Doctoral Program in Mathematics
Ph.D. degree in Mathematics.
Students without a masters degree or similar postbacheloreate 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 wellprepared 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 oneonone 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
Universitywide 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 
Comprehensive Examination
The comprehensive examination consists of two written examinations and a breadth course requirement.
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 


Analysis  Syllabus 


Differential Equations  Syllabus 


Topology  Syllabus 


Statistics  Syllabus 


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.