5000 Level - Core Courses
MTH 5310 - Advanced Abstract Algebra I
Prerequisite(s): MTH 4314 and consent of the instructor.
Finite groups, Sylow theorems, nilpotent and solvable groups, principal ideal domains, unique factorization domains, and sub rings to algebraic number fields.
MTH 5311 - Advanced Abstract Algebra II
Prerequisite(s): MTH 5310.
Field theory, Galois theory, modules, finitely generated modules, principal ideal domains, homological methods, and Wedderburn-Artin theorems.
MTH 5323 - Theory of Functions of Real Variables I
Prerequisite(s): MTH 4327.
Borel sets, measure and measurable sets, measurable functions, and the Lebesque integral.
MTH 5324 - Theory of Functions of Real Variables II
Prerequisite(s): MTH 5323.
Function spaces, abstract measure, and differentiation. MTH 5330 - Topology Prerequisite(s): Graduate standing. Topological spaces, continuous functions, metric spaces, connectedness, compactness, separation axioms, Tychenoff theorem, fundamental group, covering spaces, metrization theorems.
MTH 5331 - Algebraic Topology I
Prerequisite(s): MTH 5330.
Homology theory, simplicial complexes, topological invariance, relative homology, Eilenberg-Steenrod axioms, singular homology, CW complexes.
MTH 5350 - Complex Analysis
Prerequisite(s): MTH 4327.
Complex numbers, complex functions, analytic functions, linear fractional transformations, complex integration, Cauchy's formula, residues, harmonic functions, series and product expansions, Gamma function, Riemann mapping theorem, Dirichlet problem, analytic continuation.
MTH 5360 - Applied Mathematics I
Prerequisite(s): Graduate standing.
Dynamical systems (ODE and PDE, discrete and continuous), linear and nonlinear systems theory, transform methods, control theory and optimization, calculus of variations, stability theory.
MTH 5361 - Applied Mathematics II
Prerequisite(s): Graduate standing.
Eigenvalue theory, projections for linear equations iterations and multilevel methods, fast Fourier transforms, approximations of differential equations, grid adaption and numerical stability, weak solutions and Sobelov space, wavelets with applications.