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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 Precandidates 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 Precandidates should choose coursework to fulfill the Cognate Requirement and the Distribution Requirement.
Cognate Requirement
Rackham Graduate Level Approved Courses (Click the subject, then "Graduate Courses.") offered by a Department other than the Mathematics Department, including crosslisted courses such as Math 625/Stats 625. Please register for these courses under the Cognate Department i.e. Stats 625 rather than Math 625.
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 five major areas in the table below. Eligible courses include those at the 600 level or above. Math 700 reading courses may not be used toward the distribution requirement. 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 uptodate information. The first priority for the incoming firstyear 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.
Area  Discipline  Introductory  Foundations  Advanced  Topics 
1  Algebra  593, 594  612615, 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, 701704, 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  682684  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.