# Ph.D. Programs

We offer Ph.D. programs in both mathematics and applied mathematics. Concentrations are available in computational and interdisciplinary mathematics.

A candidate for the Ph.D. degree must (i) complete course requirements, (ii) pass written qualifying exams, (iii) pass a preliminary oral examination and (iv) conduct thesis research, write a thesis and pass a final oral examination. These steps are described below.

## Coursework

All students take a minimum of 30 credit hours of graduate courses in mathematics, numbered 500 and up. Among these should be at least one course each in continuous, discrete and computational mathematics.

Continuous mathematics | Discrete mathematics | Computational mathematics |
---|---|---|

Complex Variables (MA 513) | Linear Programming (MA 505) | Computer Algebra (MA 522) |

Analysis (MA 515) | Linear Algebra (MA 520) | Uncertainty Quantification (MA 540) |

Control (MA 531) | Abstract Algebra (MA 521) | Graph Theory (MA/CSC 565) |

PDEs (MA 534) | Combinatorics (MA 524) | Modeling (MA 573) |

Probability (MA 546) | Matrix Theory (MA 523) | Numerical Analysis (MA 580) |

ODEs (MA 532) | Algebraic Geometry (MA 526) | |

Topology (MA 551) | ||

Manifolds (MA 555) |

Students who have not had the equivalent of MA 405 (Introduction to Linear Algebra and Matrices), MA 407 (Introduction to Modern Algebra), or MA 425–426 (Mathematical Analysis I and II) must make up these courses. Students who have not had the equivalent of MA 426 may take it for graduate credit by registering for MA 591M.

## Qualifying Exams

The Ph.D. written qualifying examinations are written exams in three subjects selected by the student from a list of twelve possibilities. The purpose of the exams is to ensure that each Ph.D. student studies three subjects to a depth that gives adequate preliminary background to begin a Ph.D. project. Each subject is represented by a two-semester sequence of courses.

Select three one-year sequences from among the following.

MA 515--715 | Analysis |

MA 520--720 | Linear Algebra and Lie Algebras |

MA 521--721 | Abstract Algebra |

MA 522--722 | Computer Algebra |

MA 523--723 | Applied Matrix Theory |

MA 524--724 | Combinatorics |

MA 531--731 | Systems and Control |

MA 534--734 | Partial Differential Equations |

MA 546--747 | Applied Probability and Stochastic Process |

MA 555--753 | Geometry and Topology |

MA 573--574 | Modeling |

MA 580--780 | Numerical Analysis |

- Each exam is written and graded by two faculty members.
- At the end of the Spring Semester, the two exam writers for a qualifying exam will come up with a study guide for the qualifying exams that year. That study guide will be given to the students in the course, and will be given to the DGP so that any students who wants to take the exam and who were not in the course will be able to prepare for the exam.
- Each exam is a three hour long written exam.
- It is recommended that students attempt at least one to two exams by the end of their first year. By the end of their second year (first year for students entering with a MS in Mathematics), students have to attempt a number of exams equal to the number of exams they have yet to pass.
- One retake is allowed for each of the three examinations (with exceptions described below). Students are advised to schedule retakes as soon as possible; retakes must be done within 12 months of the date the examination is first taken. If a student fails an examination twice, he/she is considered to have failed the written qualifying examinations.
- The retake does not have to be the same exam as the one initially failed. However, if an examination for a specific sequence is failed and retaken later, the second examination must be considered a retake of the first.
- Incoming students are allowed to take qualifying exams in the August that they arrive. A fail at that time does not count against the student.
- The number of examinations taken at any given exam period cannot exceed the remaining number of passes need to reach a total of three.
- January exams are reserved for retakes. If a student takes a January retake in the first year or second year (first year only for students with a previous master’s degree), fails that happen at that time do not count against the student.
- Students must complete the exams by January of the third year (January of the second year for students with a previous masters).
- It is possible to “drop” an exam one has registered for any time up to two weeks before the exam (no questions asked). After that, we recommend students talk to the director or the administrator of the programs. Dropping an exam only applies to students who do have the option to “wait”, i.e., are not required to take the exam at that exam session.
- For part-time students, years will be counted using credit hours, with one year equal to 18 credit hours. For full-time students, calendar years are used. Students who start the graduate program in the spring semester should have a statement from the DGP put into their file specifying the date by which their exams must be taken. Depending on prior coursework, whether they are a transfer student, etc., it will be after three, four, or five semesters.

Faculty have provided these “suggested paths” of courses that students can take, depending on their area of interest. Note, however, that the particular courses most relevant for an individual student will vary depending on which faculty the student works with, and what project the student works on. Hence, it’s important to talk to faculty early to figure out the best courses that are right for each student.

Interest | Qualifying exams | Important courses (if not taken as quals) |
---|---|---|

Algebra | MA 520-720 (Linear/Lie Alg) MA 521-721(Abstract Algebra) and one of MA 524-724 (Combinatorics) MA 555-753 (Geometry/Topology) | MA 725 (Representation Theory) |

Analysis and differential equations | MA 515--715 (analysis) MA 534--734 (PDEs) | MA 532 (ODE) MA 716 (Functional Analysis) MA 719 (Vector Space Methods) MA 732 (ODE II) MA 748 (Stochastic DE) |

Biomath | MA 573-574 (Modeling) MA 580-780 (Numerical Analysis) MA 531-731 (Control Theory) MA 523-723 (Matrix Theory) | MA 771-773 (Biomath) MA 540 (Uncertainty Quantification) ST 511-512 (Statistics for Biology) |

Combinatorics | MA 521-721 (abstract algebra) MA 524-724 (combinatorics) and one of MA 515-715 (analysis) MA 520-720 (Linear/Lie Theory) MA 555-753 (geometry/topology) | MA 726 (Algebraic geometry) MA 725 (Representation theory) |

Financial Math | MA 546--747 (probability) MA 534--734 (PDEs) and one of MA 580--780 (numerical analysis) MA 531--731 (control) | MA 515 (Analysis) MA 547 (Stochastic Calculus in Finance) MA 548 (Monte Carlo Methods for Financial Mathematics) MA 549 (Financial Risk Analysis) MA 580 (Numerical Analysis) MA 748 (stochastic differential equations) |

Geometry and topology | MA 555-753 (geometry/topology) MA 521-721 (abstract algebra) and one of MA 515-715 (analysis) MA 520-720 (Linear/Lie Theory) MA 524-724 (combinatorics) MA 534-734 (PDEs) | MA 515 (analysis) MA 726 (algebraic geometry) MA 754 (algebraic topology) MA 755 (Riemannian geometry) |

Modeling and control | MA 531--731 (control) MA 573--574 (modeling) and one of MA 515--715 (analysis) MA 523--723 (matrix theory) MA 534--734 (PDEs) MA 580--780 (numerical analysis) | MA 515 (analysis) |

Numerical analysis and scientific computing | MA 580--780 (numerical analysis) and two of MA 515--715 (analysis) MA 573--574 (modeling) MA 546--747 (probability) MA 534--734 (PDEs) | MA 515 (analysis) |

Probability | Main : MA 546 - 747 (Probability) MA 515 - 715 (Analysis) Others : MA 523 - 723 (Theory of Matrices) MA 531 - 731 (Dynamical Systems) MA 534 - 734 (PDEs) | MA 544 (Computer Experiments in Mathematical Probability) MA 547 (Financial Mathematics) MA 716 (Advanced Functional Analysis) MA 748 (Stochastic Differential Equations) |

Symbolic computation | MA 522-722 (Computer algebra) and two of the followings: MA 520-720 (Linear/Lie Theory) MA 521-721 (Abstract Algebra) MA 523-723 (Applied Matrix Theory) MA 580-780 (Numerical Analysis) | CSC 505 (Design and Analysis of Algorithms) MA 505 (Linear Programming) MA 706 (Nonlinear Programming) MA 726 (Algebraic Geometry) |

## Thesis Research and Defense

Our Ph.D. programs typically take five years. More information about timetable and milestones, including the preliminary and final oral examinations, can be found in our road to graduation.

## Summer Internships

We encourage Ph.D. students at all stages of their studies to seek summer employment at government or industry facilities. Through such experiences you will expand your understanding of the mathematical sciences, discover possible areas for thesis research, and enhance your career options.

In the past few years our students have worked as summer interns at **national laboratories** including Argonne, Lawrence Livermore, Los Alamos, Oak Ridge, PNNL and Sandia; at **federal agencies** such as the Environmental Protection Agency, the National Security Agency, the Naval Research Laboratory, the Naval Sea System Command and the Army Corps of Engineers; at **international research institutions** such as the Chinese Academy of Science, INRIA-Rocquencourt (France), Osaka University (Japan), RIKEN (Japan), the University of Canterbury (New Zealand) and Xiamen University (China); and at **companies** such as Chimerix, Merck, M&T National Bank and SAS.

Advisors, faculty members and the director of the graduate programs can assist in finding summer internships.