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Math Ph.D. Course Registration

This page provides general guidance regarding course choices for the students in the Ph.D. program in Mathematics at the University of Michigan.  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:

  1. algebra, algebraic geometry, algebraic number theory
  2. analysis, analytic number theory, probability
  3. topology, differential geometry
  4. applied analysis, numerical analysis
  5. 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. 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).


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, 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
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.