math phd admission requirements

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Ph.D. Admissions

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• M.S. Admissions

The application window for Fall 2024 is closed.

The Mathematics Department offers two programs to obtain a Ph.D. Applicants can pursue a Ph.D. in  Applied & Interdisciplinary Mathematics or Mathematics.  Please use the "Programs" link at the left to explore our offerings.

• Three Letters of Recommendation  (May submit up to 5 letters, but only 3 are required.)
• Curriculum Vitae or Resume
• Academic Statement of Purpose (concise - no limit)
• Personal Statement (500 word limit)
• TOEFL iBT Special Home Edition and IELTS Indicator online test are accepted.
• Exemption rules:   https://rackham.umich.edu/admissions/applying/tests/
• List of  International English Exclusive Institutions  approved by Rackham

GRE General Test scores are no longer included in the admission process in accordance with a  policy of the Rackham Graduate School .

GRE Mathematics Subject Test scores are strictly optional. However, if an applicant chooses so, they may submit them as a combined pdf file with their transcript or personal statement.

Application Timeline

The Mathematics Department's graduate programs only accept applications for Fall semesters. 

General Requirements for Admission

A student must have completed a bachelor's degree at an accredited college or university by the time of entry in order to be considered for admission.

Applied & Interdisciplinary Mathematics (AIM) Ph.D. Admissions Requirements

Successful AIM Ph.D. applicants will demonstrate an interest in an interdisciplinary area of applied mathematics in addition to substantial mathematical ability. Two types of students are generally considered for admission to the AIM Ph.D. program:

• Mathematics majors with excellent grades in mathematics courses and excellent letters of recommendation. The admissions committee will also take into account other scholarly activities such as summer research experience, published papers, or courses in other fields.
• Non-mathematics majors from the physical, life, or engineering sciences, or from other appropriate areas of study. Such students are expected to have completed at least two upper division mathematics courses, and/or have substantial exposure to mathematics in other courses, and may submit a GRE mathematics subject test score. Other experience in working with mathematics (for instance, summer research positions) will also be taken into account, as well as grade point average and letters of recommendation.

Mathematics Ph.D.  Admission Requirements The undergraduate major need not be mathematics, but a student should have mastered material roughly equivalent to the undergraduate mathematics major at The University of Michigan including:

• three semesters of calculus
• one or two semesters of differential equations
• one semester courses in modern algebra, linear algebra, geometry or topology
• advanced calculus of one and several variables

In addition, a student should have completed at least three additional mathematics courses and at least two courses in related fields such as statistics, computer science, or the physical sciences. Students with strong records in less comprehensive programs will be considered for admission but if admitted should expect to spend the first one or two semesters in graduate school completing their undergraduate preparation in mathematics. Based on historical data, we expect that successful applicants to the Ph.D. program will have an overall GPA of at least 3.3 on a 4.0 scale.

Application Requirement Details

GRE, TOEFL, and IELTS Tests

• GRE General test scores are not required. 
• GRE Mathematics Subject Test scores may be submitted as a combined pdf file with your transcript or personal statement. (Optional)
• TOEFL or IELTS exam is required for students whose native language is not English
• TOEFL and IELTS exams should not be older than two years as of the Admission Deadline.
• Minimum TOEFL and IELTS scores must meet Rackham's requirements here .
• TOEFL or IELTS Exemptions are only given per Rackham's rules here .
• ETS school code for the University of Michigan Rackham Graduate School is 1839

Letters of Recommendation

Letters of recommendation play an especially crucial role in the admission process. At least three letters are required, and up to five may be submitted. Applicants should choose as recommenders people who know their strengths and weaknesses relevant to graduate study in mathematics. The most useful letters are those which list in some detail the accomplishments of the student and make direct comparisons with other students who have succeeded at major U.S. graduate schools. International students already in the U.S. should submit letters from their U.S. institution, whenever possible.  Please register your recommenders for the electronic Letters of Recommendation when using the Online Application.  Letters received after the application deadline will be accepted, but should be received within 1 week of that deadline.

Those students who will have completed a Master's degree in Mathematics by the time they begin studies at the University of Michigan must apply to the Ph.D. program. Others may apply to either program. 

Academic Statement of Purpose

Focus your academic statement on your mathematical interests, research experience, published papers, math camps, teaching & tutoring experience etc. Be sure to mention any specific faculty with whom you wish to work.

Personal Statement:

Focus your personal statement on what makes you unique, any struggles you have experienced and overcome, and why you feel U-M Math is the right place for you.  Be sure to include any hardships you have experiencedand how you overcame them. These could be financial, familial, or personal.

Transcript Submission:

The Mathematics Admissions Committee will review uploaded transcripts with university logos during the application process.  While these are considered "unofficial" transcripts because they have been opened from their original sealed envelopes, they are acceptable.  If an applicant receives an offer of admission, an official transcript in a sealed envelope will need to be mailed from the institution directly to the Rackham Graduate School.

Please submit your most current transcript with your online application by the due date.  If you would like the Admissions Committee to see your Fall term scores, you may email them to [email protected] after the due date, and they will be included with your application.

Additonal Application Materials: If you have additional materials you would like to submit with your application, you may email them to [email protected].  Be sure to include your name and umid number in the email and attach files in pdf format.

Note:  All credentials submitted for admission consideration become the property of the University of Michigan and will not be returned in original or copy form.

Additional Information:  Please visit the admissions page of the Rackham Graduate School for additional information regarding admission including: minimum graduate school requirements, residency, and application fees.    Unfortunately, application fee waivers are not available for international students.

Financial Support for Ph.D. Students

Ph.D. Programs

Most students enrolled in the Ph.D. program in Mathematics are granted full financial support including an annual stipend, tuition waiver, and health insurance for a period of five years, subject to satisfactory progress. The Department offers aid in the form of Graduate Student Instructorships, Research Assistantships, and Fellowships.

All entering Ph.D. students will be considered for Graduate Student Instructorships, which normally require four classroom hours of teaching per week plus additional office hours during the Fall and Winter terms. The stipend for such an appointment in 2021-2022 is $11,598 per term. In addition, Graduate Student Instructors receive a full tuition waiver. Teaching duties may involve teaching a section of a first-year calculus or pre-calculus course or serving as an instructor for recitation sections attached to a faculty lecture in multivariable calculus or elementary differential equations. The Department of Mathematics has many fellowship opportunities, including the Copeland, Glover, Rainich, and Shields Fellowships which may provide a stipend, tuition waiver and in some cases a reduced teaching load. Other fellowships administered by the Rackham Graduate School can be found at their  Fellowships office .  The University of Michigan is part of the CIC consortium, which also awards fellowships to outstanding underrepresented applicants. Also available are prestigious Rackham Science Award’s given out by the Rackham Graduate School.

After Admission

All new Graduate Student Instructors are required to attend an orientation and training program which is held the week before classes begin. New Graduate Student Instructors whose Undergraduate Degree is not from an English speaking University must pass an English Evaluation which tests the specific oral skills needed for classroom teaching and are required to attend a three-week cultural orientation program starting in July.

Research Assistantships are awarded mainly to senior Ph.D. students to relieve them of teaching duties during the final part of their dissertation research. Students at this point may also compete for Rackham Dissertation Fellowships, which provide full support for one year, or Research Partnerships. A small number of positions as paper-graders for the larger advanced courses is available each term.

Some additional funds are often available for support during the summer. More advanced students who are actively involved in research may be supported from NSF grants awarded to faculty members. For other students there is a limited number of Departmental fellowships and a few teaching positions are available. No advanced graduate courses are offered in either the Spring or Summer half-terms and students are encouraged to spend some of their summers attending workshops, doing research, working in government, or seeking internships in industry.

Marjorie Lee Browne (MLB) Scholars Program - an MS bridge to PhD program for diverse students

The Department of Mathematics at the University of Michigan is pleased to offer the Marjorie Lee Browne (MLB) Scholars Program. The program is named for Dr. Marjorie Lee Browne, who in 1949 became the first African-American woman to earn a Ph.D. in Mathematics at the University of Michigan. The MLB Scholars Program is an enhanced option for the M.S. degree in either Mathematics or Applied and Interdisciplinary Mathematics that is designed to give students professional knowledge of pure or applied mathematics in order to prepare them for continuing toward a Ph.D.  Please see this Marjorie Lee Brown Scholars webpage for eligibility and details.

If you have any questions regarding the application process, please contact the Department of Mathematics at [email protected].

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Overview of the Application Procedure

Welcome to the MIT Mathematics Graduate Admissions page. This page explains the application process in general. For complete details, go to the on-line application which is available mid-September to December. These instructions are repeated there.

MIT admits students starting in the Fall term of each year only. Admission is to the PhD program only; there is no Masters program. There is no separate application for financial support; all admitted students are offered support.

Submitting GRE scores is entirely optional: We will accept scores if submitted (and are most interested in the Math Subject test result, if any) but it will not hurt your application if not included.

To apply, follow these steps:

Fill out the on-line application by 23:59, EST, December 15.

You will be submitting:

• Field(s) of interest
• Personal information/addresses
• International student data
• Three or more names and e-mail addresses of letter writers
• Educational and work history, including IELTS/TOEFL scores (preferably from this year), and honors
• Grades in math/science/engineering courses and overall
• Statement of objectives
• Outside financial support and potential outside support
• Credit/debit card payment of $75
• The Math department requires applicants to upload an electronic copy of undergraduate transcripts. Hard copies of official transcripts are not required at the time of application.

Arrange for submission of (official reports only)

• Letters of recommendation
• For international students, IELTS (or TOEFL iBT)

We recommend that before November 15 you notify your letter writers that you will be needing evaluations from them, so that they have time to prepare them and submit them by December 15. Once you have submitted your on-line application, instructions to your letter writers will be generated for you. You are responsible for making sure that your letter writers have copies of these instructions.

You self-reported your grades in step 1, but we require an official transcript for all admitted students. If/when we request this, arrange for an official copy of your college transcript to be sent to:

Academic Services, Room 2-110 Dept of Mathematics, MIT 77 Massachusetts Ave. Cambridge MA 02139-4307 USA

TOEFL reporting codes Institution code: MIT = 3514 Mathematics Department code: 72

International Students

IELTS is the English language proficiency test we prefer, but we also accept the TOEFL iBT . (On the other hand, we generally do not accept the TOEFL PBT.) To have IELTS results reported, indicate "Mathematics Department, Massachusetts Institute of Technology (MIT)" on your IELTS test application; no code or address is needed. To have TOEFL iBT results reported, use the codes above (3514 for MIT, and 72 for Mathematics).

If you are an international student, you should take the IELTS (or TOEFL iBT) by December 31. If you will receive an undergraduate degree from an English-language university in an English-speaking country after attending it for at least three years, then the Math Department will waive the English language proficiency test requirement.

Paper Forms

If for some reason, you are unable to use the on-line system, you may use paper forms. But note that on-line documents allow us to consider your application more quickly and conveniently. Your letter writers may also use paper forms, if necessary.

Please address questions about the application process to [email protected] . You can find more information about MIT graduate admissions in general at the MIT Graduate Admissions site .

Computational Science and Engineering

Students with primary interest in Computational Science may also consider applying to the interdisciplinary Computational Science and Engineering (CSE) program, with which the Mathematics Department is affiliated. For more information, see https://oge.mit.edu/programs/computational-science-and-engineering-phd/ .

To apply for admissions and financial aid, or for additional information on admissions requirements for the Ph.D. program in pure mathematics, please go to the appropriate Harvard Kenneth C. Griffin Graduate School of Arts and Sciences website listed below. All other inquiries may be directed to the Graduate Program Administrator of the Mathematics Department.

• Harvard Kenneth C. Griffin Graduate School of Arts and Sciences (Harvard Griffin GSAS)
• Mathematics Graduate Studies
• Financial Support

Graduate Program Administrator

The Department values diversity among its members, is committed to building a diverse intellectual community, and strongly encourages applications from women and minorities.

Preparing the Application The statement of purpose for graduate applications is carefully weighted by the admissions committee. The applicant’s statement should convince the committee that they are able to communicate effectively and with a deep understanding of mathematics. It is not intended to be a biographical sketch or a reflection on one’s decision to enter the field.

Three letters of recommendation are required. Letter writers should be faculty or others qualified to evaluate the applicant’s potential for graduate study in mathematics. The letters must be submitted online and by the application deadline.

Applicants should include any research papers, publications, and other original works they would like to have evaluated by the admissions committee.

The department requests that applicants submit GRE Mathematics Subject Test scores if practical. Applicants should check on the ETS website for test dates in their area to ensure the scores will be submitted before the application deadline. An official score report should be sent to Harvard Kenneth C. Griffin Graduate School of Arts and Sciences using code 3451.

While the admissions committee reviews all applications submitted before the deadline, missing math subject test scores provide one less data point available to evaluate the application. Depending on the strength of the application, the missing subject test scores may put the application at a disadvantage.

Applicants who are non-native English speakers and who do not hold an undergraduate degree from an institution at which English is the primary language of instruction must submit scores from the Internet Based Test (IBT) of the Test of English as a Foreign Language (TOEFL) or the International English Language Testing System (IELTS) Academic test.

Harvard Griffin GSAS requires applicants to upload an electronic copy of undergraduate transcripts. Hard copies of official transcripts are not required at the time of application.

Ph.D. Program in Pure Mathematics The department does not grant a terminal Master’s degree, but the Master’s can be obtained “on the way” to the Ph.D. by fulfilling certain course and language exam requirements.

In general, there is no transfer status application to the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences or to the Department of Mathematics. No formal credit is given for an MSc or MA earned elsewhere. All applicants are considered to be applying as first-year graduate students. The only difference Master’s study may make is to better prepare students for the Qualifying Exam.

All graduate students are admitted to begin their studies in the fall term. The department plans on an entering class of about twelve students. Since the admissions committee receives a few hundred applications, the competition is keen.

Funding Graduate Study Applicants are urged to apply for all funding available to them. If no outside funding is available to the applicant, financial aid in the form of scholarships, research assistantships, and teaching fellowships is available. In general, students who do not have outside support will get scholarship support in their first year, but students are required to act as a teaching fellow for one-half course (i.e. for a one-term course) in their second through fifth years.

The department strongly recommends applicants to seek out and apply for all sources of financing available to them for graduate study. Recommended sources for funding US graduate students are NSF Graduate Fellowships and NDSEG Fellowships . Applicants from the UK are urged to also apply for the Kennedy fellowships and applicants from UK, New Zealand, Canada and Australia for Knox fellowships . International students may apply for the Fullbright IIE or any home country fellowships available for study abroad.

Harvard John A. Paulson School of Engineering and Applied Sciences The Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS) offers programs for both the Master’s degree and the Ph.D. degree in Applied Mathematics. Please visit the SEAS website for more information on degrees in applied mathematics at www.seas.harvard.edu

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The department offers the doctor of philosophy degree with a major in mathematics and a master of arts degree in mathematics.

The Ph.D. degree requires proficiency in basic and advanced graduate mathematics and the completion of a dissertation containing a significant piece of original research in some area of mathematics. The scope of the research program in mathematics is broad. The Ph.D specialty and dissertation may be in any area of mathematics, including but not limited to algebra, algebraic geometry, applied mathematics, combinatorics, computational mathematics, complex analysis, differential equations, differential geometry, dynamical systems, harmonic analysis, logic, mathematical biology, number theory, probability, and topology. A complete list of faculty and their areas of expertise is available through the department website .

Students in the Ph.D. program also have the option to earn a master of arts degree .

Please consult the table below for key information about this degree program’s admissions requirements. The program may have more detailed admissions requirements, which can be found below the table or on the program’s website.

Graduate admissions is a two-step process between academic programs and the Graduate School. Applicants must meet the minimum requirements of the Graduate School as well as the program(s). Once you have researched the graduate program(s) you are interested in, apply online .

Admission is competitive. Applicants to the Ph.D. program are automatically considered for financial support. For more information about application to the Ph.D. and M.A. programs, see the department's admission website .

Graduate School Resources

Resources to help you afford graduate study might include assistantships, fellowships, traineeships, and financial aid.  Further funding information is available from the Graduate School. Be sure to check with your program for individual policies and restrictions related to funding.

Program Resources

Prospective students should see the program website for funding information .

Minimum Graduate School Requirements

Major requirements.

Review the Graduate School minimum academic progress and degree requirements , in addition to the program requirements listed below.

MODE OF INSTRUCTION

Mode of instruction definitions.

Accelerated: Accelerated programs are offered at a fast pace that condenses the time to completion. Students typically take enough credits aimed at completing the program in a year or two.

Evening/Weekend: ​Courses meet on the UW–Madison campus only in evenings and/or on weekends to accommodate typical business schedules.  Students have the advantages of face-to-face courses with the flexibility to keep work and other life commitments.

Face-to-Face: Courses typically meet during weekdays on the UW-Madison Campus.

Hybrid: These programs combine face-to-face and online learning formats.  Contact the program for more specific information.

Online: These programs are offered 100% online.  Some programs may require an on-campus orientation or residency experience, but the courses will be facilitated in an online format.

CURRICULAR REQUIREMENTS

Required courses.

Take a total of 51 graduate credits, or generally 18 courses . This includes courses in math and in breadth. In the Ph.D. program, math courses numbered above 700 are for graduate credit. Math courses below 700 must be approved by the academic advisor.

There are five general and overlapping areas of specialization 1 within the department:   

• Algebra, Algebraic Geometry, Combinatorics and Number theory
• Analysis, Differential Equations and Probability
• Applied and Computational Mathematics
• Geometry, Topology, and Dynamics

There is also a specialty in Mathematics Education 1 . The course requirement is the same as for the other specialties except that the required 51 credits should include 18 credits in courses that relate to mathematics education, and at least one of the courses must be on research techniques in education. The 18 credits may come (wholly or in part) from courses included in breadth.

These pathways are internal to the program and represent different curricular paths a student can follow to earn this degree. Pathway names do not appear in the Graduate School admissions application, and they will not appear on the transcript.

Graduate School Policies

The  Graduate School’s Academic Policies and Procedures  provide essential information regarding general university policies. Program authority to set degree policies beyond the minimum required by the Graduate School lies with the degree program faculty. Policies set by the academic degree program can be found below.

Major-Specific Policies

Prior coursework, graduate work from other institutions.

With program approval, students in the Ph.D. program are allowed to count no more than 19 credits of graduate coursework from other institutions. Coursework earned ten years or more prior to admission to a doctoral degree is not allowed to satisfy requirements.

UW–Madison Undergraduate

No more than 7 credits from a UW–Madison undergraduate degree are allowed to count toward the degree. Coursework earned ten years or more prior to admission to a doctoral degree is not allowed to satisfy requirements.

UW–Madison University Special

With program approval, students are allowed to count no more than 15 credits of coursework numbered 300 or above taken as a UW–Madison Special student. Coursework earned ten years or more prior to admission to a doctoral degree is not allowed to satisfy requirements.

This program follows the Graduate School's Probation policy.

• Good standing (progressing according to standards; any funding guarantee remains in place).
• Probation (not progressing according to standards but permitted to enroll; loss of funding guarantee; specific plan with dates and deadlines in place in regard to removal of probationary status).
• Unsatisfactory progress (not progressing according to standards; not permitted to enroll, dismissal, leave of absence or change of advisor or program).

ADVISOR / COMMITTEE

Students who are not yet working with a dissertation advisor are required to meet semiannually with their academic advisor. All students must have a dissertation advisor by the end of the sixth semester.

CREDITS PER TERM ALLOWED

15 credits. Minimum of 6/semester, other than dissertators.

Time Limits

Students must complete all program requirements within eight years of beginning the program . Extensions have to be approved by the program.

A candidate for a doctoral degree who fails to take the final oral examination and deposit the dissertation within five years after passing the preliminary examination may be required to take another preliminary examination and to be admitted to candidacy a second time.

Doctoral degree students who have been absent for ten or more consecutive years lose all credits that they have earned before their absence. Individual programs may count the coursework students completed prior to their absence for meeting program requirements; that coursework may not count toward Graduate School credit requirements.

grievances and appeals

These resources may be helpful in addressing your concerns:

• Bias or Hate Reporting  
• Graduate Assistantship Policies and Procedures
• Office of the Provost for Faculty and Staff Affairs
• Dean of Students Office (for all students to seek grievance assistance and support)
• Employee Assistance (for personal counseling and workplace consultation around communication and conflict involving graduate assistants and other employees, post-doctoral students, faculty and staff)
• Employee Disability Resource Office (for qualified employees or applicants with disabilities to have equal employment opportunities)
• Graduate School (for informal advice at any level of review and for official appeals of program/departmental or school/college grievance decisions)
• Office of Compliance (for class harassment and discrimination, including sexual harassment and sexual violence)
• Office of Student Conduct and Community Standards (for conflicts involving students)
• Ombuds Office for Faculty and Staff (for employed graduate students and post-docs, as well as faculty and staff)
• Title IX (for concerns about discrimination)

Students should contact the department chair or program director with questions about grievances. They may also contact the L&S Academic Divisional Associate Deans, the L&S Associate Dean for Teaching and Learning Administration, or the L&S Director of Human Resources.

Take advantage of the Graduate School's  professional development resources to build skills, thrive academically, and launch your career. 

• Learn a substantial body of mathematics in introductory and research level graduate courses in mathematics.
• Complete a dissertation under the guidance of an advisor. The dissertation should make an original and substantive contribution to its subject matter.
• Demonstrate breadth within the learning experiences.
• Present research in seminar talks, conferences or publications.
• Communicate complex ideas in a clear and understandable manner.
• Foster ethical and professional conduct.

Please visit the Math Department website for a complete list of faculty and instructional academic staff .

• Requirements
• Professional Development
• Learning Outcomes

Contact Information

Mathematics College of Letters & Science math.wisc.edu

Kathie Brohaugh, Graduate Coordinator [email protected] 608-263-8884 203 Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706

Autumn Kent, Director of Graduate Studies [email protected]

Tullia Dymarz, Director of Admissions [email protected]

Graduate Program Handbook View Here

Graduate School grad.wisc.edu

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Admission to the PhD program is based on primarily on academic records, letters of recommendation, and a personal statement. Application materials for the fall of 2024 must be received by December 15, 202`3. However, applicants who wish to be considered for the Vivien Thomas Scholars Initiative must submit their application by December 1, 2023. Preliminary admissions decisions will be made in January 2024. All applicants will be notified of the admissions committee’s decision by email.

The Department of Mathematics values diversity among its members, is committed to building a diverse intellectual community, and strongly encourages applications from all interested parties.

Via the online application , applicants should submit:

• A Statement of Purpose
• An optional Personal Statement
• Transcripts from all institutions attended
• Three letters of recommendation
• Official TOEFL scores (if English is not your first language)

Submission of GRE Math subject scores is optional. The department views GRE Math subject scores as useful information. For applications which do not include these scores, more weight will be placed on the other elements of the application. Applicants who choose to self-report a GRE Math Subject score need not send an official score report at time of application; we will request verification only if you are accepted to the program.

The required Statement of Purpose discusses your academic interests, objectives, and preparation. The optional Personal Statement describes your personal background, and helps us create a more holistic understanding of you as an applicant. If you wish you may also discuss your personal background in the Statement of Purpose (e.g. if you have already written a single essay addressing both topics), instead of submitting separate statements.

Application fee waivers are available based on financial need and/or participation in certain programs .

No application materials should be mailed to the department. All application materials are processed by the Graduate Admissions Office.

Senior Academic Program Coordinator 410-516-4178 [email protected]

Undergraduate Background

The following is an example of what the math department would consider a good background for a student coming out of a four-year undergraduate program at a college or university in the U.S. (assuming a semester system):

• Calculus in one variable (two semesters, or AP credits)
• Multivariable Calculus (one semester)
• Linear Algebra (one semester)
• Complex analysis (one semester)
• Real analysis (two semesters)
• Abstract algebra (two semesters)
• Point-set topology (one semester)

Many admitted students have taken upper-level undergraduate mathematics courses or graduate courses. Nevertheless, the department does admit very promising students whose preparation falls a little short of the above model. In such cases, we strongly recommend that the student start to close the gap over the summer, before arriving for the start of the fall semester.

Financial Support    

Students admitted to the PhD program receive full tuition fellowships and teaching assistantships. Teaching assistant salaries for the 2023–2024 academic year are $34,000. Students making satisfactory progress can expect to be supported for six years. Exceptional applicants are considered for supplementary fellowships of $6,000 each year for three years.

Students from underrepresented groups may be eligible for other university-wide supplemental fellowships . Summer teaching is available for students seeking extra income.

We recognize that it can be financially burdensome to relocate to a new city to attend a PhD program. Students who are accepted to PhD programs at JHU can apply to receive a $1,500 need-based grant to offset the costs of relocating to JHU. These grants provide funding to a portion of incoming students who, without this money, may otherwise not be able to afford to relocate to JHU for their PhD program. This is not a merit-based grant. Applications will be evaluated solely based on financial need. Find out how to apply for the grant .

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Additional Information for International Students

Student Visa Information:  The Office of International Services at Homewood  will assist admitted international students in obtaining a student visa.

English Proficiency: Johns Hopkins University requires students to have adequate English proficiency for their course of study. Students must be able to read, speak, and write English fluently upon their arrival at the university. Applicants whose native language is not English must submit proof of their proficiency in English before they can be offered admission and before a visa certificate can be issued. Proficiency can be demonstrated by submitting results from either the Test of English as a Foreign Language (TOEFL) or the IELTS . Johns Hopkins prefers a minimum score of 100 on the TOEFL or a Band Score of 7 on the IELTS. Results should be sent to Johns Hopkins directly by TOEFL or IELTS. Applicants taking the IELTS must additionally upload a copy of their score through the application system. However, do not send the student copy or a photocopy of the TOEFL.

Vivien Thomas PhD Scholars

The  Vivien Thomas Scholars Initiative (VTSI ) is an endowed fellowship program at Johns Hopkins for STEM PhD students. It provides full tuition, stipend, benefits, targeted mentoring, and professional development. Students who have attended a historically black college and university or other minority serving institution for undergraduate study are eligible.

There is no additional application. To be considered for the VTSI, all components of the PhD application, including supplemental components, special VSTI questions, and letters of recommendation must be completed by  December 1, 2023 , even if the program deadline is later.

Graduate Admissions

Application information.

• The deadline for the application for admission to Autumn 2024 has passed.
• Applications are not accepted after the deadline.
• The application for admission to Autumn 2025 will open by October 2024 and the deadline will be in early December 2024. 
• Applications are accepted for admission to the Autumn quarter only.
• Applications are accepted to the PhD program only.
• The department does not offer admission to a Master’s degree program to applicants from outside Stanford or current Stanford graduate students.

For current Stanford undergraduate students only:  The department accepts applications to the Coterminal Master’s degree program. Please see the  Coterminal Master’s Degree page for more information.

• Please read the Steps for Applying section below.

The Department of Mathematics recognizes that the Supreme Court issued a ruling in June 2023 about the consideration of certain types of demographic information as part of an admission review. All applications submitted during upcoming application cycles will be reviewed in conformance with that decision.

The Department of Mathematics welcomes graduate applications from individuals with a broad range of life experiences, perspectives, and backgrounds who would contribute to our community of scholars. Review of applications is holistic and individualized, considering each applicant’s academic record and accomplishments, letters of recommendation, and admissions essays in order to understand how an applicant’s life experiences have shaped their past and potential contributions to their field.

Contact Information

• Before contacting the department, read the information on this page, including the Frequently Asked Questions at the bottom of this page. 
• Before contacting the department, read the information on the  University Graduate Admissions website .
• Please do not contact the department to check if your application materials are complete. Instead, you may  Visit Your Status Page  to check your application status. 
• Only if your question is not answered on this page or the pages mentioned above, please send an email to: mathgradadmissions [at] stanford.edu (mathgradadmissions[at]stanford[dot]edu) .

Steps for Applying

• Review all information and instructions on this page. Read through the Frequently Asked Questions at the bottom of this page.
• Review the information and instructions on the University Graduate Admissions website .
• Submit your application online. Make sure you have all required items listed below. The link to the online application is on the  University Graduate Admissions website .
• Have ETS send your official test scores to Stanford University (if applicable, see below).

Required Items:

Statement of Purpose See the Graduate Admissions page on statements for more information.

• Soft copy of unofficial transcripts/academic records for every post-secondary institution where you were previously or are currently enrolled in an undergraduate or graduate degree program. See the Graduate Admissions page on transcripts for transcript requirements and the  Minimum Education Requirements .
• Names and contact information of your recommenders, minimum of 3, maximum of 4. Your recommenders will receive an email with instructions for submitting their letters online after you enter their information in the application. The admissions committee will start their review shortly after the deadline so ask your recommenders to submit their letters by the application deadline so they will be included in the review.  See the Graduate Admissions page on recommendations for more information.
• This is required of applicants whose first language is not English. If your first language is not English, you may be eligible for an exemption or a waiver. Review the  TOEFL information on the University Graduate Admissions website  for more information including exemption and waiver information.
• The Department is not able to provide fee waivers. Please see the link above for the available fee waivers and how to submit a request. Requests are due 2 weeks before the application deadline. 

Optional Item:

• GRE Mathematics Subject Test Score ( Submitting the score is optional and your application will not be negatively impacted if you do not submit the score.)

Items that are NOT required:

• GRE General Test Scores (Not required or accepted as part of the application.)
• Official transcripts (If you are offered admission to Stanford and accept the offer, you will be required to submit official transcripts at that time.)
• Please  do not  upload research papers to the online application. Instead, please ensure that you have a letter of recommendation that addresses your papers, and/or provide a short description of the paper in your statement of purpose. You may also include a link to your papers in your statement of purpose if you would like.

Fellowship Opportunities

Students in the PhD program who make satisfactory progress are fully funded for five years. Funding sources include department fellowships, teaching assistantships and research assistantships.

In addition, applicants are encouraged to apply for fellowships outside the department. See below for some of the opportunities available, including some of the fellowships received by our students in recent years.

Join dozens of  Stanford School of Humanities and Sciences students  who gain valuable leadership skills in a multidisciplinary, multicultural community as  Knight-Hennessy Scholars  (KHS). KHS admits up to 100 select applicants each year from across Stanford’s seven graduate schools, and delivers engaging experiences that prepare them to be visionary, courageous, and collaborative leaders ready to address complex global challenges. As a scholar, you join a distinguished cohort, participate in up to three years of leadership programming, and receive full funding for up to three years of your PhD studies at Stanford. Candidates of any country may apply. KHS applicants must have earned their first undergraduate degree within the last seven years, and must apply to both a Stanford graduate program and to KHS. Stanford PhD students may also apply to KHS during their first year of PhD enrollment. If you aspire to be a leader in your field, we invite you to apply. The KHS application deadline is October 11, 2023. Learn more about  KHS admission .

The  National Defense Science and Engineering Graduate (NDSEG) Fellowships  are awarded to applicants who will pursue a doctoral degree in, or closely related to, an area of DoD interest within fifteen disciplines including mathematics.

The  Hertz Graduate Fellowship Award  supports graduate students working towards the Ph.D. degree in the applied physical, biological and engineering sciences including mathematics.

The   NSF Graduate Research Fellowship Program  recognizes and supports outstanding graduate students in NSF-supported science, technology, engineering, and mathematics disciplines who are pursuing research-based Master’s and doctoral degrees at accredited United States institutions.

Natural Sciences and Engineering Research Council of Canada (NSERC) Scholarships  provide financial support for studies in a master’s or doctoral program.

Frequently Asked Questions

Graduate Program Questions

For non-Stanford applicants, the Mathematics Department offers admission to the PhD program only.

Please see the Explore Graduate Programs page for other departments that offer a Master's degree.

Both pure and applied mathematics can be studied. There is no separate applied mathematics program. See the  Research Areas page  for information about areas of faculty research.

Five years is the typical amount of time it takes to complete the program. See the  PhD Program page  for the timeline of progression through the degree.

All admitted PhD students who make satisfactory progress are fully funded for five years. Funding sources include department fellowships, teaching assistantships and research assistantships.

In addition, applicants are encouraged to apply for fellowships outside the department. See the Fellowships Opportunities section above for some of the opportunities available, including some of the fellowships received by our students in recent years.

Application Questions 

We are not able to accommodate prospective applicant visits. Please review the information on our website and on the University Graduate Admissions website which should answer almost all questions about the application process and our program. If there are specific questions that are not answered on our website please email them to mathgradadmissions [at] stanford.edu (mathgradadmissions[at]stanford[dot]edu) .

No, we do not accept late applications.

A committee of several math faculty members reads the files. The composition of the committee varies each year.

By and large, we like to see that you are eager to learn and can handle difficult mathematics. In most cases, this is evidenced by taking and doing well in advanced math courses (admitted students usually have close to perfect GPAs in math courses), as well as having close interaction with faculty who can testify to this.

If you are applying from a school from which we rarely get applicants, it is helpful if some of your recommenders can compare you to others who have gone on to comparable math graduate schools.

Although you’re certainly welcome to include letters from faculty mentors about research experience, we don’t expect you to have such experience or to have attended an REU.

Unfortunately the admissions committee doesn't have the time that would be needed to look at research papers in any detail. Instead, please ensure that you have a letter of recommendation that addresses your paper(s). If you wish, include a weblink to the paper(s) in your statement of purpose.

Although we do not formally require an undergraduate degree in math, we do expect you to have a math background that is comparable to it. In such cases, please explain your experience carefully in your statement of purpose.

Applicants must hold, or expect to hold before enrollment at Stanford, a bachelor’s degree from a U.S. college or university accredited by a regional accrediting association. 

Applicants from institutions outside the U.S. must hold the equivalent of a U.S. bachelor’s degree from a college or university of recognized standing.

See the Minimum Education Requirements page on the Graduate Admissions website for full details.

Yes. The committee begins reviewing applications shortly after the application deadline so it's important to have all materials submitted by the deadline, including letters of recommendation. 

No. Your application will be reviewed with the transcripts you uploaded to the online application. We are not able to update the transcripts of applicants.

No. Students are admitted into the PhD program for Autumn quarter only.

ETS Exam and Score Questions

Submitting the GRE Mathematics subject score is optional and  your application will not be negatively impacted if you do not submit a score .

If you do submit a score, we do not have a cutoff. Most admitted applicants who submit scores have a GRE Mathematics subject score of above 800.

GRE Mathematics Subject scores are optional. You may still apply if you do not have a score to report and  your application will not be negatively impacted if you do not submit the score.

The TOEFL requirement is set by the University and is required for applicants whose first language is not English unless they qualify for an exemption or a waiver of the requirement. Please see the  University Graduate Admissions TOEFL information  to determine if you need to submit TOEFL scores or qualify for an exemption or waiver.

For complete TOEFL requirements please see the  University Graduate Admissions TOEFL information .

Official scores must be reported directly to Stanford from the Educational Testing Service (ETS). Stanford’s ETS institutional code is 4704. A department code is not necessary, but it is fine if you gave ETS a department code in addition to the institutional code.

Please see the ETS (Educational Testing Service) website ( www.ets.org ) for scoring information.

Your application will be reviewed with your self-reported scores until your official scores are received. The review of your application will not be affected by a delay in receiving official scores. 

Ideally you will have taken the exam before the application deadline so that you will be able to provide your self-reported scores on the application. If you do not know your scores before the application deadline, you can indicate your test date or a future test date on the application without listing your scores. You will then need to send your self-reported scores to mathgradadmissions [at] stanford.edu (mathgradadmissions[at]stanford[dot]edu) as soon as you know them so that they can be added to your application.

It is OK if the official scores arrive after the application deadline. 

Yes. You must submit a complete application and application materials each time you apply to the PhD program. This includes letters of recommendation, test scores, and transcripts.

Application Status Questions

Official scores are sent electronically to Stanford and matched to applicants’ records around once per week. If any identifying information on your ETS score report is different from your online application, your scores may not be matched to your record. As long as you self-reported your scores on your application, those will be used for the admissions committee's review and it will not adversely affect the review. Please do not contact the department to see if your official scores have been received.

Admission Decision Questions

Admission decisions are made between late January to late February.

We will notify all applicants via email once decisions have been made. Please do not contact the department to see if a decision has been made unless you haven’t heard by early March.

No, there is not a process to appeal the admissions committee's decision, it is final.

No, we are not able to provide individualized feedback to applicants. 

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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.

Mathematical Sciences

Mellon college of science.

The Department of Mathematical Sciences at Carnegie Mellon University invites applications to our graduate program in Mathematics. To be admitted to this program applicants must document competence equivalent to graduation from a recognized U.S. four-year college, university or institute of technology.

There is no application fee. All application materials should be received by December 20 to be assured of full consideration.

The following materials are required or optional (as indicated) as part of your application:

• Curriculum Vitae
• Statement of Purpose Your statement of purpose should describe your mathematical background and interests, previous research experiences, as well as directions and goals you would like to pursue for your graduate studies. Information about your background or activities that helps identify your potential to grow and achieve will also be considered. Typical statements of purpose are about one or two pages long.
• Transcripts You need to submit transcripts from undergraduate and graduate programs you attended. Transcripts that are not in English should be submitted together with English translations. You may submit an unofficial transcript with your application; in this case official transcripts will be required later should you be accepted into the program.
• Reference letters You must submit contact information for at least three academic references. Your references will be directly asked to submit their letters of recommendation through this website.
• Subject GRE score (optional) We do not collect scores for the GRE General Test. The GRE Mathematics Subject Test is recommended but not required. Your application would receive full consideration without a score. You may submit an unofficial score with your application. If so, the official test report would be required later should you be accepted into the program.
• Language Proficiency Test Scores (non-native English speakers) We accept scores from the Test of English as a Foreign Language (TOEFL), International English Language Testing System (IELTS), or Duolingo English Test (DET). The minimum acceptable scores for admission are cumulative TOEFL score 86/120, IELTS 6.5, or DET 110/160. However, it is recommended that you have the cumulative TOEFL score 100/120, IELTS 7.5, or DET 120/160. Test scores should be no more than 2 years old. If you are a non-native English speaker, you need to provide a score even if you attended a U.S. undergraduate institution. If you receive an offer of admission, you will need to provide an official score. Once enrolled, non-native English speakers must obtain the ITA Language Certification from CMU's Student Academic Success Center . It can be obtained either by achieving a sufficiently high TOEFL Speaking score (26/30) or by passing the ITA test once at CMU.
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Further info: Christine Gilchrist +1 412 268 7154 [email protected]

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Graduate Program

Thinking of applying to graduate school in mathematics.

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?

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• How to Apply

The Duke Mathematics Graduate Program offers admission to students interested in obtaining a PhD in many research areas spanning pure and applied mathematics.  

Application Instructions

Applications for Fall 2024 admission are due by Wednesday,  December 13, 2023 .  Submission of the General and Math Subject GRE exams are both optional.

• For questions about the application process and requirements (including the GRE and TOEFL), or the online application, consult the The Graduate School's list of  Frequently Asked Questions .  If your answer is not addressed there, you can contact The Graduate School from the FAQ page.
• Applications to the program are administered by the Graduate School (not the Math Department) and all logistical questions should be directed there (eg: whether or not materials have been received and added to your file). See the Graduate School's  Admissions FAQs , especially "Can I update my application after I submit it?" and send email to grad-admissions-center at duke edu if appropriate.
• The Math Director of Graduate Studies (DGS) can answer questions about academic aspects of the program.  (You will find answers to many common questions on the math department website.)  Email [email protected] .
• An application fee waiver may be available to you if you meet certain criteria set out by The Graduate School. To request a fee waiver for your Duke Graduate School application, you will first need to have an application in progress. The fee waiver option is built into the application system. For more information about fee waivers please visit the Graduate School webpage . Please be aware that the Mathematics Department does not have fee waivers and all waiver requests should be through the application portal. 

Please note that the department has neither a Masters program, nor a separate track for students primarily interested in teaching.

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Graduate Admissions

Last updated on September 8, 2023 for Fall 2024 Admissions. 

Master’s Program    

We are in the process of revising the role of our Master’s Program.  While we revise the Master's Program, it will mainly serve s tudents who are enrolled in other graduate programs at the University of Washington and who also want training in mathematics. The department provides no financial support for these students.

PhD Program    

The Department of Mathematics offers a PhD Program  for students interested in careers using advanced mathematics. We typically admit 15-20 new PhD students each year.  Most of our students are funded by teaching assistantships (TA's), some are funded by research assistantships (RA's), and some have fellowships from outside of UW.   In addition to the degree program requirements linked above, applicants requesting funding should review the guidelines for TA and RA appointments.   

Here are some reasons to consider graduate study at the University of Washington:

• We offer courses and research opportunities in most major fields of pure mathematics , as well as opportunities to collaborate with local companies, government labs, and with the departments of Statistics , Biostatistics , Applied Math , and Computer Science . 
• Almost all of our PhD students are supported through teaching assistantships , which come with union negotiated salaries, tuition waivers, and free health insurance. TAs receive extensive training and mentoring, and a wide variety of different assignments are available, including opportunities to teach one’s own courses.
• We encourage collaboration, and our graduate program is deliberately structured so that students are not in any sense competing against each other.
• Our students' success rate is high.  Over the past decade or so, about 80% of our entering PhD students have graduated with PhDs.   
• Most of our graduate students go on to jobs in academia or go into rewarding careers in industry, government, etc. using their math skills.  
• We have a seminar series focusing on  career transitions  to help students find a good path through the program and into their next role.  We regularly invite back alumni to offer advice and support to current grad students about their current jobs in industry and academia.   So far, everyone invited has accepted  enthusiastically.  
• There is  an Advanced Data Science Option for the PhD, which can be very useful when students are applying for jobs in industry.  
• The Mathematics Department believes that excellence in graduate education is enhanced by assembling a diverse and inclusive graduate student community that reflects a wide range of differences of culture and circumstance. Such differences may include race, ethnicity, religion, gender, age, abilities/disabilities, gender identity, sexual orientation, and socioeconomic status. See the Math Department’s Diversity Commitment statement  for more information.
• Seattle is an attractive city, with great coffee,  endless opportunities for exploration, good public transportation, bike paths, abundant natural beauty, cultural events, and outdoor activities.   Temperatures are mild year-round, and the city is not as rainy as its reputation suggests.   We get only 30 inches per year on average, but many winter days are partly cloudy and drizzly.  Many people who move here like the mild climate and decide they never want to leave.
• We recognize that rent in Seattle is relatively high, but students do find housing in the city that they can afford within a reasonable commuting distance to campus.  

Applying to the PhD Program

Since all entry-level graduate courses are year-long sequences beginning in the autumn, applications for admission are normally accepted only for autumn quarter. Applications must be complete by December 8  for admission the following autumn.

Follow the links below to read about admission requirements and application procedures.

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Graduate Admission

Application deadline:  december 15, required application materials:.

• Electronic application
• Statement of Academic Purpose
• Three letters of recommendation
• Transcript(s) - One transcript from every college/university from which you have earned, or expect to earn a degree, must be uploaded with your electronic application. The Graduate School does not accept mailed transcripts. The transcript you obtain does NOT have to be official.
• English language proficiency:  TOEFL/IELTS

Note: GREs, both general and subject test scores, will  not  be accepted for Fall 2024 admissions.

Program Length: Four years; however, a fifth year is usually granted if approved in advance.

If your interest is in applied and computational mathematics, you must apply directly to the Program in Applied and Computational Mathematics (PACM).

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Georgia Institute of Technology College of Sciences

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PhD in Mathematics

Here are the requirements for earning the PhD degree in Mathematics offered by the School of Math. For requirements of other PhD programs housed within the School, please see their specific pages at  Doctoral Programs . The requirements for all these programs consist of three components:  coursework ,  examinations , and  dissertation  in accordance to the guidelines described in the  GT Catalogue .

Completion of required coursework, examinations, and dissertation normally takes about five years. During the first one or two years, students concentrate on coursework to acquire the background necessary for the comprehensive examinations. By the end of their third year in the program, all students are expected to have chosen a thesis topic, and begin work on the research and writing of the dissertation.

The program of study must contain at least 30 hours of graduate-level coursework (6000-level or above) in mathematics and an additional 9 hours of coursework towards a minor. The minor requirement consists of graduate or advanced undergraduate coursework taken entirely outside the School of Mathematics, or in an area of mathematics sufficiently far from the students area of specialization.

Prior to admission to candidacy for the doctoral degree, each student must satisfy the School's comprehensive examinations (comps) requirement. The first phase is a written examination which students must complete by the end of their second year in the graduate program. The second phase is an oral examination in the student's proposed area of specialization, which must be completed by the end of the third year.

Research and the writing of the dissertation represent the final phase of the student's doctoral study, and must be completed within seven years of the passing of comps. A final oral examination on the dissertation (theses defense) must be passed prior to the granting of the degree.

The Coursework

The program of study must satisfy the following  hours ,  minor , and  breadth  requirements. Students who entered before Fall 2015 should see  the old requirements , though they may opt into the current rules described below, and are advised to do so.

Hours requirements.  The students must complete 39 hours of coursework as follows:

• At least 30 hours must be in mathematics courses at the 6000-level or higher.
• At least 9 hours must form the doctoral minor field of study.
• The overall GPA for these courses must be at least 3.0.
• These courses must be taken for a letter grade and passed with a grade of at least C.

Minor requirement.  The minor field of study should consist primarily of 6000-level (or higher) coursework in a specific area outside the School of Math, or in a mathematical subject sufficiently far from the student’s thesis work. A total of 9 credit hours is required and must be passed with a grade of B or better. These courses should not include MATH 8900, and must be chosen in consultation with the PhD advisor and the Director of Graduate Studies to ensure that they form a cohesive group which best complements the students research and career goals. A student wishing to satisfy the minor requirement by mathematics courses must petition the Graduate Committee for approval.  Courses used to fulfill a Basic Understanding breadth requirement in Analysis or Algebra should not be counted towards the doctoral minor. Upon completing the minor requirement, a student should immediately complete the  Doctoral Minor form .

Breadth requirements.  The students must demonstrate:

• Basic understanding of 2 subjects must be demonstrated through passing the subjects' written comprehensive exams.  At least 1 of these 2 exams must be in Algebra or Analysis.
• Basic understanding of the third subject may be demonstrated either by completing two courses in the subject (with a grade of A or B in each course) or by passing the subject's written comprehensive exam.
• A basic understanding of both subjects in Area I (analysis and algebra) must be demonstrated.
• Earning a grade of A or B in a one-semester graduate course in a subject demonstrates exposure to the subject.
• Passing a subject's written comprehensive exam also demonstrates exposure to that subject.

The subjects.  The specific subjects, and associated courses, which can be used to satisfy the breadth requirements are as follows.

• Area I subjects:​
• Area II subjects:​

Special Topics and Reading Courses.

• Special topics courses may always be used to meet hours requirements.
• Special topics courses may be used to meet breadth requirements, subject to the discretion of the Director of Graduate Studies.
• Reading courses may be used to meet hours requirements but not breadth requirements.

Credit Transfers

Graduate courses completed at other universities may be counted towards breadth and hours requirements (courses designated as undergraduate or Bachelors' level courses are not eligible to transfer for graduate credit).  These courses do not need to be officially transferred to Georgia Tech. At a student’s request, the Director of Graduate Studies will determine which breadth and hours requirements have been satisfied by graduate-level coursework at another institution.  

Courses taken at other institutions may also be counted toward the minor requirement, subject to the approval of the Graduate Director; however, these courses must be officially transferred to Georgia Tech.

There is no limit for the transfer of credits applied toward the breadth requirements; however, a maximum of 12 hours of coursework from other institutions may be used to satisfy hours requirements. Thus at least 27 hours of coursework must be completed at Georgia Tech, including at least 18 hours of 6000-level (or higher) mathematics coursework.

Students wishing to petition for transfer of credit from previous graduate level work should send the transcripts and syllabi of these courses, together with a list of the corresponding courses in the School of Math, to the Director of Advising and Assessment for the graduate program.

Comprehensive Examinations

The comprehensive examination is in two phases. The first phase consists of passing two out of seven written examinations. The second phase is an oral specialty examination in the student's planned area of concentration. Generally, a student is expected to have studied the intended area of research but not necessarily begun dissertation research at the time of the oral examination.

Written examinations.  The written examinations will be administered twice each year, shortly after the beginning of the Fall and Spring semesters. The result of the written examination is either pass or fail. For syllabi and sample exams see the  written exams page .

All students must adhere to the following rules and timetables, which may be extended by the Director of Graduate Studies, but only at the time of matriculation and only when certified in writing. Modifications because of leaves from the program will be decided on a case-by-case basis.

After acceptance into the PhD Program in Mathematics, a student must pass the written examinations no later than their fourth administration since the student's doctoral enrollment. The students can pass each of the two written comprehensive exams in separate semesters, and are allowed multiple attempts.

The Director of Graduate Studies (DGS) will be responsible for advising each new student at matriculation of these rules and procedures and the appropriate timetable for the written portion of the examination. The DGS will also be responsible for maintaining a study guide and list of recommended texts, as well as a file of previous examinations, to be used by students preparing for this written examination.

Oral examination.  A student must pass the oral specialty examination within three years since first enrolling in the PhD program, and after having passed the written portion of the comprehensive exams. The examination will be given by a committee consisting of the student's dissertation advisor or probable advisor, two faculty members chosen by the advisor in consultation with the student, and a fourth member appointed by the School's Graduate Director. The scope of the examination will be determined by the advisor and will be approved by the graduate coordinator. The examining committee shall either (1) pass the student or (2) fail the student. Within the time constraints of which above, the oral specialty examination may be attempted multiple times, though not more than twice in any given semester. For more details and specific rules and policies see the  oral exam page .

Dissertation and Defense

A dissertation and a final oral examination are required. For details see our  Dissertation and Graduation  page, which applies to all PhD programs in the School of Math.

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.  

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Earn admissions to a nationally ranked program.

Incoming Ph.D. students are expected to pass  qualifying examinations  by the end of their third semester in the Ph.D. program.

After passing the qualifying examinations, students are expected to select a thesis adviser and form a doctoral committee. The committee administers the comprehensive examination (no later than the end of the sixth semester of study) and offers counsel to the student as his research progresses.

• No credit will be given for any course in which a grade of less than B is received.
• A minimum grade point average of 3.0 is required for graduation for all advanced degrees.

Requirements

• American English Oral Communicative Proficiency Test (AEOCPT)-International students only  ( Graduate School requirement) This exam must be taken upon arrival during the week of orientation. Students who pass the exam may teach without restrictions; marginal or failing students must register for the appropriate ESL course .  
• SARI/CITI (Scholarship and Research Integrity/Collaborative Institutional Training Initiative) ( Graduate School requirement ) The  SARI@PSU program is composed of two parts: an online course, and an interactive, discussion-based component; and encompasses content that is both interdisciplinary and discipline-specific. The online portion Part 1 ), offered through the Collaborative Institutional Training Initiative (CITI), provides a common language and understanding of the history and principles of the responsible conduct of research. This requirement must be completed within the first two weeks of the semester of arrival. The discussion-based component Part 2)provides an opportunity for in-depth exploration of important issues unique to each field of study. Part 2 is completed by attendance at the Graduate Student Seminar (see below).  
• Qualifying Examinations  ( Departmental requirement ) Ph.D. students are expected to pass four qualifying examinations out of the following areas: real analysis, linear algebra, abstract algebra, complex analysis, functional analysis, topology.  All students are required to pass real analysis and complex analysis. Students may then choose between linear and abstract algebra and functional analysis and topology.  Exams must be completed by the end of their third semester in the Ph.D. program. The system of qualifying examinations is explained in detail  here:  Qualifying Exams .  
• Colloquium Attendance Requirement  ( Departmental requirement ) Students year one (1) through the semester of the Ph.D. Oral Comprehensive Examination are required to attend at least 12 colloquium talks each academic year. Four (4) of the talks may be substituted by Departmental Seminars. First-year students (only) may substitute Student Colloquium talks for some of the required Departmental Colloquium talks. Students post-comprehensive are required to attend six (6) colloquium talks with NO substitutions.  Colloquium Attendance Form for 2023-24.  
• Graduate Student Seminar Requirement ( Departmental requirement ) This is a three credit course offered every spring. It is a requirement that every student successfully completes the seminar before his or her third year of study.  
• Ph.D. Candidacy  ( Graduate School requirement ) The Department of Mathematics will recommend Ph.D. candidacy after a student has passed all qualifying examinations and successfully completed 18 credits of Mathematics courses. Admission to candidacy is conferred by the Graduate School.  
• Oral and Written English Competency  ( Graduate School and Departmental requirement ) A candidate for the degree of Doctor of Philosophy is required to demonstrate high-level competence in the use of the English language, including reading, writing, and speaking, as part of the language and communication requirements for the Ph.D. Oral competency is assessed by the GTA (Graduate Teaching Assistant) Oversight Committee. Written competency is evaluated by the students' advisor. For additional detail, please see  Oral and Written English Competency .  
• Advisers and Doctoral Committees  ( Graduate School requirement ) Consultation or arrangement of the details of the student's semester-by-semester schedule is the function of the adviser. General guidance of a doctoral candidate is the responsibility of a doctoral committee consisting of four or more active members of the Graduate Faculty, which normally includes at least two faculty in the major field and is chaired by the student's adviser. This committee is appointed by the Graduate Dean through the Office of Graduate Programs, upon recommendation of the Director of Graduate Studies. A student should have an adviser by the end of their fourth semester and must have an adviser by the end of their fifth semester in the Ph.D. program.  
• Course Requirements  ( Departmental requirement ) Students must receive a minimum grade of B in at least eleven 3-credit 500-level mathematics courses. Students must take the Graduate Student Seminar before the third year of study.  

Starting 5/8/23:  All Modes Allowed The comprehensive examination may be held fully in-person, fully remote, or hybrid with some individuals participating in-person while others participate remotely. Student preference for delivery mode should be strongly considered, but the student and adviser must agree on the mode. If the student and adviser cannot agree on the mode, the Graduate Program Head will make the final decision. Either the student or adviser can appeal the decision of the Graduate Program Head to the ( insert appropriate graduate education administration role for the unit role, e.g. Associate Dean for Graduate Education, Director of Academic Affairs, etc. ).

• Ph.D. Thesis  ( Graduate School requirement ) The ability to do independent research and competence in scholarly exposition must be demonstrated by the preparation of a thesis on some topic related to the major subject. It should represent a significant contribution to knowledge, be presented in a scholarly manner, reveal an ability on the part of the candidate to do independent research of high quality, and indicate considerable experience in using a variety of research techniques. The contents and conclusions of the thesis must be defended at the time of the final oral examination. A draft of the thesis must be submitted to the doctoral committee a month before the final oral examination.  

Starting 5/8/23:  All Modes Allowed

The final oral examination (dissertation defense) may be held fully in-person, fully remote, or hybrid with some individuals participating in-person while others participate remotely.  Student preference for delivery mode should be strongly considered, but the student and adviser must agree on the mode. If the student and adviser cannot agree on the mode, the Graduate Program Head will make the final decision. Either the student or adviser can appeal the decision of the Graduate Program Head to the ( insert appropriate graduate education administration role for the unit role, e.g. Associate Dean for Graduate Education, Director of Academic Affairs, etc. ).

• Residency Requirement  ( Graduate School requirement ) After being admitted to candidacy, the student must be a full-time graduate student as defined by the Graduate Bulletin for two consecutive semesters (excluding summers) before comprehensive examinations can be scheduled.  
• Continuous Registration  ( Graduate School requirement ) After a Ph.D. candidate has passed the comprehensive examination and has met the two-semester full-time residency requirement (above), the student must register continuously for each fall and spring semester (beginning with the first semester after both of the above requirements have been met) until the Ph.D. thesis is accepted and approved by the doctoral committee.

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Apply to PhD Program

The Fall 2024 deadline for PhD applications is  December 15, 2023 . All application materials and letters should be submitted by this date. Late applications, or applications that are incomplete by the deadline, will in general not be considered. Applications are not accepted for Spring 2024.

How to Apply

First complete the online portion of the application , including payment of the application fee. Under-represented U.S. minority applicants can apply for an application fee waiver .   Applicants who are currently enrolled in a master’s program in the Illinois Department of Mathematics may use this form instead (to avoid paying another application fee).

We require transcripts from your undergraduate (and, if applicable, graduate) institutions, three letters of recommendation, and a personal statement.  (Please see this link for more details from our Graduate College; section 1 details the application requirements, and section 2 details the minimum requirements for admission.)

As part of the online application process you will be asked to upload a copy of your undergraduate transcript (and also graduate transcript if applicable) showing grades for the most recent semester at the time of application. We typically do not accept students with a grade point average below 3.25 on a 4.00 scale, and GPA is a significant factor in our decision.

Please arrange for three letters of recommendation to be uploaded to our online application system. Letters from professors who have taught you in mathematics classes or who have supervised you in a research project are the most useful. Both the content and the source of the letters can play a significant role in our decision.

Please include a personal statement as part of your online application. We prefer statements that are honest, specific, and brief, and explain why you are applying to our program.

The Department of Mathematics does not require any program specific questions or writing samples.

The GRE general test and mathematics subject test are not required .

Information for International Students

English requirements for international applicants.

Students with high TOEFL or IELTS scores have a better chance of admission.

1. Minimum Requirement for Admission

All international applicants must satisfy the Graduate College English Proficiency Requirement for Admission . The Mathematics Department requires a minimum TOEFL iBT speak score of 22 (or IELTS 6.5 speaking score) and a minimum TOEFL total of 95 for admission to the PhD program. Some applicants are  exempt from the English requirement for admission . Even those students will increase their chance of admission if they submit recent TOEFL or IELTS scores. If you do not have an exemption as listed above and have a TOEFL SPEAK score less than 22, then do not apply.

2. Minimum Requirement for Teaching Assistantship Funding

Teaching assistants must meet a higher standard: proficiency in spoken English , as measured by the Internet Based TOEFL (iBT), the IELTS, the TSE (Test of Spoken English), or the university-administered EPI (English Proficiency Interview). Students who do not satisfy this requirement can still be granted a Teaching Assistantship in Mathematics with duties outside the classroom (e.g. grading) for the first two years, provided they enter with a TOEFL iBT speaking sub-section score of 22 or 23. After two years, continuation of the Teaching Assistantship depends on whether the student has successfully demonstrated proficiency in spoken English.

Declaration of Finances

International students must submit the Declaration and Certification of Finances form before a visa can be issued. We do not need this form in order to decide on your application. If you are later offered admission and accept the offer, then you must submit the form promptly along with the required bank statement, before you can be admitted to the university and get a visa. The bank statement must show resources for the full cost of education and living expenses. If you are offered a teaching assistantship, then a bank statement showing about US$1,700 to cover any additional cost of living will be sufficient. Please note that you are not required to pay this amount, only to show evidence of such finances in a bank account . To repeat: we do not need this documentation now in order to review your application.

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Ph.D. Program in Mathematics

Degree requirements.

A candidate for the Ph.D. degree in mathematics must fulfill a number of different departmental requirements .

NYU Shanghai Ph.D. Track

The Ph.D. program also offers students the opportunity to pursue their study and research with Mathematics faculty based at NYU Shanghai. With this opportunity, students generally complete their coursework in New York City before moving full-time to Shanghai for their dissertation research. For more information, please visit the  NYU Shanghai Ph.D. page .

Sample course schedules (Years 1 and 2) for students with a primary interest in:

Applied Math (Math Biology, Scientific Computing, Physical Applied Math, etc.)

Additional information for students interested in studying applied math is available here .

Probability

PDE/Analysis

The Written Comprehensive Examination

The examination tests the basic knowledge required for any serious mathematical study. It consists of the three following sections: Advanced Calculus, Complex Variables, and Linear Algebra. The examination is given on three consecutive days, twice a year, in early September and early January. Each section is allotted three hours and is written at the level of a good undergraduate course. Samples of previous examinations are available in the departmental office. Cooperative preparation is encouraged, as it is for all examinations. In the fall term, the Department offers a workshop, taught by an advanced Teaching Assistant, to help students prepare for the written examinations.

Entering students with a solid preparation are encouraged to consider taking the examination in their first year of full-time study. All students must take the examinations in order to be allowed to register for coursework beyond 36 points of credit; it is recommended that students attempt to take the examinations well before this deadline. Graduate Assistants are required to take the examinations during their first year of study.

For further details, consult the page on the written comprehensive exams .

The Oral Preliminary Examination

This examination is usually (but not invariably) taken after two years of full-time study. The purpose of the examination is to determine if the candidate has acquired sufficient mathematical knowledge and maturity to commence a dissertation. The phrase "mathematical knowledge" is intended to convey rather broad acquaintance with the basic facts of mathematical life, with emphasis on a good understanding of the simplest interesting examples. In particular, highly technical or abstract material is inappropriate, as is the rote reproduction of information. What the examiners look for is something a little different and less easy to quantify. It is conveyed in part by the word "maturity." This means some idea of how mathematics hangs together; the ability to think a little on one's feet; some appreciation of what is natural and important, and what is artificial. The point is that the ability to do successful research depends on more than formal learning, and it is part of the examiners' task to assess these less tangible aspects of the candidate's preparation.

The orals are comprised of a general section and a special section, each lasting one hour, and are conducted by two different panels of three faculty members. The examination takes place three times a year: fall, mid-winter and late spring. Cooperative preparation of often helpful and is encouraged. The general section consists of five topics, one of which may be chosen freely. The other four topics are determined by field of interest, but often turn out to be standard: complex variables, real variables, ordinary differential equations, and partial differential equations. Here, the level of knowledge that is expected is equivalent to that of a one or two term course of the kind Courant normally presents. A brochure containing the most common questions on the general oral examination, edited by Courant students, is available at the Department Office.

The special section is usually devoted to a single topic at a more advanced level and extent of knowledge. The precise content is negotiated with the candidate's faculty advisor. Normally, the chosen topic will have a direct bearing on the candidate's Ph.D. dissertation.

All students must take the oral examinations in order to be allowed to register for coursework beyond 60 points of credit. It is recommended that students attempt the examinations well before this deadline.

The Dissertation Defense

The oral defense is the final examination on the student's dissertation. The defense is conducted by a panel of five faculty members (including the student's advisor) and generally lasts one to two hours. The candidate presents his/her work to a mixed audience, some expert in the student's topic, some not. Often, this presentation is followed by a question-and-answer period and mutual discussion of related material and directions for future work.

Summer Internships and Employment

The Department encourages Ph.D. students at any stage of their studies, including the very early stage, to seek summer employment opportunities at various government and industry facilities. In the past few years, Courant students have taken summer internships at the National Institute of Health, Los Alamos National Laboratory, Woods Hole Oceanographic Institution, Lawrence Livermore National Laboratory and NASA, as well as Wall Street firms. Such opportunities can greatly expand students' understanding of the mathematical sciences, offer them possible areas of interest for thesis research, and enhance their career options. The Director of Graduate Studies and members of the faculty (and in particular the students' academic advisors) can assist students in finding appropriate summer employment.

Mentoring and Grievance Policy

For detailed information, consult the page on the Mentoring and Grievance Policy .

Visiting Doctoral Students

Information about spending a term at the Courant Institute's Department of Mathematics as a visiting doctoral student is available on the Visitor Programs  page.

graduate admissions

Frequently Asked Questions

More information for international students

Dear Prospective Applicant,

Thank you for your interest in graduate studies in the Department of Mathematics at UCLA. Applications are accepted for Fall quarter matriculation only. The application deadline is December 15. Application review process begins in late December; to ensure full consideration, applications should be complete, with all supporting material submitted, by that time.  Please note that admissions to the M.A. and M.A.T. programs have been suspended indefinitely, so if you’re interested in doing graduate work in our department you must apply for admission to the Ph.D. program.

You must submit the UCLA Graduate Admissions Application (see  http://www.gdnet.ucla.edu/gasaa/admissions/applicat.htm ). You also must submit the Application Fee for the UCLA Graduate Admissions Application. Instructions for this payment are given online. As part of the Graduate Division Application, you will be required to submit:

• Transcripts (scans and originals)
• Statement of Purpose
• Letters of Recommendation (see online UCLA application)
• List of Mathematics Courses taken
• Applicants whose native language is not English must take the TOEFL/IELTS exam and submit their TOEFL/IELTS score.

Each applicant must select an area (pure or applied) and a field of interest as part of the application. The two areas have slightly different course requirements, but both allow for flexibility to take many combinations of courses and qualifying exams throughout the department. Students can ask to switch areas or fields after starting the program if their interests change.

Transcripts (and TOEFL/IELTS scores, if required) should be mailed to:

Graduate Advisor UCLA Mathematics Department 520 Portola Plaza Los Angeles, CA 90095-1555

Admissions Requirements

The minimum Department requirements for applicants, in addition to the minimum University requirements, are:

*Applicants for the PhD program  must have a 3.5 GPA in upper division math courses.

Prospective students do not need to have an undergraduate mathematics major, but must complete at least 12-quarter, or 8-semester courses in substantial upper division mathematics.  Comparable Upper Division Courses at UCLA:

• Algebra 110AB : Ring of integers, integral domains, fields, polynomial domains, unique factorization. Groups, structure of finite groups.
• Linear Algebra 115AH : Abstract vector spaces, linear transformations, and matrices; determinants; inner product spaces; eigenvector theory.
• Analysis 131ABH : Rigorous introduction to foundations of real analysis; real numbers, point set topology in Euclidean space, functions, continuity. Derivatives, Riemann integral, sequences and series of functions, power series, Fourier series.
• Differential Geometry 120A : Curves in 3-space, Frenet formulas, surfaces in 3-space, normal curvature. Gaussian curvature. Congruence of curves and surfaces. Intrinsic geometry of surfaces, isometrics, geodesics, Gauss/Bonnet theorem.
• Ordinary Differential Equations 135AB : Systems of differential equations; linear systems with constant coefficients, analytic coefficients, periodic coefficients, and linear systems with regular singular points; existence and uniqueness results; linear boundary and eigenvalue problems; two-dimensional autonomous systems, phase/plane analysis, stability and asymptotic behavior of solutions.
• Applied Numerical Methods 151AB : Introduction to numerical methods with emphasis on algorithms, analysis of algorithms, and computer implementation issues. Solution of non-linear equations, numerical differentiation, integration, and interpolation. Numerical solution of differential equations.
• Three letters of recommendation with preferably all, but always at least 2, from mathematicians who are familiar with the student’s work.

The GRE Math Subject test . 

A General Recommendation:

The UCLA Mathematics Department encourages students to change their educational institution between their Bachelor and PhD degrees to broaden their horizons. In particular, no admissions advantage is given to UCLA undergraduates.

international students

Dear Prospective Applicant,

Thank you for your interest in graduate studies in the Department of Mathematics at UCLA. Applications are accepted for the Fall quarter only. The application deadline is December 15. The application review process begins late December; please ensure that all supporting documents, including reference letters and test scores, are received by that time. Please read all the information very carefully. In addition to the requirements below, return to Graduate Admissions for complete details.

Please note: THE UCLA Graduate Admissions Application fee is MANDATORY!! If the application fee is not paid, the UCLA Admissions Office cannot process the application.

Please note: DO NOT send publications.

Financial Support: International students are supported through a mix of fellowships, research assistantships, and teaching assistantships. Support is normally guaranteed for four years. A 5th year of support is usually provided if the student is in good standing and advanced to candidacy. Guaranteed support normally provides income that is enough for living expenses (exact amounts vary from year to year), and in addition provides full registration fees, health insurance, and non-resident tuition. All applicants are automatically considered for this financial support; no separate application is required. Students who have access to other means of support, for example scholarships from their own governments or international organizations, should apply for those separately.

Students whose first language is not English must pass UCLA’s “Test of Oral Proficiency” (TOP) before they can become teaching assistants at UCLA. Offers of support for international students are conditioned on passing this exam within the first year.

Required Tests

• GRE: Math Subject Test
• TOEFL/IELTS: Your TOEFL score on the paper and pencil test must be at least 560 (600 is recommended) or at least 87 on the internet-based test. Your IELTS Academic overall band score should be at least 7.0. 7.0 is the  minimum  required.
• TOP (Test of Oral Proficiency):  Minimum score of 6.4 is a provisional pass permitting the student to TA.

frequently asked questions

Thank you for your interest in graduate studies in the Department of Mathematics at UCLA. Applications are accepted for the fall quarter only. The application deadline is December 15. The application review process begins late December; please ensure that all supporting documents, including reference letters and test scores, are received by that time. Please return to Graduate Admissions for more detailed information or email  [email protected]  for very specific questions.

The on-line University application must be submitted by December 15. We will accept scores, letters and other materials after the December 15th deadline, but your application package will be considered incomplete and will not be reviewed if we do not have all materials by the end of the second week in January.

Not really.  Three letters are sufficient for the review committee to get a sense of your abilities and strengths. 

Either way is acceptable as long as they are officially sealed when we receive them.

International applicants that have received a Bachelor’s degree or higher from a university located in the United States, or from another country in which English is both the spoken language and the medium of instruction are exempted from the TOEFL.  Please include this information in your application package.

It is University policy that you score at least a 560 on the pencil and paper test, 220 on the electronic test, or 87 on the iBT.

Students who wish to apply for admission to our graduate program for the 2024-2025 academic year are to submit a GRE Math subject test score report as part of their application package. However, students are NOT required to submit the GRE General test score.

Please note that admissions to the M.A. and M.A.T. programs have been suspended until further notice. 

Be aware that when we talk about a successful applicant, numbers are only one part of the total package. That being said; the minimum GPA is 3.5 for a PhD and 3.2 for a Master’s, and although there is no minimum GRE, you must keep in mind that admission is very competitive and the higher your score, the better.  On the average, those we offer admission to have GRE subject scores in or above the 80th percentile. Our GPA range is from about 3.6 to 4.0, though again there can be exceptions. Keep in mind that we look at the entire package and although we have a minimum, the average is higher. Also, there is no score at any level, which insures admission.

The Statement is your opportunity to explain your reasons for pursuing graduate study in mathematics, and to let us get a sense of who you are.  You may mention any relevant facts about your education or experience with mathematics that are not apparent in the other materials you submit.  You might want to note what kind of mathematics you find interesting and wish to pursue in graduate school, and what you plan to do after obtaining a Ph.D. It may include information about your background, interests, or career goals, and how UCLA might help you meet those career and educational goals.  There is no specific length, but one or two pages are typical.

Your application will be accepted if we receive your scores by the end of the first week in January.

Admission decisions are based on an applicant’s entire package.  Weaknesses in one area may be offset by strengths in another. The committee weighs many factors in addition to GPAs.  These may include the breadth of your undergraduate courses, your letters of recommendation, statement of purpose and GRE subject score.

You may apply for admission to our program before your actual degree date as long as you have your Bachelor degree by the fall enrollment date.

The UCLA code is 4837 and the Mathematics department code is 0703.

One of the most important things you can do is to get to know your professors. You want to have letters of recommendation that tell us knowledgeably and personally about your qualifications and abilities. Your letters of recommendation should attest to the fact that you have the potential and mathematical background to be a research mathematician. It is already expected that all applicants to UCLA have a solid preparation in undergraduate math, with strong grades and GRE scores, that allows them at a minimum to pass our Basic Qualifying Exam in their first year. What we would also like to see in addition is that you have potential for advanced mathematical research. This can be demonstrated through courses that are heavy in proof oriented math, honors and graduate courses, a senior research project, a senior thesis, or a summer REU program.

You will receive an official letter by the first week of March.

PhD Requirements

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TA Training

First week before the beginning of and during Fall semester. During the semester, enroll in Math 900.

Qualifying Exams

Offered in August and January before the start of classes each term.

Enrollment Requirements

– At least 6 credits as a TA and 8 credits as a RA

– At least two 700+ level courses each semester

– Math 900 (TA seminar)

Prior Coursework

A graduate student can receive department conferral for up to 12 credits from another institution toward the UW-Madison PhD requirements. These courses will not appear on the UW-Madison transcript nor count toward the graduate career GPA. Coursework earned 10 or more years prior to admission to a doctoral degree is not allowed to satisfy requirements.

To confer credits, you will need to fill out the Math Transfer Credit Approval Form and submit it to the Director of Graduate Studies .

Dissertation Advisor

Begin meeting potential advisors through courses and seminars.

After completing 30 credits and 2 qualifying exams (or 1 computer science course), you have the option to obtain an MA.

– Apply to add the program of Mathematics – Foundations for Research.

Second Year

– At least one 700+ leavel course each semester; total of 3 for the year

Pass at least one exam between the Fall and Spring semester.

– Submit the warrant request to the Box Submission Folder before the end of the semester.

Non-dissertator

– At least one 700+ level course each semester

Dissertator

– Exactly 3 credits each semester. At most two semesters of 990 (not including summer)

Pass second exam between the Fall and Spring semester.

– Secure a dissertation advisor

– Notify the Graduate Coordinator

Specialty Exam

– Establish specialty exam committee and date/time/mode

– After successful completion, have warrant signed and return to the Graduate Coordinator

Fourth Year

– At least one 700+ leavel course each semester

Final Defense

– Guide to Completing Your Doctoral Degree

– Establish final defense committee and date/time/mode.

– Email your dissertation to your committee at least 2 weeks prior to the final defense

– After successful completion, deposit all documents electronically

Fifth Year & Beyond

Phd program.

• PhD Program More
• PhD Requirements More
• Graduate Student Handbook More
• Academic Exception Policy More
• Qualifying Exams More
• Teaching Assistantship More
• Summer Information More
• Incoming Students More

Dates & Deadlines

• Academic Calendar More
• Graduate School Deadlines More
• Registrar Deadlines More

Admissions Requirements

The Penn AMCS Graduate Program is an interdisciplinary program that offers training in advanced mathematics and its applications to problems in empirical science, medicine and engineering. We offer both the Masters and PhD degrees.

We will begin accepting applications for the Fall 2024 academic year on October 1, 2023.  We typically accept PhD applications until January 1, and Masters applications until February 1.

Please note that the Graduate Division of the School of Arts and Sciences is responsible for managing Application FEE WAIVERS. Individual departments do not have the authority to grant these waivers. To apply for a fee waiver, please send a brief letter to Patricia Rea, Associate Director of Admissions, at [email protected] . The letter should state the reason for your request and your application should be in process or near completion. Please do not request a fee waiver if you haven’t started your application. Please keep in mind that you must demonstrate a clear and compelling case of financial hardship to be eligible for the waiver.

We welcome applications from interested and dedicated students who have done substantial work in mathematics, and/or its applications of superior quality as undergraduates. Applicants for admission are expected to have a mathematical background that is comparable to an undergraduate mathematics major.  All applicants are expected to take courses, beyond calculus,  in mathematical analysis (2 semesters), linear algebra (1 semesters), probability (1 semesters),  and complex analysis (1 semesters). In general, the more advanced mathematics you have studied the better. While we are willing to consider applications from students in applied fields, with less complete mathematical preparation,  we need to have solid evidence of your aptitude for work in advanced mathematics.  Undergraduate Mathematics course grades should mostly be As or A-s (or the equivalent).

A written Preliminary Exam , which covers the foundational mathematics expected of all students, is given before the staart of the first year.  Students who do not pass are expected to take foundational courses as necessary to bring their knowledge to the level expected.  Ph.D students not passing the prelim at the start must pass at the end of the first year.  Masters students get one additional opportunity at the start of their second year.

Applications must be submitted   electronically .

GRE requirements: The Math subject GRE is required but not the General GRE Test.  Please take this exam in person at a test center.  Scores from a prior year are fine.  If you cannot take the GRE subject test for some unavoidable reason, please state the reason in your personal statement.

Applicants whose native language is not English must take the  TOEFL  (Test of English as a Foreign Language). See the end of this page if you are a non-native English speaker but have studied in the US.

The GRE and English language exams should be taken early enough for the scores to be received at Penn in time to be considered by the Admissions Committee. Note that it can take a month or more after taking the exam before the score is reported, and that advanced registration is required for these tests.  To see that we receive your standardized test scores: for the GRE, use the institution code  2926. For the  TOEFL the institution code is 2926, and the Department code is 72.

Applicants are considered on the basis of their courses, grades, letters of recommendation, test scores (basic and advanced GRE, TOEFL, and TSE), and other supporting material. Letters of recommendation should preferably be from mathematics professors, or faculty from another scientific discipline,  who know the applicant well, and who can comment on the applicant’s abilities, accomplishments, and potential for research in applied mathematics and computational science .

Applicants are required to submit a personal statement with their application. This personal statement should address the applicant’s specific interest in AMCS as well as how the applicant’s interests align with the research interests of the AMCS faculty. Applicants should mention specific AMCS faculty members whose research is of interest to them. This personal statement will serve as a writing sample; no additional writing sample is required.

Applicants should indicate the final degree that they hope to be awarded from the Penn AMCS Graduate Program (either M.A., or Ph.D.). Students who intend to obtain an M.A. on the way to a Ph.D. at Penn should indicate that they intend to get a PhD. Masters students are not generally eligible to receive financial aid.

Please Note:

• If you have graduated from a 4-year program in the US or other English speaking countries, your TOEFL requirement is waived for purposes of applying to AMCS.
• If you have graduated from a 4-year program in the US and wish to be a TA in some department at Penn, you must either pass Penn’s internal IPT or present a TOEFL of at least 27, or an IELTS score of at least 8.

Graduate Studies

Letters to the Editor: Higher math is everywhere. But should UC require all applicants to learn it?

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To the editor: As a physics professor who uses Algebra 2 (and higher math) every day, I felt compelled to respond to the debate over how much math the University of California system should require for applicants .

Math is a logical and rigorous foundation of all STEM fields. You can’t build bridges, program computers, manufacture medicines, maneuver boats, communicate with satellites or fly planes and rockets without math, which is a key means that we use to connect our creativity with reality.

On top of that, math, as a rigorous platform for analysis (starting with algebra, which involves solving for unknowns), trains our minds to solve other problems. In today’s high- tech age, which depends on myriad achievements in science and technology, we need algebra and higher math to properly use these miracles and to continue innovation in those fields.

We do need to address the failures in math education; otherwise, the U.S. will fall behind other countries to a point where it will lose its superpower status to those who are better trained at math.

Michael Pravica, Henderson, Nev.

To the editor: Thank you to The Times’ editorial board for the thoughtful article on flexibility on the math requirement.

I taught high school for more than 40 years, most of it in a large, urban school. As I read the editorial, I was gratified to see that the writer addressed each concern or comment it raised for me. I found the article balanced and compelling, with a largely accurate assessment of students’ interests and needs.

It motivated me to look for ways I could be involved with helping bring about some of the changes that the article advocates for.

Tassie Hadlock-Piltz, Los Angeles

To the editor: Data science courses? Alas, 60 years too late!

How I wish I could have had that option instead of Algebra 2 at South Gate High School in the 1950s. Mr. Carden, our Algebra teacher, would hold up my report card with mostly A’s and say ruefully, “Done it again, ruined a young lady’s card with an F.”

So, I took Algebra 2 again in summer school at Huntington Park High. I’d hoped to bring my grade up to at least a C. Wrong! I got a D, as in “devastated.”

But with the encouragement of some terrific, dedicated teachers, plus my own stubborn streak, I still made it to UCLA. I eventually graduated with an education major and became an elementary school demonstration teacher. (How I wish Mr. Carden could have seen me then!)

Ultimately I became a school psychologist. Not surprisingly, I sometimes counseled anxious kids that failing Algebra was not the end of the world. My 40-year career with the Los Angeles Unified School District was the proof.

Let’s hope a good data science course with its “application to real-life scenarios” will become an accepted alternative to the “devastating” Algebra 2 requirement. (By the way, while in graduate school, I got an A in statistics.)

Josie Levy Martin, Montecito

More to Read

Editorial: Not every student needs Algebra 2. UC should be flexible on math requirement

March 28, 2024

Letters to the Editor: Hey, UC, who has ever thought, ‘I sure wish I had taken more algebra’?

March 22, 2024

UC stirs furious debate over what high school math skills are needed to succeed in college

March 19, 2024

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