Doctor of Philosophy in Mathematics

The doctoral program provides an opportunity to gain deep expertise, conduct research and write a dissertation in the areas of computational algebra, cryptology, decision making under uncertainty, group theory, numerical analysis, optimization, partial differential equations (PDEs), reliability theory, and stochastic modeling under the guidance of world-class experts. Its main outcome is producing an original and significant result in mathematics and its applications. In addition to rigorous coursework, doctoral candidates are expected to explore and learn on their own, e.g., by attending and organizing seminars, and to build effective communication skills. The ability to conduct scientific research requires not only intimate knowledge of the chosen area of specialization but also qualities such as scientific curiosity, initiative, critical thinking, independence, and tenacity. Upon completion of the program, the students can pursue careers in academia, technology industries (pharmaceuticals, insurance, financial services, etc.), scientific and management consulting companies, and military and government research labs.

Degree Requirements

Degree requirements include:

Coursework and credits,
General Exam, and
Dissertation (proposal and defense).

1. Coursework and Credits

The doctoral program requires 84 credits beyond the bachelor’s degree. A prior master’s degree may be transferred for up to 30 credits without specific course descriptions and with approval of the department and the Office of Graduate Education. Up to one-third of additional course credits may be transferred with the approval of the thesis committee and the Office of Graduate Academics & Student Success. The grade of “B” (3.0 GPA) or better is required for such courses and such courses may not have been already used to obtain an academic degree. The remaining 54 credits must include:

Three-credit “Signature” course PRV961
30 research credits (some of those credits can be substituted by course credits with approval from the thesis advisor)

Academic Standing

Doctoral students are expected to maintain the Institute’s required cumulative grade-point average. Students failing to meet this requirement may be placed on probation at the discretion of (follow the institute policy on academic probation).

2. General Exam

The general exam is offered once a year, typically, in the second half of September, and in case of failure, needs to be retaken in the same academic year, typically, in the first half of February. The general (qualifying) exam tests the knowledge of three subjects: real analysis and two subjects chosen in consultation with the student’s academic advisor. The real analysis subject is based on two courses: Functional Analysis I and II (MA 635, MA 636), and each chosen subject is based on two closely related courses. Subjects and corresponding courses include but are not limited to:

Algebra:

MA 605	Foundations of Algebra I	3
	And
MA 606	Foundation of Algebra II	3

Discrete Mathematics:

MA 627	Combinatorial Analysis	3
	And
MA 620	Intro Network & Graph Theory	3

ODEs and Numerical Analysis:

MA 615	Numerical Analysis I	3
	And
MA 649	Intermediate Differential Equations	3

Optimization:

MA 629	Nonlinear Optimization	3
	And Either
MA 630	Advanced Optimization Methods	3
	Or
MA 661	Dynamic Programming and Reinforcement Learning	3
	Or
MA 662	Stochastic Programming	3
	Or
MA 655	Optimal Control Theory	3

PDEs and Complex Analysis:

MA 650	Intermediate Partial Differential Equations	3
	And Either
MA 681	Complex Analysis with Applications	3
	Or
MA 653	Numerical Solutions of Partial Differential Equations	3
	Or
MA 711	Inverse Problems in Science and Engineering	3

Probability and Statistics:

MA 611	Probability	3
	And Either
MA 612	Mathematical Statistics	3
	Or
MA 623	Stochastic Processes	3
	Or
MA 641	Time Series Analysis I	3
	Or
MA 720	Advanced Statistics	3

A student and his/her academic advisor can propose different course combinations for the above subjects or propose other subjects along with corresponding courses. Such proposals must be submitted to the graduate committee for approval three months prior to taking the qualifying exam. Students admitted to the Ph.D. program with BS/MS degrees should attempt the qualifying exam no later than the end of their fourth/second semester.

Students pass the qualifying exam and are admitted to Ph.D. candidacy if they score at least 70 out of 100 on each subject. Students failing all three subjects will not be admitted to Ph.D. candidacy. Students failing at most two subjects are allowed a second attempt to pass exams on the failed subjects. This second attempt is to take place in the following semester. Students are admitted to Ph.D. candidacy only if they pass all remaining subjects on the second attempt.

3. Proposal Defense and Dissertation

The final and most important step of the Ph.D. program is writing a dissertation of publishable quality. This will embody the results of the student’s original research in mathematics, and the dissertation will be presented by the student at a public defense. If the suitably appointed Dissertation Committee approves the defense, the student will be recommended to the Office of Graduate Academics & Student Success for the Ph.D. degree. At least one manuscript based on dissertation work should be submitted to a peer-reviewed journal at the time of the dissertation defense.

(a) Dissertation Advisory Committee

Upon entering the Ph.D. program, a doctoral student is assigned an advisor who helps devise the study plan for the first academic year and consults the student on academic matters. The Ph.D. students are urged to determine their research interests and choose a thesis adviser before taking the qualifying exam. Within six months of becoming a doctoral candidate, the student and the Research Advisor should have agreed on a research topic. Refer to the Graduate Student Handbook for further details on the requirements and process for approving the Advisory Committee.

(b) Dissertation Proposal

Within one year from the date of successfully passing the qualifying exam, the Ph.D. candidate will prepare and defend a research proposal. The research proposal should address (i) problem motivation and literature review, (ii) problem formulation, (iii) approaches and methods to be used and (iv) expected results and potential impact. A few days prior to the defense, the candidate will deliver to the advisory committee a brief (two-page) summary of the proposal.

(c) Dissertation Defense (Institute requirements from the Doctoral Handbook)

Within two weeks after the start of the last semester in which the student expects to complete the requirements for the doctorate, the student needs to submit an Application for Program Completion to the Office of the Registrar.
Six weeks before the date on which the student expects to defend the dissertation, the student distributes copies to his or her advisory committee.
After the dissertation has been accepted and approved by the student’s advisory committee, the student, in conjunction with the appropriate department, shall schedule the final public oral examination. The dissertation abstract shall be submitted to the Office of Graduate Academics & Student Success to publicize the "Public Defense of Doctoral Dissertation" at least ten working days before the examination.