**Courses in the Mathematics Ph.D. Program at the University of Michigan (General Guidelines)**

This page provides general guidance regarding course choices for the students in the Ph.D. program in Mathematics at the University of Michigan. (For the AIM program/courses, please consult this page.) Its main goal is to help incoming graduate students design their first-year curriculum. In the table below, the courses are categorized, very roughly, into four tiers:

*Introductory*courses aimed at advanced undergraduates, M.S. students, and Ph.D. students from other programs- Courses presenting
*foundations*of the respective fields of mathematics at the beginning graduate level; among them, the**core**, “**alpha**”, courses forming the basis of the Qualifying Review (QR) exams are shown in**boldface**. *Advanced*courses which require considerable familiarity with the foundations of the corresponding subject, either by succeeding in the foundational courses or by passing the QR exams in that subject- more specialized courses dedicated to
*Topics*of current research interest

This classification is by no means rigid, and is provided as a general guidance only. Course syllabi may vary depending on year, instructor, and choice of topic. Please consult the current course descriptions for up-to-date information. The first priority for the incoming first-year student is to pass the QR, which requires passing approved combinations of written exams and **core** courses. The syllabi for the core courses and past QR exams can be found here.

Furthermore, in order to achieve candidacy, you must fulfill the Distribution Requirement as stated below.

**II. Distribution Requirement**

Each student must earn a grade of at least B- in six advanced mathematics courses. These must be chosen from three of the following five areas:

- algebra, algebraic geometry, algebraic number theory
- analysis, analytic number theory, probability
- topology, differential geometry
- applied analysis, numerical analysis
- applied discrete mathematics, combinatorics, logic

Eligible courses include those at the 600 level or above. Certain advanced 500 level courses may also be eligible, subject to prior approval by the Chair of the Doctoral Committee. **However, core "alpha" courses are not eligible, nor is any course used to satisfy the QR requirement.** With the approval of the Chair of the Doctoral Committee, certain courses taken outside the department, for example in physics, may be allowed to count for the distribution requirement (under area 4).

Area |
Discipline |
Introductory |
Foundations |
Advanced |
Topics |

1 | Algebra | 593, 594 |
612-615, 711 | 619, 637, 715 | |

1 | Algebraic Geometry | 631, 632 | 731, 732 | ||

1 | Algebraic Number Theory | 575 | 675*, 676*, 679 | 775*, 776* | |

2 | Analysis | 555 | 596, 597, 656*, 657 |
602, 604, 605, 650 | 609, 701-704, 707, 710, 793, 794 |

2 | Analytic Number Theory | 675*, 676*, 677, 678 | 775*, 776* | ||

2 | Probability | 525, 526* | 625, 626 | ||

3 | Geometry/Topology | 531, 590 | 592 |
691, 692, 695, 696 | 697, 791, 792 |

3 | Differential Geometry | 537 | 591 |
635, 694 | 636 |

4 | Applied Analysis | 526*, 550*, 556, 557, 558, 654, 656* |
651, 756 | ||

4 | Numerical Analysis | 571, 572 |
671 | ||

5 | Applied Discrete Mathematics | 550*, 526*, 565*, 567 | |||

5 | Combinatorics | 565*, 566 | 664 | 665, 669 | |

5 | Logic | 582 | 681 | 682-684 | 781 |

*** Courses listed with an asterisk may fit in a different area based on the instructor/semester offered. More generally, there is sometimes flexibility in the classification of these courses depending on the semester and instructor. Talk to the Doctoral Committee Chair if you have questions.

Additional information can be found on the graduate page. See in particular the list of math graduate courses by term, and the list of all courses by area.

Additional information can be found on the graduate page. See in particular the current course offerings, the list of math graduate courses, and the list of all courses by area (including those not intended for graduate students in math). Some of those lists may be out of date; ditto for the table above. Please send corrections to* *math-grad-office@umich.edu.