Doctor of Philosophy (PhD)

Program Synopsis and Training

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

Math PhD Poster 2019

Our PhD program offers two tracks, one for Theoretical Mathematics and one for Applied Mathematics. The tracks differ only in the course and qualifying requirements during the first two years. Specifically, the qualifying requirements for the theoretical track consist of our Abstract Algebra (Math 6111, Math 6112) and Real Analysis (Math 6211, Math 6212) course sequences that can be fulfilled by passing each with at least an A- (or a respective examination). Those for the applied track combine a mandatory Scientific Computing (Math 6601) course, one of the algebra and analysis courses, three additional courses from Math 6602, Math 6411, and Math 6451, as well as the algebra and analysis courses. The breadth course requirements in the applied track, as compared to those in the theoretical track, offer more flexibility but also include a mandatory graduate course in a non-math STEM department from an approved list. 

Applicants are required to decide on one of the tracks and applications will be evaluated subject to respective criteria described below. Once students have passed their pre-candidacy requirements the two tracks merge and there is no distinction in later examinations and research opportunities. In particular, the candidacy exam for both tracks consists of a research proposal, the graduate faculty available for advising is the same, and the final degree and thesis defense are independent of the initially chosen track.

Expected Preparations and Admission Criteria

General: All applicants need to carefully follow our application instructions and meet all university minimum requirements. Incomplete and non-compliant applications will normally not be considered, and our program does not petition university requirements.

Course Preparations: Students applying to the theoretical track should have strong foundations in Real Analysis and Abstract Algebra, equivalent to our Math 5201-5202 and Math 5111-5112 sequences. Expected preparations for the applied track include the equivalents of a rigorous Real Analysis course (such as Math 5201), a strong background in Linear Algebra, as well as an introductory course in Scientific Computing. Besides these basic requirements competitive applicants submit evidence for a broad formation in mathematics at the upper-division or beginning graduate level. Relevant coursework in other mathematical or quantitative sciences may also be considered, especially for the applied track.

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

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

GRE Subject Test in Mathematics: The subject test is generally required for both tracks. It serves mainly as a reassurance of basic skills, particularly for applicants from less familiar schools or with unconventional career paths, and tends to weigh more for the theoretical track. There is no minimum cut-off score. The distribution of percentiles of subject test scores of recently admitted PhD applicants provides a rough idea of the typical range of scores our committee considers. Students who have compelling reasons to have the subject test requirement waived may submit a justification through our waiver request form, which will be forwarded to our committee. Applications that contain neither a subject test score nor a justification for a waiver are automatically removed from consideration. 

Other Test Scores: General GRE Test scores are required for international students and only recommended for domestic students. They tend to have only little weight in admission decisions but may be helpful in fellowship competitions. Students whose native language is not English should score at least a 20 on the speak portion of the TOEFL. 

Opportunities & Outcomes

Graduate Faculty & Research Areas: Currently 84 graduate faculty members are available to advise doctoral students in our mathematics program, including 19 from our regional campuses. 

  • Algebra and Number Theory
  • Algebraic Geometry
  • Applied Mathematics PDF icon [pdf]
  • Real and Complex Analysis, Operator Algebra
  • Combinatorics and Group Theory
  • Differential Geometry
  • Dynamical systems and Ergodic Theory
  • Financial and Actuarial Mathematics
  • Logic
  • Probability Theory, Statistical Mechanics
  • Mathematical Biology
  • Ordinary and partial differential equations
  • Representation theory
  • Scientific computing
  • Topology, Topological Data Analysis

Nearly half of the graduate population consists of domestic students coming from both larger universities and smaller liberal arts colleges with a solid math curriculum. International students come from all parts of the world and different educational backgrounds. About a quarter of our current graduate students are female.

Time to degree varies significantly among our students, ranging from as little as four years for well-prepared students to more than six years for students working on challenging topics. The limit of funding for each student in the program is seven years. We conduct an annual academic review ensuring sufficient progress is being made toward the completion of the degree. Currently, the average time to complete the degree is just above six years.

A more detailed overview of our PhD program can be found in the Graduate Program Prospectus on our Prospective Students page.

Most of our graduates continue their careers in academia. Many find placements in very competitive post-doctoral research positions which, in recent years, include Princeton University, IAS, University of Chicago, Yale University, University of Michigan, Cal-Tech, Northwestern University, University of Texas, Duke University, SUNY Stony Brook, Purdue University, University of North Carolina - Chapel Hill, and Indiana University.

Some of our graduates are awarded regular faculty teaching positions at established four-year colleges across the United States. A few enter the private and public sectors, finding employment (often in leadership positions) in the financial or software development industries and for US Federal government agencies such as the National Security Agency (NSA).


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