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This page provides general guidance regarding course choices for the students in the Ph.D. program in Mathematics at the University of Michigan.
Stage 1 Pre-candidates should register for Alpha core courses to prepare for the corresponding Qualifying Review exams:
- 556 - Applied Functional Analysis
- 572 - Numerical Methods for Differential Equations
- 591 - General & Differential Topology
- 592 - Algebraic Topology
- 593 - Algebra I
- 594 - Algebra II
- 596 - Complex Analysis
- 597 - Real Analysis
Stage 2 Pre-candidates should choose coursework to fulfill the Cognate Requirement and the Distribution Requirement.
Rackham Graduate Level Approved Courses (Click the subject, the "Graduate Courses.") offered by a Department other than the Mathematics Department, including cross-listed courses such as Math 625/Stats 625. Please register for these courses under the Cognate Department i.e. Stats 625 rather than Math 625.
Each student must earn a grade of at least B- in six advanced mathematics courses. These must be chosen from three of the five major areas in the table below. 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. Core "alpha" courses (or any substitute) are typically not eligible, but a student who has already taken four alpha courses may count one or two additional alpha courses towards the distribution 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).
The table below is designed to help students meet the Distribution Requirement. 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.
|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, 709, 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|
*** 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.