terms.

View Full Catalog Listing

Doctoral Degrees

All candidates for the Ed.D., Ed.D.C.T., or Ph.D. degrees are expected to demonstrate both mathematics and mathematics education competencies through a series of certification examinations taken upon the completion of 60 graduate points. Certification examinations test the student’s knowledge of current research and theory in mathematics education and mathematics content. Examinations are offered once in the fall, spring, and summer terms. Courses recommended as preparation for the examinations in mathematics education include MSTM 6037,  MSTM 4019, and other mathematics education courses; Courses recommended as preparation for the examinations in mathematics are 6000- level mathematics content courses.

Students must demonstrate acceptable proficiency in at least three of the following six mathematics content areas: algebra, analysis, discrete mathematics, foundations of mathematics, geometry and topology, and probability and statistics. Students may sit for the examination in mathematics content during the regular certification examination times. Alternatively, they may register for advanced content courses and, with permission of the program, sit for the content area certification examination upon completion of the course. Incoming doctoral candidates should register for MSTM 6037 Professional Seminar in Mathematics during the first year of doctoral studies.

Doctoral students whose dissertations require statistical analysis should include appropriate statistics courses in their programs. These points can be included either in the mathematics/mathematics education requirement or can be taken as research electives.

Doctor of Philosophy in Mathematics Education

The Doctor of Philosophy (Ph.D.) degree emphasizes research competencies. The degree program requires a scholarly dissertation of intellectual merit and sound research methodology. Dissertation research may include analytical studies of the process of teaching or experimental studies of the teaching-learning process, including studies of verbal learning and laboratory practice or historical studies.

Candidates are encouraged to develop an association with a faculty member early in their studies to identify a problem area of mutual interest to plan a course of studies that leads to the competencies needed to complete dissertation research and prepare for a professional role. Further details are available in the brochures on doctoral studies and in the general descriptions of doctoral programs available from the Office of Doctoral Studies (ODS).

A program of study for the Doctor of Philosophy degree must include at least 45 points taken under Teachers College registration. In order to permit the acquisition of broad and basic scholarship, each program of study should include at least 60 points in mathematics, mathematics education, statistics, and computing. At least 35 points should be in advanced courses – including research courses (MSTM 6500 or 6501 and MSTM 7500). (Any Teachers College course at the 6000 level or above, any Columbia University Graduate School of Arts and Sciences course with a “G” prefix, any “W” course numbered above 4000, or any transferred course with a graduate-level prerequisite will be considered an advanced course.) Further, 15 points in the philosophical, psychological, and curricular foundations of education must be included in every Ph.D. degree program. Students whose dissertations require statistical analysis should include appropriate statistics courses in their programs. These points can be included either in the mathematics/mathematics education requirement or can be taken as research electives.

Candidates for the Ph.D. degree are required to demonstrate competency in two languages chosen from among French, German, and Russian. Students who require other languages for the preparation of their dissertation may petition the program to request one substitution. Students in mathematics may not use computer languages or statistics to satisfy the language requirement.

The Ph.D. dissertation is a scholarly study contributing new theoretical knowledge to the field and should be planned early in the program when sufficient advanced courses have been completed to permit the candidate to enroll in relevant research courses. Ph.D. dissertations in mathematics education should be (1) experimental studies in learning, (2) analytical studies in policy theory in mathematics education, or (3) other scholarly investigations of problems and issues of broad significance in the field.

The website of the Program offers a list of Topic study groups which doctoral students are recommended to join.

  • View Other Degrees

Program Director : Professor Alexander Karp

Teachers College, Columbia University 323 Thompson

Phone: (212) 678-3381 Fax: (212) 678-8319

Email: tcmath@tc.edu

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

All listings (alpha order) All listings by state

Institution name

Program type, masters programs (check all that apply), phd specialties (check all that apply), financial support available, gre required, online options available, number of phds awarded in the last year, enrollments, canadian province, list or edit your graduate program in the mathematical sciences.

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math phd curriculum

Find Graduate Programs Find the right graduate program in the mathematical sciences

Find Graduate Programs (FGP) in the Mathematical Sciences offers comparative information on over 500 graduate programs in the mathematical sciences in the US and Canada. This web service provides only an overview of the programs offered; departments should be contacted directly for more detailed information. Currently 740 graduate programs are listed.

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

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PhD Degree Requirements

Updated on August 15, 2023

A PhD candidate must satisfy the requirements for a doctoral degree as laid out by the UW Graduate School.  Some of the key Graduate School requirements and additional requirements from the Mathematics Department are listed below.  If you have questions about the requirements, please contact your preliminary advisor, Student Services or the Graduate Program Chair.  

Three years of full time study, two of which must be at the University of Washington.

All PhD students must take twelve 500-level numerically graded courses in Mathematics, Applied Mathematics, or Statistics in their first two years of study.  Students are expected to take seven quarters of core courses; at least five of them in their first year. The current list of core courses consists of MATH 504, MATH 505, MATH 506; MATH 524, MATH 525, MATH 534; MATH 544, MATH 545, MATH 546.  Courses from other departments may be included in the twelve with approval of the Graduate Program Coordinator.  Students who have already met their core course requirement may take additional core courses for a grade of S/NS. It is the student’s responsibility to select S/NS credit if they w ish to use this option . 

The seven quarters of core courses requirement only applies to students who are entering our PhD Program in September 2023 or later.  

Writing Milestone :

By the end of the second year, each student must produce a carefully written mathematical document. This could be in any area, not just a presumed area of specialization, with any faculty member, not just a potential advisor. Such a document can be an expository paper coming from deep study of material in a reading course with a faculty member, a write-up of a project done under the supervision of a faculty member, an existing research paper that the student has done could also count, if a faculty member agrees. In general, this is flexible but the resulting document must be substantial .  

The writing milestone requirement only applies to students who are entering our PhD Program in September 2023 or later.  

General Examination:

The General Examination is an oral examination on a special area of intended research, given by a committee after the student has passed the preliminary exams. This exam can be given only after two years of graduate study, to students who have formally declared an advisor, and have formed their thesis committee with the Graduate School. Normally, it should be taken by the middle of the student's fourth year. In extenuating circumstances, the exam can be postponed if approved by the Graduate Committee. In addition, the students must prepare a written General Paper and distribute it to the committee at least two weeks before the date of the General Exam. The content of this paper is decided upon in consultation with the committee. For example, this might be a 10-20 page expository account of the student's research area, culminating in a problem or list of problems to be studied, together with a discussion of some of the relevant literature.

A PhD thesis must be a well-written original contribution to the scientific literature.  This document will be archived by the University of Washington Library.   During the preparation of the thesis, a student must take 27 credits of Math 800 required over a period of at least three quarters.

Final examination:.

The PhD thesis defense is the Final Examination.  The defense is an oral exam given by the thesis committee consisting of the advisor as the chair of the committee, two additional members of the department who have been appointed to the Graduate School, plus one member of the committee from outside the Math Department called the Graduate School Representative (GSR).  Successful completion of the exam is required for the PhD.

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Contact the mathematics department.

Learn more about the mathematics department.

Application Requirements

All application materials must be submitted directly through the online application system. We do not accept paper application materials. Official transcripts should not be sent to the Guarini School office during the application process.

Application Fee

English language proficiency .

  • Language proficiency test scores are required for non-US citizens, with the exception of those who are earning or have earned a degree from institutions in the US or Canada, or whose primary language of instruction at their non-US institution was English.
  • We accept TOEFL, IELTS, and Duolingo 
  • The ETS code for the Guarini School is 3351

Personal Statements

  • 2 required personal statement prompts.

Recommendation Letters

  • 3 required, up to 4 accepted.

Transcripts

  • Your most recent unofficial transcripts should be uploaded as part of your application. Official transcripts are not needed during the application process.

Program Supplement

Please upload a list of upper-level mathematics courses you have taken, specifying which textbooks you have used.

Research Areas

Indicate your interest in pursuing each research area during your graduate study. For more information about our various research groups, click here .

Learn more about the mathematics department faculty.

Questions About the Admissions Process

Admissions information can be found here.

Mathematics Department Website

Application Deadline: December 15, 2024

Degree Offered: PhD

Mathematics Education, PHD

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

  • Location: Tempe campus
  • Second Language Requirement: No

Program Description

Degree Awarded: PHD Mathematics Education

This transdisciplinary PhD program in mathematics education accommodates students from a variety of academic backgrounds. It provides students with a solid foundation in graduate-level mathematics as well as research skills and perspectives that enable them to incorporate mathematics into such core educational areas as:

Conducting individual and collaborative research in the learning and teaching of mathematics is an integral part of the program.

Degree Requirements

Curriculum plan options.

  • 84 credit hours, a written comprehensive exam, an oral comprehensive exam, a prospectus and a dissertation

Required Core (12 credit hours) MTE 501 Research in Undergraduate Mathematics Education I (3) MTE 502 Research in Undergraduate Mathematics Education II (3) MTE 503 Research in Undergraduate Mathematics Education Ill (3) MTE 504 Research in Undergraduate Mathematics Education IV (3)

Electives (42 credit hours)

Area Courses (12 credit hours)

Research (6 credit hours) MTE 792 Research (6)

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

Additional Curriculum Information Four to five graduate-level (500 and above) elective courses from mathematics, cognitive science, psychology, educational technology, philosophy or research should be taken as approved by the advisor.

For the area courses, students are required to take four graduate-level courses from the following areas of interest: mathematics, applied mathematics or statistics. Students should see the academic unit for the approved course list.

Students should see the school's website for information about qualifier and comprehensive examinations based on math coursework.

The doctoral dissertation culminating experience consists of a dissertation prospectus, oral dissertation defense and the submission of a final revised, formatted dissertation document to the Graduate College. Dissertations are composed under chair- and committee-supervised research, including literature review, research, data collection and analysis, and writing.

When approved by the student's supervisory committee and the Graduate College, up to 30 credit hours from a previously awarded master's degree may be used for this program. If students do not have a previously awarded master's degree, the remaining coursework is made up of electives and research.

Admission Requirements

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

Applicants are eligible to apply to the program if they have earned a bachelor's or master's degree in mathematics or a closely related area, with exceptionally high grades in advanced coursework in mathematics, from a regionally accredited institution.

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

All applicants must submit:

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

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

At least two of the letters of recommendation must be from faculty.

Next Steps to attend ASU

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

  • Able to complete original research in applied mathematics.
  • Able to incorporate mathematical concepts into novel teaching methods.
  • Address an original research question in mathematics education.

Career Opportunities

Graduates of the doctoral program in mathematics education have opportunities in Arizona, the U.S. and internationally. Opportunities are typically at research universities and liberal arts colleges, community colleges, and education consulting firms and in roles such as:

  • faculty-track academic
  • education consultant or analyst
  • mathematics professor, instructor or researcher

Program Contact Information

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

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Graduate Studies in Mathematics

Our department offers Masters degrees in Mathematics, Applied Mathematics, and Statistics as well as a Ph.D. Degree in Mathematics, which can have an emphasis in any of the three areas mentioned. The Masters degrees emphasize different aspects of theory and applications in order to prepare the students for either continuing studies at the Ph.D. Level or employment. The Ph.D. Program prepares the students for a career in research at a university, a government agency or in industry. Our faculty members are actively engaged in research and mentoring so that doctoral students can choose a faculty advisor according to the students’ interests.

Tulane is a privately endowed university located in New Orleans, Louisiana. At present it has an enrollment of about 10,000 students from almost every state and from 60 foreign countries.

The Mathematics program has, if anything, been strengthened by the reorganization of Tulane University in the aftermath of Hurricane Katrina.

Tulane's mathematical tradition can be traced back to the late nineteenth century, when Florian Cajori, later an expert in the history of mathematical notation, and the most famous translator of Isaac Newton's Principia, was the recipient of the first doctorate in mathematics from Tulane University (1894). Four undergraduates from the period up to the end of World War II (as well as Cajori) later became presidents of the Mathematical Association of America (Duren, McShane, Moise, Young); one (McShane) was a member of the National Academy of Sciences. In the 1950's Tulane became a major center in mathematical research. From 1970 to 2021, 207 Ph.D.'s were awarded.

The mathematics life at Tulane is enlivened by the distinguished mathematicians who visit each year for long or short periods, and by the international symposia which are held here from time to time. The department holds the annual Clifford Lectures, a week-long series of talks by a distinguished mathematician. A mini-conference supported by the National Science Foundation is held in conjunction with each of the Clifford lecture series. The first Clifford lecturer, in 1984, was Fields Medal recipient Charles Fefferman of Princeton University. In 1985 and 1986, the Clifford lecturers were Fields Medal winners, S. T. Yau of UC San Diego and William Thurston of Princeton University. The Clifford lecturers from 1987 through 1990 were Saharon Shelah of Hebrew University in Jerusalem, Clifford Taubes of Harvard University, Charles Peskin of Courant Institute and Haim Brezis of Université de Paris and Rutgers. From 1991 to 1996 they were Sylvain Cappell of Courant Institute of Mathematical Sciences, Nigel Hitchin of the University of Warwick and Persi Diaconis of Harvard University, Peter Sarnak of Princeton University and Dan Voiculescu of UC Berkeley. In 1994, a special conference on semigroups was held in honor of Alfred H. Clifford. In 1997 the Clifford lecturers were Paul Fife of University of Utah (Spring) and Peter Kronheimer of Harvard University (Fall). The speakers from 1998 to the present were Peter Bickel and Alexander Chorin of UC Berkeley, Robert Friedman of Columbia University, Sergei N. Artemov of City University of New York, T. J. Pedley of Cambridge University and Yakov Eliashberg of Stanford University.

The Mathematics Department at Tulane University offers a Ph.D. degree in Mathematics as well as Master of Science degrees in Mathematics, Applied Mathematics and in Statistics. These programs are described below. Undergraduate students majoring in mathematics or other sciences (like engineering, physics or computer science) with a strong interest in mathematics are encouraged to apply for admission to any one of the graduate programs. People who already hold undergraduate degrees in mathematics or other sciences are also encouraged to apply.

Requirements for admission into the Tulane Graduate School include:

  • The GRE Test is not required as part of the graduate application for any program.

The way to apply is to fill out and submit a Web-based application form.

Web-based application form

If you have any problems receiving the application, you may inquire at:

Director of Graduate Studies

Mathematics Department Tulane University 6823 St. Charles Ave. New Orleans, LA 70118 phone: (504) 865-5727 fax: (504) 865-5063 [email protected]

Most graduate students receive tuition waivers and teaching assistantships , which carry a stipend of $26,500 approximately. Teaching Assistants typically teach two laboratories (each meets weekly), although more advanced students may teach one section of an undergraduate course. All Ph.D. students are required to teach an undergraduate course, or to serve as teaching assistants in problem sessions attached to undergraduate courses, for at least two semesters during their residence.

Ph.D. students (U.S. Citizens or Legally Permanent Residents) are strongly recommended to apply the external funding resources such as NSF GRFP (Graduate Research Fellowship Program). More details on here https://www.nsfgrfp.org . For non-U.S. Citizens, there is a bountiful funding resources from several fields. Please find the information on https://immigrantsrising.org .

The Tulane Mathematics Department is known for its friendly atmosphere and its practice of fostering close contact and cordial relations between faculty and graduate students. To us, this is a very important aspect of life here, and we strive to maintain it. The ratio of graduate students to faculty members is kept between 1.0 and 1.5. This is important to us because it allows all new graduate students to soon become familiar with everyone and feel at home. It also allows the faculty to get to know the students during their first semester.

The incoming graduate students are advised by the Director of Graduate Studies of the Mathematics Department. The Director, in consultation with the students, determines appropriate first-year courses for each student, according to their preparation and interests. Throughout the program, the Director of Graduate Studies continues to help the students plan their studies and realize their mathematical interests.

All graduate (both Master and Ph.D.) students are given a cubicle in either 6 rooms at Gibson or Stanley Thomas 4th floor equipped with desks, desktop, bookcases and chalkboards. Students also have unlimited access to the lounge and the Mathematics Library, all located within the 4th floor at Gibson Hall. The lounge, or "commons room", is a place where people gather after seminars and colloquia for refreshments and discussion.

Ph.D. in Mathematics

Completing this degree takes about 5 years, depending on the student's preparation and progress satisfying the requirements. For advanced incoming students, limited transfer credit is possible. The Ph.D. prepares the students for a research career in mathematics in industry or academia.

Mathematicians with a Ph.D. from Tulane University have been successful getting jobs in a variety of colleges, research universities, government research laboratories and industries, including biotechnology, e-commerce and financial institutions.

More details on Ph.D. Requirements

Master of Science in Mathematics

This program is designed to provide students with the opportunity to broaden and deepen their knowledge of core areas of mathematics. The course work is designed to provide both breadth of knowledge and depth in an area of interest to the student. This experience will prepare the student for further studies leading to a Ph.D. degree in mathematics.

More details on MS in Mathematics

Master of Science in Applied Mathematics

This program is designed to provide students with the opportunity to broaden and deepen their knowledge of mathematics with an emphasis on those areas that have been most important in science and engineering. Students will also examine, through seminars and case studies, examples of significant applications of mathematics to other areas. This expanded base of knowledge, together with extensive experience in problem solving, is excellent preparation for further studies leading to the Ph.D. degree or for immediate employment in many areas of industry and government.

The program is open to students who have a Bachelor's degree in mathematics or a related field, and have completed undergraduate courses equivalent to Linear Algebra, Numerical Methods, and Analytical Methods. Proficiency in a programming language is essential. Students who have not completed all of these courses may be admitted and are required to take them during the first year.

More details on MS in Applied Mathematics

Master of Science in Statistics

The Master of Science degree in Statistics combines theory and application. Students in statistics will be trained in data collection, the editing and presentation of large data sets, the analyses of these sets and the mathematical foundations upon which all of these areas are based. The training has the two-fold purpose of preparing the student to enter commercial, governmental and other work areas which extensively rely on statistical information and to prepare the student to continue in pursuit of a more advanced degree. Students with appropriate background (three semesters of Calculus and some knowledge of elementary statistics) usually complete the program in one or two academic years.

More details on MS in Statistics

Master of Data Science

The M.S. in Data Science (MSDS) program is a professional, non-thesis degree that is jointly offered by the Mathematics and Computer Science departments. The burst of data in the modern world has fundamentally changed many fields of human activity, including healthcare, energy, manufacturing and scientific research. It has also generated an ever-increasing demand for a new type of professional: the data scientist. The MSDS program aims at providing the next generation of practitioners with cutting-edge data-driven problem-solving skills. These are based on rigorous mathematical foundations, and include data management, advanced statistical modeling, as well as the practical implementation and use of state-of-the-art algorithms.

More details on M.S. in Data Science

Math 6030: Stochastic Processes

Math 6050-6060: Real Analysis I & II

Math 6070: Introduction to Probability

Math 6080: Introduction to Statistical Inference

Math 6090: Linear Algebra

Math 6110-6120: Abstract Algebra I & II

Math 6210: Differential Geometry

Math 6240: Ordinary Differential Equations

Math 6250: Mathematical Foundation of Computer Security

Math 6280: Information Theory

Math 6300: Complex Analysis

Math 6310: Scientific Computing

Math 6350: Numerical Optimization

Math 6370: Time Series Analysis

Math 6470: Analytic Methods of Applied Mathematics

Math 7010-7020: Topology I & II

Math7150: Probability Theory I

Math 7110-7120: Algebra I & II

Math 7210-7220: Analysis I & II

Math 7240: Mathematical Statistics

Math 7260: Linear Models

Math 7291-7292: Algebraic Geometry I & II

Math 7310-7320: Applied Math I & II

Math 7360: Data Analysis

Math 7510-7520: Differential Geometry I& II

Math 7530-7540: Partial Differential Equations I & II

Math 7550: Probability Theory II

Math 7560: Stochastic Processes II

Math 7570-7580: Scientific Computation II & III

Math 7710-7790: Special Topics Courses

More information on the courses

The Mathematics Department consists of 24 regular faculty members, several postdoctoral researchers and frequent visiting faculty in many areas of mathematics.

Its faculty enjoys national and international recognition in Algebra, Analysis, Differential Geometry, Mathematical Physics, Probability and Statistics, Scientific Computation, Theoretical Computer Science, and Topology. The researchers in Scientific Computation and in Statistics, and an increasing number of faculty in other areas, collaborate actively with colleagues in other units of the university such as the Schools of Engineering, Liberal Arts and Sciences, Medicine, and Public Health.

During the past five years our regular faculty have published over 100 research articles and several books. The regular faculty direct theses in very diverse areas which range through all of Pure Mathematics, Applied Mathematics, and Statistics. Detailed information can be found on the faculty page.

The Mathematics Department is housed in the upper floors of Gibson Hall, a stone structure built in 1894. Here are located faculty, graduate students, and staff offices, as well as classrooms, seminar rooms and computers linked to Tulane's main computing system. The department also contains the A. H. Clifford Mathematics Research Library, housing some 28,000 bound volumes and subscribing to 243 journals devoted to all areas of mathematics.

The department has a Microsoft Network with Windows and Mac workstations. The Math Department is connected by a cloud based network and the ability to store your files on Box (a cloud base storage).  Graduate students are provided with adequate computing resources, Ethernet connections, Matlab, Mathematica, Microsoft office and other essential programs that are aimed to help aid in your success.

Tulane University is located in America's most exciting and most visited city. Our department is on St. Charles Avenue, across from Audubon Park, in a quiet residential area full of majestic oak trees and fine old antebellum homes. Often-photographed streetcars provide an easy ride to the picturesque French Quarter. New Orleans has a rich cultural life, with a symphony orchestra, operas, ballets, plays, a noted art museum, many art galleries, excellent jazz, a major jazz festival and many other events. During Mardi Gras (40 days before Easter) the town fills with parades and revelry. New Orleans is also famous for its cuisine; it boasts a number of great restaurants, and many more with good inexpensive meals.

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