The typical schedule for a first-year Ph.D. student is as follows:
Measure Theory, Hilbert Space and Fourier Theory. Possible topics from: Lebesgue measure starting on R, convergence and Fubini theorems, generalizing to locally compact spaces and groups.
Local and global theory of analytic functions of one variable.
Algebra I and II
Group Theory (Group actions, Sylow, Nilpotent/Solvable, simple groups, Jordan-Holder series, presentations), Commutative algebra (uniqueness of factorization, Jordan decomposition, Dedekind rings, class groups, local rings, Spec), finite fields, algebraic numbers, Galois theory, Homological algebra, Semisimple algebras.
Geometry-Topology I and II
Point-set topology, fundamental group and covering spaces, smooth manifolds, smooth maps, partitions of unity, tangent and general vector bundles, (co)homology, tensors, differential forms, integration and Stokes' theorem, de Rham cohomology.
Number Theory I and II
Possible topics include: Factorization of ideals, local fields, local-vs-global Galois theory, Brauer group, adeles and ideles, class field theory, Dirichlet L-functions, Chebotarev density theorem, class number formula, Tate's thesis.
Geometry-Topology III and IV
Possible topics include: differential geometry, hyperbolic geometry, three dimensional manifolds, knot theory.
Advanced topics in number theory/representation theory and geometry/topology.
The research seminar is an opportunity for students to present their own research or give lectures on advanced topics. Participation in the research seminar is encouraged by the department. A student may be required by their advisor to participate and/or speak in the research seminar.