Pure and Applied Mathematics Doctoral Program Curriculum Overview

The doctoral program provides opportunity to gain deep expertise, conduct research and write a thesis in the areas of stochastic modeling, optimization, graph theory, group theory, cryptography, probability, and mathematical finance under the guidance of world-class experts. Its main outcome is producing an original and significant result in mathematics and its application. In addition to rigorous coursework, doctoral candidates are expected to explore and learn on their own outside of class, e.g., by attending and organizing seminars.

By the end of this program, you will be able to:

  • Take leadership and initiative in scientific projects and devise research programs related to their areas of expertise
  • Develop mathematical models within their areas of expertise, explain the underlying assumptions that were used in constructing a model, and understand the limitations of a particular model
  • Identify and apply appropriate analytical and numerical methodologies for investigating a model and develop and implement suitable numerical algorithms, as needed
  • Effectively communicate mathematical concepts and results and use reasoned arguments to defend those results through both written reports and oral presentations
  • Teach undergraduate and graduate-level courses within their areas of expertise
  • Serve as referees for scientific work and proposals relevant to their areas of expertise

Degree Requirements

The doctoral program requires 84 credits beyond the bachelor’s degree (54 credits beyond the master’s) of which at least 30 credits must be doctoral research credits (MA 960). This credit total includes the three-credit “Signature” course, PRV961. Some of the 30 research credits can be substituted by course credits with approval from the thesis advisor. 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 Dean of Graduate Academics. Up to one-third of additional course credits may be transferred with the approval of the thesis committee and the Dean of Graduate Academics. 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 preliminary requirements for the doctorate are regarded not as ends in themselves, but rather as preparation for the dissertation in which the student demonstrates ability.

If you have existing graduate credits or experience in this area of study, please contact [email protected] to discuss opportunities to include it in the curriculum.

Qualifying Examination

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: Real Variables 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: Foundations of Algebra I and II (MA 605, MA 606)
  • Discrete Mathematics: Combinatorial Analysis (MA 627) and Introduction to Network & Graph Theory (MA 620)
  • ODEs and Numerical Analysis: Numerical Analysis (MA 615) and Intermediate Differential Equations (MA 649)
  • Optimization: Nonlinear Optimization (MA 629) and either Advanced Optimization Methods (MA 630) or Dynamic Programming & Reinforcement Learning (MA 661) or Stochastic Optimization (MA 662) or Optimal Control (MA 655)
  • PDEs & Complex Analysis: Partial Differential Equations (MA 650) and either Functions of a Complex Variable I (MA 681) or Numerical Solutions of PDEs (MA 653) or Inverse Problems in Science and Engineering (MA 711)
  • Probability & Statistics: Probability (MA 611) and either Mathematical Statistics (MA 612) or Stochastic Processes (MA 623) or Time Series Analysis I (MA 641) or Multivariate Statistics (MA 720)

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.

Doctoral Dissertation and Advisory Committee

The primary requirement for a doctoral degree in mathematics is that you produce a dissertation containing an original and significant result in mathematics. You will work under the guidance of a faculty advisor who is an expert in your area of research.

Preparation for dissertation work includes both courses in mathematical fundamentals and practice in communicating mathematics orally and in writing. The courses you take will not necessarily include everything you will need to know. As a doctoral student you will be expected to learn some mathematics on your own outside of class. Seminars afford a means to that end. They can be organized informally among students or more formally with a faculty advisor. Seminars of the latter type may be taken for academic credit. Students are encouraged to identify subjects they would like to study and to seek out faculty advisors.

For additional information about courses, please review the academic catalog.