Coursework and Prelims for Ph.D. Students

Students admitted in the fall of 2012 or later must follow the new prelim rules. Students admitted before the fall of 2012 can choose either the old or new prelim system.

New Prelim Rules

In their first two years, students must take all 8 core courses listed below, or demonstrate proficiency. During their graduate career, but not necessarily in the first two years, students must take 5 formal courses at the 500 level.

Secondly, students must pass the 6 written preliminary exams ("prelims"). Prelims are given during the year, as part of the final exams of the corresponding courses, and again in August. Students will receive either a Ph.D. pass, a Masters pass, or a fail. Students must pass at least 4 of the 6 exams at the Ph.D. level by January of their second year. By the end of their second year, students must pass all 6 exams listed below.

Algebra I (MTH 436)
Algebra II (MTH 437)
Real Analysis (MTH 471)
Complex Analysis (MTH 467)
General Topology (MTH 440)
Geometry (MTH 453)

Suggested Course Sequence Under the New System

If at all possible, students should take advanced courses in their chosen area starting in the second year. The following schedule allows students to pass the prelims at least by August at the end of their first year, and start research quickly.

First year
Fall	Spring
MTH 436, Algebra I	MTH 437, Algebra II
MTH 471, Real Analysis	MTH 467, Complex Analysis
MTH 440, General Topology	MTH 453, Differentiable Manifolds

Second year
Fall	Spring
MTH 472, Functional Analysis	MTH 443, Algebraic Topology
Courses in your research area

Third and Fourth year
Do your research

Fifth year
Write your thesis, submit papers, apply for jobs

Old Prelim Rules

Secondly, students must pass the written preliminary exams ("prelims") by the end of their second year. To pass the prelims a student must pass two different exams, not necessarily at the same time.There is no penalty for failure, and students are encouraged to take the exams early for the experience. Prelims are given in January and in August.

Prelims will consist of three separate exams on three consecutive days:

(I) Algebra (MTH 436, 437)
(II) Analysis- Real analysis, Complex analysis, Advanced Calculus (MTH 471, 467)
(III) General Topology and Geometry (MTH 440, 453)

Core courses

MTH436 and MTH437: Algebra. Topics include group theory, ring theory, modules, Galois theory, and linear algebra.
MTH440: General Topology. Topics include continuity, compactness, connectedness, metrizability, product spaces.
MTH453: Differentiable Manifolds. Topics include manifolds, vector fields and vector bundles, differential forms, partitions of unity, homogeneous spaces, inverse function theorem and submanifolds. Other topics may be chosen from tensors, Lie derivatives, deRham cohomology, integrable flows (fundamental theorem of ODE for manifolds), and Frobenius theorem.
MTH443: Algebraic Topology I. Topics include the combinatorial structure of complexes and the homology of polyhedra, algebraic techniques for the classification of topological spaces, fixed point theory.
MTH471: Real analysis. Topics include Lebesgue measure on the real line, measure spaces, Lebesgue integration, convergence theorems, Radon-Nikodym theorem, differentiation, Fubini's theorem, L^p spaces.
MTH467: Theory of analytic functions. Topics include Cauchy's theorem, Taylor and Laurent series, residues, conformal mapping, analytic continuation.
MTH472: Functional Analysis. Hilbert space, operators on Hilbert space, spectral theorem for compact self-adjoint operators. Banach spaces, Hahn-Banach theorem, open mapping theorem, closed graph theorem, principle of uniform boundedness. Weak topologies.

Past Prelims

Here are some past prelims in pdf form.

The following exams are from the new system instituted in 2011.

The following exams are from the old system.