Coursework for Ph.D. Students

To prepare for the preliminary exams and to help fulfill the coursework requirements, most graduate students enroll in the following core courses in their first and second years.

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.
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.
MTH467: Theory of analytic functions. Topics include Cauchy's theorem, Taylor and Laurent series, residues, conformal mapping, analytic continuation.
MTH471: Measure and integration. Topics include Lebesgue measure on the real line, measure spaces, Lebesgue integration, convergence theorems, Radon-Nikodyn theorem, differentiation, Fubini's theorem, L_p spaces.
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.

Written Preliminary Examinations

The written preliminary exams ("prelims") cover the material treated in the core courses above. Students are required to pass the prelims by the end of their second year. Students may take them without penalty for failure and are encouraged to do so 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 467, 471)
(III) General Topology and either (Algebraic topology or Functional analysis). (MTH 440, (443 or 472))
To pass the prelims a student must pass two different exams, not necessarily at the same time. Students must successfully complete coursework for any area in which they do not pass a prelim exam.

Here are some old exams in pdf form.