Coursework and Prelims for Ph.D. Students
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. By January of their second year,
students must pass at least 4 exams at the Ph.D. level. By the end of
their second year, students must pass all 6 exams listed below, with at least 4
of the 6 passes at the Ph.D. level.
Each prelim has 5 problems, and grades are based on the number of questions which are
substantially correct. A score of 3 or higher is good for a Ph.D. pass, and a score of
2 is good for a Masters pass.
Suggested Course Sequence
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.
|MTH 436, Algebra I
||MTH 437, Algebra II
|MTH 471, Real Analysis
||MTH 467, Complex Analysis
|MTH 440, General Topology
||MTH 453, Differentiable Manifolds
|MTH 472, Functional Analysis
||MTH 443, Algebraic Topology
|Courses in your research area
|Third and Fourth years
|Do your research
|Write your thesis, apply for jobs. A paper or two will greatly boost your job chances.
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.
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.