# Doctoral Program in Pure and Applied Mathematics

**Degree Requirements**

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.

**Coursework and Credits**

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 Graduate Academics & Student Success. Up to one-third of additional course credits may be transferred with the approval of the thesis committee and the 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.

## 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 | Foundation of Algebra I | 3 |

And | ||

MA 606 | Foundation of Algebra II | 3 |

### Discrete Mathematics:

MA 627 | Combinatorial Analysis | 3 |

And | ||

MA 620 | Introduction to Network and 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.

**Dissertation Advisory Committee**

Upon entering the Ph.D. program, a doctoral student is assigned an adviser 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.

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.

**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 for the Ph.D. degree.