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 hr./wk.
Cauchy-Riemann equations, conformal mapping, elementary, entire, meromorphic, multiple-valued functions, Cauchy integral theorems, series expansion.
4 hr./wk.
Lebesgue measure and integration on the real line, differentiation of real functions and the relation with integration, classical Lp spaces.
4 hr./wk.
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 hr./wk.
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 hr./wk.
The propositional calculus, the sentential calculus, normal forms, first order theories, consistency, categoricity, decidability, Godel's incompleteness theorem, the Loewenheim-Skolem theorem.
4 hr./wk.
Dynamical systems in one and more dimensions, symbolic dynamics, chaos theory, hyperbolicity, stable manifolds, complex dynamics.
4 hr./wk.
Linear systems, matrix decompositions, inner product spaces, self-adjoint transformations, spectral theory, discrete Fourier Transforms.
C or better in
MATH 34600 or departmental permission.
4 hr./wk.
4 hr./wk.
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 hr./wk.
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 hr./wk.
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.
C or better in
MATH 34700 or departmental permission.
4 hr./wk.
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 hr./wk.
Special topics in probability such as stochastic processes, Markov chains.
4 hr./wk.
The multivariate normal distribution, multiple and partial correlation, regression and least squares, the analysis of variance.
4 hr./wk.
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.
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.
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.
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.
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 hr./wk.
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 hr./wk.;
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 hr./wk.
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 hr./wk.
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 hr./wk.
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 hr./wk.
Topics will be chosen from the areas of ergodic theory, topological dynamics, differentiable dynamics, complex dynamics and symbolic dynamics.
4 hr./wk.
Field extensions, Galois theory, vector spaces and modules, category theory, special topics.
4 hr./wk.
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 hr./wk.
Markov chains, limit theorems, renewal equations, random walks, Brownian motion, branching processes, queuing theory.
4 hr./wk.
The general decision problem, decision-making principles, application to hypothesis testing and estimation, minimax and Bayes solutions, utility theory, sequential procedures.
MATH A7800
4 hr./wk.;
A program of independent study under the direction of a member of the Department, with approval of the Graduate Advisor.
Independent Study. This course can be repeated at most 2 times for a maximum of 3 credits total.
1 hr./wk.
Independent Study. This course can be repeated at most 2 times for a maximum of 6 credits total.
2 hr./wk.
Independent Study. This course can be repeated at most 2 times for a maximum of 9 credits total.
3 hr./wk.
Independent Study. This course can be repeated at most 2 times for a maximum of 12 credits total.
4 hr./wk.