Real Analysis I (Math 515):
Lebesque measure and Lebesgue integral, one variable differentiation
theory, product integration, Lp spaces.
Real Analysis II (Math 516):
Metric spaces, topological spaces, compactness, abstract theory of
measure and integral, differentiation of measures, Banach spaces,
distribution theory, Fourier transform, functional analysis.
Abstract Algebra (Math 504): Algebraic
systems and their morphisms, including groups, rings, modules, and
Advanced Linear Algebra (Math
Advanced topics in linear algebra including canonical forms; unitary,
normal, Hermitian and positive-definite matrices; variational
characterizations of eigenvalues, and applications to other branches of
Complex Analysis (Math 511): Theory
of analytic functions, integration, topology of the extended complex
plane, singularities and residue theory, maximum principle.
Stochastic Processes (Math 554):
chains on discrete spaces in discrete and continuous time (random
walks, Poisson processes, birth and death processes) and their
long-term behavior, branching processes, renewal theory, introduction
to Brownian motion.
Advanced Stochastic Processes (Math 645): Weak
convergence, Random walks, and Brownian motion. Martingales, Stochastic
integration and Ito's Formula. Stochastic differential equations and