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PhD Preliminary Exams

Last updated May 2, 2022.

The preliminary exams, or "prelims," are the examinations required by the department for admission to official candidacy for the PhD degree. Old prelims are available on-line.

Starting in 2018, the prelims will be administered twice a year: once in September (1st try) and once over the spring break (2nd try).  See “Exam Process” section below for changes to the eligibility criteria.

Exam Dates

In 2022, the prelims will tentatively be offered September 12-16

  • Algebra: Monday, September 12
  • Analysis: Wednesday, September 14
  • Manifolds: Friday, September 16

The second try for all of the prelims for 2023 will be announced at a later date.

Exam Process

These exams are offered every year in the mathematical subjects treated by the designated core courses: algebra,  analysis, and geometry/topology. They are intended to serve as an objective measure of a student's mastery of basic graduate level mathematics, as a milestone on the road to becoming an independent mathematician.

The written preliminary exams are given twice a year: once during the week in September that precedes the start of Autumn Quarter, and once during the break that follows Winter Quarter. Only the students who have completed the corresponding year-long core sequence with a grade of at least 3.0 each quarter are eligible for the second try.

Each exam is written and graded by a group of faculty that is kept confidential. Examiners arrive at a recommendation that is given to the Graduate Program Committee, which makes a final decision on who has passed the exam. These decisions are communicated to students in brief memos roughly a week after the exam. 

Students who wish to have additional feedback on an exam are encouraged to do so by requesting a review with a faculty member in the area of the exam, and submitting a request that the Graduate Program Coordinator release the student's exam to that faculty member. If, as a result of this process, it is believed that an exam was graded incorrectly, then a student may submit a written request that the Graduate Program Committee reconsider its decision. The final decision on prelim results rests with the Graduate Program Committee.

A student may substitute completion of a full three-quarter sequence of a designated core course with a high enough grade for the passing of the corresponding preliminary exam.

Normally, students take two exams in September of the beginning of their second year, but students are welcome to attempt exams in September of year one.  A student is expected to pass two of the three exams by the end of the student's second year in the Ph.D. program. If you are concerned about not meeting this milestone, please speak with your preliminary advisor and/or the Graduate Program Coordinator.

Students' experiences in the program and on the tests are important to us. If you have a temporary health condition or permanent disability that requires accommodations for exams (conditions include but not limited to mental health, attention-related, learning, vision, hearing, physical or health impacts), you are welcome to contact DRS at 206-543-8924 or or DRS offers resources and coordinates reasonable accommodations for students with disabilities and/or temporary health conditions.  Reasonable accommodations are established through an interactive process between you, DRS, and the GPC.   If you have already established accommodations with Disability Resources for Students (DRS), please communicate your approved accommodations to the GPC at your earliest convenience.

It is the policy and practice of the University of Washington to create inclusive and accessible learning environments consistent with federal and state law.

Exam Topics

There are three exams:

  • Algebra: Topics at the level of 402-3-4 and 504-5-6.
  • Analysis: Topics at the level of 424-5-6, 524-5, and 534.
  • Manifolds: Topics at the level of 544-5-6.

Each syllabus below lists certain topics that have appeared on the exams. This list is advisory only – it is intended to suggest the level of the exams, not to prescribe exactly the material that will appear. Past exams can be a useful source of practice questions, but a student need not master all material that has been covered on these exams. A student who knows the material in the syllabus and who has spent some time solving problems should do well on the exams. 



  • Linear algebra: vector spaces and linear operators, characteristic and minimal polynomials, eigenvalues and eigenvectors, Cayley-Hamilton theorem, Jordan canonical form, rational canonical form.
  • Commutative rings: PIDs, UFDs, modules over PIDs, prime and maximal ideals, Noetherian and Artinian rings and modules, Hilbert basis theorem, local rings and Nakayama lemma, localization, Integral extensions, Noether normalization lemma, Hilbert Nullstellensatz, prime ideal spectrum.
  • Rings and modules: simple modules, composition series, Jordan-Holder theorem for modules, semi-simple rings, Artin-Wedderburn theorem, tensor product.
  • Group theory: nilpotence, solvability and simplicity, composition series, Sylow theorems, group actions, free groups, simple groups, permutation groups, and linear groups, direct and semi-direct product of groups, presentations in terms of generators and relations.
  • Representation theory: group algebras, irreducible representations, Schur's lemma, Maschke's theorem, character theory.
  • Field theory: roots of polynomials, finite and algebraic extensions, algebraic closure, splitting fields and normal extensions, finite fields, Galois groups and Galois correspondence, solvability of equations.
  • Category theory and homological algebra: categories and functors, natural transformations, universal properties, products and coproducts, exact and  split exact sequences, 5-lemma and snake lemma, projective and injective modules, resolutions, (left and right) exact functors, adjoint functors, adjointenss of Hom and Tensor, Tor and Ext.

References: Dummit and Foote, Abstract Algebra, second edition; Lang, Algebra; Herstein, Topics in Algebra; Hungerford, Algebra; J.-P. Serre, Linear Representations of Finite Groups; M. Atiyah, I. Macdonald, Introduction to Commutative Algebra; M. Reid, Undergraduate Commutative Algebra.


Topics in Real Analysis: Metric spaces. General measure and general integration theory, Lebesgue integral, convergence theorems. Banach spaces, Hilbert spaces, Lp-spaces. Differentiation and its relation to integration in Rn, signed measures, the Radon-Nikodym Theorem, representation of bounded linear functionals on C_0(X) for locally compact Hausdorff spaces X.

References: Folland, Real Analysis; Royden, Real Analysis; Rudin, Real and Complex Analysis; Stein and Shakarchi, Real Analysis.

Topics in Complex Analysis: Basic theory of analytic functions from complex numbers to power series to contour integration, Cauchy's theorem and applications such as the maximum principle, Schwarz Lemma, argument principle, Liouville theorem etc. 

References: Ahlfors, Complex Analysis; Conway, Functions of One Complex Variable, vol. 1; Marshall, Complex Analysis; Rudin, Real and Complex Analysis (the chapters devoted to complex analysis).


Topics: Elementary manifold theory; the fundamental group and covering spaces; submanifolds, the inverse and implicit function theorems, immersions and submersions; the tangent bundle, vector fields and flows, Lie brackets and Lie derivatives, the Frobenius theorem, tensors, Riemannian metrics, differential forms, Stokes's theorem, the Poincare lemma, de Rham cohomology; elementary properties of Lie groups and Lie algebras, group actions on manifolds, the exponential map.

References: Lee, Introduction to Topological Manifolds, 2nd ed. (Chapters 1-12) and Introduction to Smooth Manifolds, 2nd ed. (all but Chapters 18 and 22);