# MATH - Mathematics Course Descriptions

## MATH A1200 Topics in Mathematics

Topics to be chosen from graduate mathematics and related fields. This course can be repeated at most 2 times for a maximum of 12 credits total.

4

4 hr./wk.

## MATH A3200 Theory of Functions of a Complex Variable I

Cauchy-Riemann equations, conformal mapping, elementary, entire, meromorphic, multiple-valued functions, Cauchy integral theorems, series expansion.

4

### Prerequisites

MATH 32500 or MATH 32404.

4 hr./wk.

## MATH A3400 Theory of Functions of a Real Variable I

Lebesgue measure and integration on the real line, differentiation of real functions and the relation with integration, classical Lp spaces.

4

### Prerequisites

MATH 32500 or MATH 32404.

4 hr./wk.

## MATH A3500 Partial Differential Equations I

First order equations, shock waves; classification and canonical forms of higher order equations, characteristics, the Cauchy problem for the wave equation: Huygens' principle; the heat equation; Laplace's equation; the Dirichlet and Neuman problems; harmonic functions; eigenvalue expansions; Green's functions.

4

### Prerequisites

(MATH 32500 or MATH 32404) and MATH 39100.

4 hr./wk.

## MATH A4300 Set Theory

Axioms of Zermelo-Fraenkel set theory; relations, functions, equivalences and orderings, cardinal numbers and cardinal arithmetic; well-ordered sets, ordinal numbers, transfinite induction and recursion, The Axiom of Choice and the Continuum Hypothesis.

4

MATH 32300.

4 hr./wk.

## MATH A4400 Mathematical Logic

The propositional calculus, the sentential calculus, normal forms, first order theories, consistency, categoricity, decidability, Godel's incompleteness theorem, the Loewenheim-Skolem theorem.

4

### Prerequisites

MATH 32300 and either MATH 34700 or MATH 44900.

4 hr./wk.

## MATH A4500 Dynamical Systems

Dynamical systems in one and more dimensions, symbolic dynamics, chaos theory, hyperbolicity, stable manifolds, complex dynamics.

4

### Prerequisites

MATH 32404 or its equivalent.

4 hr./wk.

## MATH A4600 Linear Algebra

Linear systems, matrix decompositions, inner product spaces, self-adjoint transformations, spectral theory, discrete Fourier Transforms.

4

### Prerequisites

C or better in MATH 34600 or departmental permission.

4 hr./wk.

## MATH A4900 Modern Algebra I

Groups, rings, fields.

4

### Prerequisites

MATH 32300 and MATH 34600.

4 hr./wk.

## MATH A6100 Differential Geometry

The theory of curves and surfaces in three-dimensional space: frames, fundamental forms, geodesics, curvature of surfaces, surface area, surfaces with boundary, the Gauss-Bonnet Theorem, introduction to Riemannian metrics.

4

### Prerequisites

MATH 32500 or MATH 32404.

4 hr./wk.

## MATH A6300 Topology I

A course in general topology. Topological spaces: metric spaces, subspaces, continuous maps, connectedness, separation axioms; topological vector spaces: Hilbert spaces, Banach space, Frechet spaces; the quotient topology or identification spaces: the classification of two-dimensional manifolds; fundamental group and covering spaces; covering spaces of graphs: applications to group theory.

4

### Prerequisites

MATH 32500 or MATH 32404.

4 hr./wk.

## MATH A6400 Number Theory

A first course in algebraic number theory which assumes some abstract algebra. Topics include: unique factorization in the integers and Euclidean domains, structure of the groups Z/mZ and their multiplicative units, quadratic residues and quadratic reciprocity, algebraic number fields, finite fields.

4

### Prerequisites

C or better in MATH 34700 or departmental permission.

4 hr./wk.

## MATH A6800 Combinatorial Analysis

Permutations, combinations, generating functions and recurrence relations, inclusion and exclusion, applications to matching theory, linear and dynamic programming, Polya's theory of counting, introduction to graph theory and coloring theory.

4

### Prerequisites

MATH 30800 or MATH 32300.

4 hr./wk.

## MATH A7700 Stochastic Processes I

Special topics in probability such as stochastic processes, Markov chains.

4

### Prerequisites

MATH 32300, MATH 34600 and MATH 37500.

4 hr./wk.

## MATH A7800 Advanced Mathematical Statistics

The multivariate normal distribution, multiple and partial correlation, regression and least squares, the analysis of variance.

4

### Prerequisites

MATH 32300, MATH 34600 and MATH 37600.

4 hr./wk.

## MATH A9801 Independent Study

A program of independent study under the direction of a member of the Department, with approval of the Graduate Advisor.

### Credits

1 repeatable 2 times for a total of 3 credits

1 hr./wk.

## MATH A9802 Independent Study

A program of independent study under the direction of a member of the Department, with approval of the Graduate Advisor.

### Credits

2, repeatable 2 times for a total of 6 credits

2 hr./wk.

## MATH A9803 Independent Study

A program of independent study under the direction of a member of the Department, with approval of the Graduate Advisor.

### Credits

3, repeatable 2 times for a total of 9 credits

3 hr./wk.

## MATH A9804 Independent Study

A program of independent study under the direction of a member of the Department, with approval of the Graduate Advisor.

### Credits

4, repeatable 2 times for a total of 12 credits

4 hr./wk.

## MATH B1100 Selected Topics in Pure Mathematics

Topics to be chosen from the areas of algebra, analysis, topology, geometry, and logic. This course can be repeated at most 2 times for a maximum of 12 credits total.

4

4 hr./wk.

## MATH B1200 Selected Topics in Classical Analysis

Topics to be chosen from applied mathematics and related fields. Typical subjects are: asymptotic methods, wave propagation, mathematical biology. This course can be repeated at most 2 times for maximum of 12 credits total.

4

4 hr./wk.;

## MATH B1300 Selected Topics in Probability and Statistics

Topics to be chosen from the areas of probability, statistics, game theory, combinatorial analysis, etc. This course can be repeated at most 2 times for a maximum of 12 credits total.

4

4 hr./wk.

## MATH B3200 Theory of Functions of a Complex Variable II

A continuation of MATH A3200, including such topics as analytic continuation, conformal mapping, Dirichlet problem, meromorphic functions, entire functions, Picard's Theorem, elliptic functions.

4

MATH A3200.

4 hr./wk.

## MATH B3400 Theory of Functions of a Real Variable II

Abstract measure and integration theory, abstract Lebesgue measure and integral, signed measures, Radon-Nikodym derivative, Lp spaces, product spaces, Daniell integral. Special topics such as Stieltjes integrals, Denjoy integral, Haar measure, measure rings, applications to probability.

4

MATH A3400.

4 hr./wk.

## MATH B3500 Partial Differential Equations II

First order quasi-linear and nonlinear equations, Cauchy-Kowalewsky Theorem, well-posed problems, Cauchy problem for hyperbolic systems, the wave equation in n-dimensions, boundary value problems for elliptic equations, Laplace's equation, parabolic equations, heat equation.

4

MATH A3500.

4 hr./wk.

## MATH B4500 Dynamical Systems II

Topics will be chosen from the areas of ergodic theory, topological dynamics, differentiable dynamics, complex dynamics and symbolic dynamics.

4

### Prerequisites

MATH A4500 or departmental permission.

4 hr./wk.

## MATH B4900 Introduction to Modern Algebra II

Field extensions, Galois theory, vector spaces and modules, category theory, special topics.

4

MATH A4900.

4 hr./wk.

## MATH B6300 Topology II

An introduction to algebraic topology. Homology: simplicial and singular, computations and applications; categories and functors; cohomology of groups, cup product, and Poincare duality; universal coefficients for homology; homotopy theory: homotopy groups, calculating them, connections with cohomology.

4

### Prerequisites

MATH A4900 and MATH A6300.

4 hr./wk.

## MATH B7700 Stochastic Processes II

Markov chains, limit theorems, renewal equations, random walks, Brownian motion, branching processes, queuing theory.

4

MATH A7700.

4 hr./wk.

## MATH B7800 Advanced Topics in Statistics

The general decision problem, decision-making principles, application to hypothesis testing and estimation, minimax and Bayes solutions, utility theory, sequential procedures.

4

MATH A7800

4 hr./wk.;

## MATH B9800 Independent Study

A program of independent study under the direction of a member of the Department, with approval of the Graduate Advisor.

Variable credit.

## MATH B9801 Independent Study

Independent Study. This course can be repeated at most 2 times for a maximum of 3 credits total.

1

1 hr./wk.

## MATH B9802 Independent Study

Independent Study. This course can be repeated at most 2 times for a maximum of 6 credits total.

2

2 hr./wk.

## MATH B9803 Independent Study

Independent Study. This course can be repeated at most 2 times for a maximum of 9 credits total.

3

3 hr./wk.

## MATH B9804 Independent Study

Independent Study. This course can be repeated at most 2 times for a maximum of 12 credits total.

4

4 hr./wk.