The department offers the doctor of philosophy degree with a major in mathematics and a master of arts degree in mathematics.
The PhD 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 PhD 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 PhD program also have the option to earn a master of arts degree.
Admissions
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.
| Fall Deadline | December 15 |
| Spring Deadline | This program does not admit in the spring. |
| Summer Deadline | This program does not admit in the summer. |
| GRE (Graduate Record Examinations) | Not required but may be considered if available. |
| English Proficiency Test | Refer to the Graduate School: Minimum Requirements for Admission policy: https://policy.wisc.edu/library/UW-1241. |
| Other Test(s) (e.g., GMAT, MCAT) | The GRE subject test in Mathematics is optional but may be considered if available. |
| Letters of Recommendation Required | 3 |
Admission is competitive. Applicants to the PhD program are automatically considered for financial support.
Application Checklist
A complete application includes the following items:
- Statement of purpose: Your essay should be a concise description of your reasons for choosing to study this field at the University of Wisconsin-Madison (not to exceed two pages). Please include your research interests and career goals as well as a description of your preparation for graduate study including relevant coursework, related employment, research experience, publications, presentations, awards, and honors. (samples)
- Supplementary application:
- Coursework (different from transcript): In a list or table format, state all advanced mathematics courses you have completed, are currently enrolled in, or intend to take. For each course, give the name of the school where you took it, the course number, the title, description, books used, and the grade you received (for completed courses). If you have more than 20 courses, please include ONLY the most advanced courses. (sample)
- CV/Resume: Your CV/resume should include your educational background as well as teaching and research experience, publications, and accomplishments. (sample)
- Unofficial transcripts: Unofficial transcripts from all previous postsecondary studies are required. International academic records must be submitted in the original language and accompanied by an English translation. If an applicant is recommended for admission, the Graduate School will ask applicants to request official transcripts sent to the Graduate School from the undergraduate institution.
- Letters of recommendation: Three (3) letters of recommendation that address the question of mathematical promise. These letters should be from mathematics faculty, or other faculty familiar with the demands of graduate work in mathematics.
Background Education
Successful applicants have a bachelor’s degree in mathematics/applied mathematics/statistics.
Preparatory Coursework
| Code | Title | Credits |
|---|---|---|
| Analysis I/II 1 | ||
| Examples of courses (or equivalent): | ||
| MATH 521 | Analysis I | 3 |
| MATH 522 | Analysis II | 3 |
| Algebra I/II 1 | ||
| Examples of courses (or equivalent): | ||
| MATH 541 | Modern Algebra | 3 |
| MATH 542 | Modern Algebra | 3 |
| Linear Algebra | ||
| Example of course (or equivalent): | ||
| MATH 341 | Linear Algebra | 3 |
| Electives | ||
| Four electives among advanced undergrad or grad math courses. Incoming students usually, but not always, have some exposure to graduate level mathematics through graduate coursework, independent study, or research experience. This background often exceeds the minimum requirements for a math major. | ||
Faculty Advisors
You do not need an advisor prior to applying. Incoming students are assigned academic advisors at the beginning of their programs. PhD students have until the end of their 6th semester to secure a dissertation advisor.
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Candidates with interest in Applied Math may have a different mathematics background, typically including coursework in ordinary and partial differential equations, and computational science with programming experience in Python, Matlab, or similar language. That coursework should be at a commensurately advanced undergraduate level.
Funding
Graduate School Resources
The Bursar’s Office provides information about tuition and fees associated with being a graduate student. 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
Review the Graduate School minimum degree requirements and policies, in addition to the program requirements listed below.
Major Requirements
Mode of Instruction
| Face to Face | Evening/Weekend | Online | Hybrid | Accelerated |
|---|---|---|---|---|
| Yes | No | No | No | No |
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
| Minimum Credit Requirement | 51 credits |
| Minimum Residence Credit Requirement | 32 credits |
| Minimum Graduate Coursework Requirement | 51 credits must be graduate-level coursework. Refer to the Graduate School: Minimum Graduate Coursework (50%) Requirement policy: https://policy.wisc.edu/library/UW-1244. |
| Overall Graduate GPA Requirement | 3.00 GPA required. Refer to the Graduate School: Grade Point Average (GPA) Requirement policy: https://policy.wisc.edu/library/UW-1203. |
| Other Grade Requirements | No additional grade requirements. |
| Assessments and Examinations | Students are required to pass at least two qualifying exams. At least one qualifying exam must be passed by the beginning of the fourth semester (the spring semester of the second year). Both qualifying exams must be passed by the beginning of their sixth semester (the spring semester of the third year). Students must pass a preliminary exam (specialty exam) by the end of the eighth semester (end of fourth year). Students must satisfy all the requirements for dissertator status by the end of the eighth semester (end of fourth year). |
| Language Requirements | No language requirements. |
| Graduate School Breadth Requirements | All doctoral students are required to complete at least a 12-credit doctoral minor or graduate/professional certificate. Refer to the Graduate School: Breadth Requirement in Doctoral Training policy: https://policy.wisc.edu/library/UW-1200. |
Required Courses
Students must take a total of 51 graduate credits, or generally 18 courses. This includes courses in math and in breadth. In the PhD program, math courses numbered above 700 are for graduate credit. Math courses below 700 must be approved by the academic advisor.
Area of Specialization
There are five general and overlapping areas of specialization1 within the department:
- Algebra, Algebraic Geometry, Combinatorics and Number theory
- Analysis, Differential Equations and Probability
- Applied and Computational Mathematics
- Logic
- Geometry, Topology, and Dynamics
Additional Area of Specialization: Mathematics Education1
The course requirement is the same as for the other specialties except that the required 51 credits should include 18 credits in courses related 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.
There are many different paths that one may take to 51 credits while pursuing a PhD in Mathematics at Wisconsin. Below are a few possible paths in different subject areas. These are just possibilities, and are not meant to prescribe any particular path.
Algebra, Algebraic Geometry, or Number Theory1
Students may complete the following sequence of courses.
| Code | Title | Credits |
|---|---|---|
| MATH 741 | Abstract Algebra | 3 |
| MATH 742 | Abstract Algebra | 3 |
| MATH 747 | Lie Algebras | 3 |
| MATH 748 | Algebraic Number Theory | 3 |
| MATH 750 | Homological Algebra | 3 |
| MATH 751 | Introductory Topology I | 3 |
| MATH 752 | Introductory Topology II | 3 |
| MATH 761 | Differentiable Manifolds | 3 |
| MATH 763 | Introduction to Algebraic Geometry | 3 |
| MATH 764 | Introduction to Algebraic Geometry | 3 |
| MATH 843 | Representation Theory | 3 |
| MATH 844 | Arithmetic Geometry | 3 |
| MATH 845 | Class Field Theory | 3 |
| MATH 848 | Advanced Topics in Number Theory | 3 |
| MATH 851 | Topics in Geometric Topology | 3 |
| MATH 900 | Graduate Teaching Seminar | 1 |
| MATH 941 | Seminar-Algebra | 1-3 |
| MATH 967 | Seminar in Number Theory | 1-3 |
| MATH 990 | Reading and Research | 1-3 |
Analysis and/or Differential Equations1
Students may complete the following sequence of courses.
| Code | Title | Credits |
|---|---|---|
| MATH 719 | Partial Differential Equations | 3 |
| MATH 720 | Partial Differential Equations | 3 |
| MATH 721 | A First Course in Real Analysis | 3 |
| MATH 722 | Complex Analysis | 3 |
| MATH 725 | A Second Course in Real Analysis | 3 |
| MATH 751 | Introductory Topology I | 3 |
| MATH 752 | Introductory Topology II | 3 |
| MATH 758 | Introduction to Ergodic Theory and Dynamics | 3 |
| MATH 761 | Differentiable Manifolds | 3 |
| MATH 821 | Advanced Topics in Real Analysis | 3 |
| MATH 823 | Advanced Topics in Complex Analysis | 3 |
| MATH 825 | Selected Topics in Functional Analysis | 3 |
| MATH 827 | Fourier Analysis | 3 |
| MATH 828 | Advanced Topics in Harmonic Analysis | 3 |
| MATH 900 | Graduate Teaching Seminar | 1 |
| MATH 921 | Seminar in Analysis | 1-3 |
| MATH 990 | Reading and Research | 1-3 |
Probability1
Students may complete the following sequence of courses.
| Code | Title | Credits |
|---|---|---|
| MATH 718 | Randomized Linear Algebra and Applications | 3 |
| MATH/COMP SCI/I SY E/STAT 726 | Nonlinear Optimization I | 3 |
| MATH/COMP SCI/I SY E 730 | Nonlinear Optimization II | 3 |
| MATH/STAT 733 | Theory of Probability I | 3 |
| MATH/STAT 734 | Theory of Probability II | 3 |
| MATH 735 | Stochastic Analysis | 3 |
| MATH 741 | Abstract Algebra | 3 |
| MATH 742 | Abstract Algebra | 3 |
| MATH 751 | Introductory Topology I | 3 |
| MATH 752 | Introductory Topology II | 3 |
| MATH 758 | Introduction to Ergodic Theory and Dynamics | 3 |
| MATH 801 | Topics in Applied Mathematics | 3 |
| MATH/STAT 833 | Topics in the Theory of Probability | 3 |
| MATH/E C E/STAT 888 | Topics in Mathematical Data Science | 1-3 |
| MATH 900 | Graduate Teaching Seminar | 1 |
| MATH 921 | Seminar in Analysis | 1-3 |
| MATH 990 | Reading and Research | 1-3 |
Applied or Computational Mathematics1
Students may complete the following sequence of courses.
| Code | Title | Credits |
|---|---|---|
| MATH 703 | Methods of Applied Mathematics 1 | 3 |
| MATH 704 | Methods of Applied Mathematics-2 | 3 |
| MATH 705 | Mathematical Fluid Dynamics | 3 |
| MATH/STAT 709 | Mathematical Statistics I | 4 |
| MATH/COMP SCI 714 | Methods of Computational Mathematics I | 3 |
| MATH/COMP SCI 715 | Methods of Computational Mathematics II | 3 |
| MATH 717 | Stochastic Computational Methods | 3 |
| MATH 718 | Randomized Linear Algebra and Applications | 3 |
| MATH 719 | Partial Differential Equations | 3 |
| MATH 721 | A First Course in Real Analysis | 3 |
| MATH 722 | Complex Analysis | 3 |
| MATH 725 | A Second Course in Real Analysis | 3 |
| MATH 751 | Introductory Topology I | 3 |
| MATH 752 | Introductory Topology II | 3 |
| MATH 900 | Graduate Teaching Seminar | 1 |
| MATH 921 | Seminar in Analysis | 1-3 |
| MATH 990 | Reading and Research | 1-3 |
Logic1
Students may complete the following sequence of courses.
| Code | Title | Credits |
|---|---|---|
| MATH 721 | A First Course in Real Analysis | 3 |
| MATH 725 | A Second Course in Real Analysis | 3 |
| MATH 740 | Enumerative Combinatorics/Symmetric Functions | 3 |
| MATH 741 | Abstract Algebra | 3 |
| MATH 742 | Abstract Algebra | 3 |
| MATH 747 | Lie Algebras | 3 |
| MATH 751 | Introductory Topology I | 3 |
| MATH 752 | Introductory Topology II | 3 |
| MATH 763 | Introduction to Algebraic Geometry | 3 |
| MATH 770 | Foundations of Mathematics | 3 |
| MATH 771 | Set Theory | 3 |
| MATH 773 | Computability Theory | 3 |
| MATH 776 | Model Theory | 3 |
| MATH 873 | Advanced Topics in Foundations | 3 |
| MATH 900 | Graduate Teaching Seminar | 1 |
| MATH 975 | Seminar-The Foundations of Mathematics | 1-3 |
| MATH 990 | Reading and Research | 1-3 |
Geometry, Topology, and Dynamics1
Students may complete the following sequence of courses.
| Code | Title | Credits |
|---|---|---|
| MATH 721 | A First Course in Real Analysis | 3 |
| MATH 722 | Complex Analysis | 3 |
| MATH 725 | A Second Course in Real Analysis | 3 |
| MATH 741 | Abstract Algebra | 3 |
| MATH 742 | Abstract Algebra | 3 |
| MATH 751 | Introductory Topology I | 3 |
| MATH 752 | Introductory Topology II | 3 |
| MATH 758 | Introduction to Ergodic Theory and Dynamics | 3 |
| MATH 761 | Differentiable Manifolds | 3 |
| MATH 765 | Differential Geometry | 3 |
| MATH 843 | Representation Theory | 3 |
| MATH 851 | Topics in Geometric Topology | 3 |
| MATH 853 | Topics in Algebraic Topology | 3 |
| MATH 873 | Advanced Topics in Foundations | 3 |
| MATH 900 | Graduate Teaching Seminar | 1 |
| MATH 951 | Seminar in Topology | 1-3 |
| MATH 990 | Reading and Research | 1-3 |
Footnotes
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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 serve as the official document of record for Graduate School academic and administrative policies and procedures and are updated continuously. Note some policies redirect to entries in the official UW-Madison Policy Library. Programs may set more stringent policies than the Graduate School. Policies set by the academic degree program can be found below.
Major-Specific Policies
Prior Coursework
Graduate Credits Earned at Other Institutions
With program approval, students in the PhD program are allowed to transfer 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.
Undergraduate Credits Earned at Other Institutions or UW-Madison
No more than 7 credits from an undergraduate degree are allowed to transfer. Coursework must be at a graduate level and must request program approval. Coursework earned ten years or more prior to admission to a doctoral degree is not allowed to satisfy requirements.
Credits Earned as a Professional Student at UW-Madison (Law, Medicine, Pharmacy, and Veterinary careers)
Refer to the Graduate School: Transfer Credits for Prior Coursework policy.
Credits Earned as a University Special Student at UW–Madison
With program approval, refer to the Graduate School: Transfer Credits for Prior Coursework policy.
Probation
Refer to the Graduate School: 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 per 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.
Refer to the Graduate School: Time Limits policy.
Grievances and Appeals
These resources may be helpful in addressing your concerns:
- Bias or Hate Reporting
- Graduate Assistantship Policies and Procedures
- Hostile and Intimidating Behavior Policies and Procedures
- 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 Student Assistance and Support (OSAS) (for all students to seek grievance assistance and support)
- 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)
L&S Policy for Graduate Student Academic Appeals
Graduate students have the right to appeal an academic decision related to an L&S graduate program if the student believes that the decision is inconsistent with published policy.
Academic decisions that may be appealed include:
- Dismissal from the graduate program
- Failure to pass a qualifying or preliminary examination
- Failure to achieve satisfactory academic progress
- Academic disciplinary action related to failure to meet professional conduct standards
Issues such as the following cannot be appealed using this process:
- A faculty member declining to serve as a graduate student’s advisor.
- Decisions regarding the student’s disciplinary knowledge, evaluation of the quality of work, or similar judgements. These are the domain of the department faculty.
- Course grades. These can be appealed instead using the L&S Policy for Grade Appeal.
- Incidents of bias or hate, hostile and intimidating behavior, or discrimination (Title IX, Office of Compliance). Direct these to the linked campus offices appropriate for the incident(s).
Appeal Process for Graduate Students
A graduate student wishing to appeal an academic decision must follow the process in the order listed below. Note time limits within each step.
- The student should first seek informal resolution, if possible, by discussing the concern with their academic advisor, the department’s Director of Graduate Studies, and/or the department chair.
- If the program has an appeal policy listed in their graduate program handbook, the student should follow the policy as written, including adhering to any indicated deadlines. In the absence of a specific departmental process, the chair or designee will be the reviewer and decision maker, and the student should submit a written appeal to the chair within 15 business days of the academic decision. The chair or designee will notify the student in writing of their decision.
- If the departmental process upholds the original decision, the graduate student may next initiate an appeal to L&S. To do so, the student must submit a written appeal to the L&S Assistant Dean for Graduate Student Academic Affairs within 15 business days of notification of the department’s decision.
- To the fullest extent possible, the written appeal should include, in a single document: a clear and concise statement of the academic decision being appealed, any relevant background on what led to the decision, the specific policies involved, the relief sought, any relevant documentation related to the departmental appeal, and the names and titles of any individuals contributing to or involved in the decision.
- The Assistant Dean will work with the Academic Associate Dean of the appropriate division to consider the appeal. They may seek additional information and/or meetings related to the case.
- The Assistant Dean and Academic Associate Dean will provide a written decision within 20 business days.
- If L&S upholds the original decision, the graduate student may appeal to the Graduate School. More information can be found on their website: Grievances and Appeals (see: Graduate School Appeal Process).
Other
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Professional Development
Graduate School Resources
Take advantage of the Graduate School's professional development resources to build skills, thrive academically, and launch your career.
Learning Outcomes
- 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.