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Doctoral Program in Mathematics

Ph.D. degree in Mathematics.

Students without a masters degree or similar post-bacheloreate experience will not be admitted directly to the doctoral program; instead they should apply to the doctoral preparation track of the masters program.

Program Mission

To train students to create, apply, and disseminate mathematical knowledge and understanding.

Program Learning Objectives

  • Graduates will be able to extend the frontier of mathematical knowledge by producing quality research with original results.
  • Graduates will be able to apply a range of mathematical tools to problems within mathematics and in other disciplines.
  • Graduates will be able to effectively disseminate mathematical knowledge and understanding through publications, seminars, classroom teaching, or other means.

Program Overview

The Ph.D. in mathematics is intended for students who wish to advance mathematical knowledge itself, apply such knowledge to problems confronting society and science, and educate others in mathematical methods and ways of thinking. The Mathematics Department offers students the possibility of designing study plans to meet their individual goals and interests. In particular, we offer a broad spectrum of possible research areas for our Ph.D. students, including algebra, analysis, coding theory, computational harmonic analysis, partial and ordinary differential equations, dynamical systems, financial mathematics, mathematical biology, numerical analysis, optimal control theory, set theory, statistics, stochastic processes, and topology.

The first phase in doctoral education in Mathematics is to understand a few subjects deeply and a range of subjects in less detail. Our program accomplishes this through a system of courses and written examinations. An exceptionally well-prepared student can attempt the examinations early and spend relatively little time doing coursework.

The second phase is to become the expert on a specific problem and produce new mathematical results on it suitable for a dissertation. In our program this phase is done one-on-one with a faculty advisor or in a small research group. The dissertation is a scholarly work demonstrating the ability to understand, organize, improve, and present mathematical ideas of outstanding importance, depth, or interest. It must include original mathematical research and be worthy of publication.

Most doctoral students are trained and financially supported as teaching assistants and have the opportunity to teach classes as the primary instructor.

Degree Requirements

University-wide policies and standards appear in the Graduate Catalog. The department additionally imposes:

  • A student whose grade point average (GPA) is below 3.0 (B) for two consecutive semesters will be dropped from the program.

The deadlines for fulfilling the requirements of the program are given below, assuming the student entered in the Fall semester. Note that these are deadlines and not a suggested timeline; in particular, students become ineligible for financial support before the deadline to defend.

Milestone Student Type

Has M.S. from elsewhere Enters from our M.S. program
Clock Starts Enters our doctoral program Entered our master's program
Pass 1 written examination winter of year 2 winter of year 3
Pass 2 written examinations winter of year 3 winter of year 4
Find advisor and form advisory committee 2 years 3 years
Breadth course requirement 3 years 4 years
Advance to Candidacy 3 years 4 years
Successfully defend dissertation 7 years from entering doctoral program

Students who miss a deadline to pass a written examination will be dismissed at the end of Spring semester. If they believe that some exceptional circumstances justify an extension to this deadline, they may request an extension from the Graduate Committee, in writing, including any supporting documentation, by March 1.

Students who miss the deadline to advance to candidacy will be dismissed at the end of Summer term. If they believe that some exceptional circumstances justify an extension to this deadline, they may request an extension from the Graduate Committee, in writing, including letters of support from their advisor and advisory committee member and any other supporting documentation, by August 1. The renewal of any departmental financial support will be contingent upon advancing to candidacy by May 15.

Comprehensive Examination

The comprehensive examination consists of two written examinations and a breadth course requirement. A student may not register for MATH 8950 Dissertation until they have completed the comprehensive examination.

Written Examinations

Students are required to pass written examinations in two of the following subjects. Their choices of subjects should be made in consultation with potential Ph.D. advisors. An examination in each subject will be offered twice each year, in late summer and in winter. A student may make no more than 5 total attempts at examinations, with a maximum of 3 attempts in any given subject area.

Subject Examination Syllabus Core Courses Prerequisite Courses
Algebra Syllabus
  • MATH 6221 Algebra I
  • MATH 6222 Algebra II
  • MATH 5221 Modern Algebra I
  • MATH 5222 Modern Algebra II
Analysis Syllabus
  • MATH 6301 Analysis I
  • MATH 6302 Analysis II
  • MATH 5301 Advanced Calculus I
  • MATH 5302 Advanced Calculus II
Differential Equations Syllabus
  • MATH 6411 Partial Differential Equations I
  • MATH 6412 Partial Differential Equations II
  • MATH 5400 Advanced Differential Equations
  • MATH 5410 Fourier Analysis and Partial Differential Equations
  • One of:
    • MATH 5330 Hilbert Spaces and Applications
      • MATH 5301 Advanced Calculus I
      • MATH 5302 Advanced Calculus II
Topology Syllabus
  • MATH 6700 Point Set Topology
  • MATH 6710 Algebraic Topology
  • MATH 5301 Advanced Calculus I
  • MATH 5302 Advanced Calculus II
  • MATH 5700 Introduction to Topology
Statistics Syllabus
  • MATH 6500 Mathematical Statistics
  • MATH 6510 Linear Models
  • MATH 5301 Advanced Calculus I
  • MATH 5302 Advanced Calculus II
  • MATH 5500 Theory of Statistics
  • MATH 5510 Applied Statistics

Breadth Requirement

Students must earn a grade of B or better in 4 courses above 5999, excluding the core courses in the subjects of their two written examinations. At least one course must be a core course in an examination subject in which the student did not take a written examination. Project, seminar, and independent study courses may not be used for this requirement.

Advisor and Advisory Committee

The student must choose (and be accepted by) a dissertation advisor. The advisor will form an Advisory Committee, consisting of the advisor and at least one other faculty member. The Advisory Committee is in charge of the student's study plan, and with determining if/when the student is prepared to undertake the independent research necessary for a dissertation. If the Advisory Committee judges that the student is not making satisfactory progress, they may recommend to the Graduate Committee that the student be withdrawn from the Ph.D. program.

Advancement to Candidacy

The student must form a Dissertation Committee, submit a dissertation proposal, and have it approved by the committee. The committee consists of at least the advisor, the College Faculty Representative, and two other members. The proposal must demonstrate that the student has developed sufficient background expertise in a specialty to conduct research in it, and present a problem or conjecture, which, if solved, would be sufficient for a dissertation. Upon completion of the comprehensive examination and approval of the dissertation proposal, the student is admitted to candidacy.