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.