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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. Its main goal is to help incoming graduate students design their firstyear 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 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.
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  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.