Algebra I-II (3-3)

Abrams

Group theory including symmetric groups, free abelian groups, finitely generated abelian groups, Sylow theorems, solvable groups. Factorization in commutative rings, rings of polynomials, chain conditions, semisimple rings, Wedderburn-Artin theorems, Galois theory.

Topics in Algebra (3)

Abrams, Schmitt

Topics chosen from Lie groups and Lie algebras, non-associative algebras, abelian groups, classical groups, algebraic number theory, representation theory, algebraic geometry, and ring theory. Prerequisite: Math 6101-2. May be repeated for credit with permission.

Real Analysis I (3)

Junghenn, Ren

A rigorous study of the real number system, metric spaces, topological spaces, product topology, convergence, continuity and differentiation. Topics include Dedekind's cuts, Tikhonov's theorem, sequences and series, Abel's theorem, continuity and differentiability of real-valued functions of a real variable. Credit may not be earned for both Math 6201 and 4239.

Real Analysis II (3)

Robinson, Roudenko

Continuation of Math 6201. Topics include Riemann-Stieltjes integrals, equicontinuity, Arzela-Ascoli theorem, Stone-Weierstrass theorem, derivatives of functions of several variables, contraction mapping theorem, inverse and implicit function theorems, differential forms, exterior differentiation, Stoke's theorem, differentiable manifolds. Credit may not be earned for both Math 6202 and 4240.

Measure and Integration Theory (3)

Robinson, Roudenko

Lebesgue measure and integration in abstract spaces. Probability measures. Absolute continuity, the Radon-Nikodym theorem, measures on product spaces, and the Fubini theorem. L^P spaces and their properties. Prerequisite: Math 4239 or equivalent.

Introduction to Functional Analysis (3)

Junghenn, Robinson

Topological and metric spaces; Tychonoff theorem; Banach spaces; linear functionals and operators; Hahn-Banach, closed graph, and open-mapping theorems; uniform boundedness; Hilbert spaces; eigenvalues, projections. Prerequisite: Math 6214 or equivalent.

Ergodic Theory (3)

Robinson

Ergodicity, mixing, the K-property and the Bernoulli property. Poincaré recurrence, the Rohlin lemma, the ergodic theorem, and entropy theory. Additional topics from isomorphism theory, spectral theory, the theory of joinings, and coding theory. Prerequisite: Math 6214 or permission of instructor.

Dynamical Systems and Chaos (3)

Robinson

Linear and nonlinear systems, flows, Poincaré maps, structural stability. Examples of chaotic systems in the physical sciences. Local bifurcations, center manifold theory, normal forms, the averaging theorem. Hyperbolic invariant sets, strange attractors, the Smale horseshoe, symbolic dynamics. Prerequisite: Math 2184 and 4240 or permission of instructor.

Complex Analysis (3)

Junghenn, Robinson

Topology of the complex plane; complex differentiation and integration; Cauchy's theorem and its consequences; Taylor and Laurent series; classification of singularities; residue theory; conformal mapping; the Riemann mapping theorem. Prerequisite: Math 4239 or equivalent.

Topics in Real and Functional Analysis (3)

Junghenn, Roudenko

Possible topics include Banach algebras, function algebras, spectral theory for bounded and unbounded operators, harmonic analysis on topological groups and semigroups, topological vector spaces and operator algebras. Prerequisite: permission of instructor. May be repeated for credit with permission.

Applied Mathematics I (3)

Baginski, Ren

Dimensional analysis, perturbation methods, calculus of variations, boundary value problems in one dimension, eigenvalue problems, stability and bifurcation in nonlinear problems. Related numerical techniques. Prerequisite: Math 2184 or equivalent.

Applied Mathematics II (3)

Baginski, Ren

Method of characteristics, shock waves, wave and heat equation, Laplace operator on a bounded region, maximum principles, Green's functions, Schrödinger's equation, spherical harmonics. Numerical methods for partial differential equations. Prerequisite: Math 2184 or equivalent.

Ordinary Differential Equations (3)

Robinson

Existence and uniqueness of solutions, continuity and differentiability of solutions with respect to initial conditions. Properties of linear systems, phase portraits, planar systems and Poincaré-Bendixson theory. Prerequisite: Math 4240.

Modern Partial Differential Equations (3)

Baginski

Emphasis on modern theory and analytical techniques applied to the solution of partial differential equations. Topics include Sobolev spaces, generalized solutions, strong solutions and regularity; Sobolev imbedding theorem; Rellich-Kondrachov theorem; Leray-Schauder fixed-point theorems; nonlinear eigenvalue problems. Prerequisite: Math 6319 or permission of instructor.

Topics in Applied Mathematics (3)

Baginski

Possible topics include, but are not limited to, the calculus of variations, control theory, nonlinear partial differential equations, and mathematical programming. May be repeated for credit with permission.

Introduction to Financial Mathematics (3)

Junghenn, Ren

Elementary finance. Basic probability. Discrete random variables. Forwards, futures, and options. Options and arbitrage. The binomial model. Cox-Ross-Rubenstein formula. Martingales. Continuous random variables. The continuous model as a limit of the binomial model. Prerequisite: Math 2184, 2233.

Stochastic Calculus Methods in Finance (3)

Junghenn, Ren

Review of probability theory. The Brownian motion. The Ito integrals. Ito's formula. Martingales. Stochastic differential equations. Kolmogorov's backward equation. The generator of an Ito diffusion. Boundary value problems and the Dirichlet problem. The Black-Scholes equation. Optimal stopping. American options. Prerequisite: Math 2184, 2233.

Introduction to Numerical Analysis (3)

Gupta, Musielak

Computer arithmetic and round-off errors. Solution of linear and nonlinear systems. Interpolation and approximations. Numerical differentiation and integration. Eigenvalues and eigenvectors. Prerequisite: Math 1232 and 2184 and knowledge of a programming language.

Numerical Solution of Ordinary and Partial Differential Equations (3)

Gupta

Initial and boundary value problems for ordinary differential equations. Error propagation, convergence and stability. Finite difference and finite element methods for partial differential equations. Prerequisite: Math 3342 and knowledge of a programming language.

Topics in Numerical Analysis (3)

Gupta

Numerical methods and software. Introductions to the methods, tools, and ideas of numerical computation. Problem solving using standard mathematical software. Interpolation; linear and nonlinear equations. Differential equations. Prerequisite: Math 3342; knowledge of a programming language.

Combinatorics (3)

Bonin, Schmitt

An introduction to fundamental methods and current research problems in partially ordered sets and enumeration. Prerequisite: undergraduate modern algebra and linear algebra or permission of instructor.

Graph Theory (3)

Ullman

Graphical enumeration, factors, planarity and graph coloring, algebraic graph theory, extremal graph theory, applications. Prerequisite: undergraduate modern algebra and linear algebra or permission of instructor.

Topics in Combinatorial Mathematics (3)

Bonin, Ullman, Schmitt

Topics selected from a wide range of research subjects in combinatorics, its relations with other areas of mathematics, and applications. Recent selections have included matroid theory, topological methods in ordered sets, algebraic methods in combinatorics, fractional graph theory, combinatorics of polytopes, the symmetric group. May be repeated for credit with permission.

Mathematical Logic (3)

Harizanov, Moses

Model theory: the relation between a formal language (syntax) and its interpretations (semantics). Consistency, completeness, and compactness. Tarski's theorem on the inexpressibility of truth. Godel's incompleteness theorem and its impact on mathematics.

Topics in Logic (3)

Harizanov, Moses

Topics selected from a broad spectrum of areas of logic and applications, based on students' suggestions and interests. May be repeated for credit with permission.

General Topology (3)

Rong, Przytycki, Shumakovitch, Wu

Topological spaces, bases, open sets and closed sets; continuous maps and homeomorphisms; connectedness and compactness; metric topology, product topology and quotient topology; separation axioms; covering spaces and fundamental groups.

Algebraic Topology (3)

Rong, Przytycki, Wu

Fundamental groups and the Van Kampen theorem; simplicial complexes, simplicial homology, and Euler characteristic; singular homology, Mayer-Vietoris sequences. Topics may include cohomology, cup products, and Poincaré duality; classification of surfaces; knots and their fundamental groups. Prerequisite: Math 6810 or permission of instructor.

Knot Theory and Low Dimensional Topology (3)

Rong, Przytycki

Introduction to fundamental methods and current research in knot theory and 3-dimensional topology. Topics include Reidemeister moves, Alexander invariants, Jones-type invariants, skein modules, Khovanov homology, incompressible surfaces, and torus decomposition. Prerequisite: Math 6810 or permission of instructor.

Topics in Knot Theory and Low Dimensional Topology (3)

Rong, Przytycki

Possible topics include, but are not limited to, topology of 3-manifolds and work of Perelman, quantum invariants and their categorizations, topology of 4-manifolds after Freedman and Donaldson, computational complexity in topology, and applications in biology, chemistry, and physics. Prerequisite: Math 6850 or permission of instructor. May be repeated for credit with permission.

Rong, Przytycki, Shumakovitch, Wu

Topics may include hyperbolic structures on surfaces and 3-manifolds; knot theory; topology of 3-manifolds; topology of 4-manifolds. Prerequisite: Math 6820 or permission of the instructor. May be repeated for credit with permission.

Reading and Research (arr.)

Staff

May be repeated for credit.

Advanced Reading and Research (arr.)

Staff

Limited to students preparing for the Doctor of Philosophy general examination. May be repeated for credit.

Dissertation Research (arr.)

Staff

Limited to Doctor of Philosophy candidates. May be repeated for credit.