Mathematics Courses [229-244]

[204-224] [229-244] [245-550]

MATH 229. Complex Analysis I. 3 crs. Linear fractional transformations, conformal mapping, holomorphic functions, Cauchy’s theorem (including the homotopic version), properties of holomorphic functions, the argument principle, residues, power series, Laurent series, meromorphic functions.

MATH 230. Complex Analysis II. 3 crs. Continuation of MATH 229. Freihet space of holomorphic functions, Montel’s theorem, normal families, Picard’s theorem, Mittag-Leffler’s theorem, Weierstrass’ theorem, simply connected domains, d-bar equation and Runges Theorem, compact Riemann surfaces, de Rham Cohomology, Zeta functions, Marmonic and subharmonic functions, Dirichlet problems.

MATH 231. Functional Analysis I. 3 crs. Banach spaces; the dual topology and weak topology; the Hahn-Banach, Krein-Milman and Alaoglu theorems; the Baire category theorem; the closed graph theorem; the open mapping theorem; the Banach-Steinhaus theorem; elementary spectral theory; and differential equations. Prereq: Graduate status.

MATH 232. Functional Analysis II. 3 crs. Continuation of MATH 231, including topological vector spaces; bounded operators; Banach algebras; spectra and symbolic calculus; Gelfand and Fourier transforms; and distributions. Prereq: MATH 231.

MATH 234. Advanced Ordinary Differential Equations I. 3 crs. Existence, uniqueness, and representation of solutions of ordinary differential equations; periodic solutions, singular points, oscillation theorems, and boundary value problems. Prereq.: Graduate status.

MATH 235. Advanced Ordinary Differential Equations II. 3 crs. Continuation of MATH 234, including qualitative theory stability and Liapunov functions; focal, nodal, and saddle points; limit sets; and the Poincare-Bendixson theorem. Prereq.: MATH 234.

MATH 236. Partial Differential Equations I. 3 crs. First-order partial differential equations, method of characteristics; Cauchy-Kovalevskaya theorem; second-order equations, classification existence, and uniqueness results; formulation of some of the classical problems of mathematical physics. Prereq.: Graduate status.

MATH 237. Partial Differential Equations II. 3 crs. Continuation of MATH 236, showing applications of functional analysis to differential equations including distributions, generalized functions, semigroups of operators, the variational method, and the Riesz-Schauder theorem. Prereq: MATH 236.

MATH 239. Fourier Series and Boundary Value Problems. 3 crs. Fourier analysis, Bessel’s inequality, Parseval’s relation, Hilbert spaces, compact operators, eigenfunction expansions, and Sturm-Liouville problems. Prereq.: Graduate status.

MATH 240. Mathematics Statistics I. 3 crs. Probability; random variables; distributions; moment generating functions; limit theorems; parametric families of distributions; sampling distributions; sufficiency; and likelihood functions. Prereq.: Graduate status.

MATH 241. Mathematical Statistics II. 3 crs. Continuation of MATH 240 including point and interval estimations; hypotheses testing; decision functions; regression; non-parametric inferences; and analysis of categorical data.

MATH 242. Stochastic Processes. 3 crs. Continuation of MATH 241 including conditional probability, conditional expectation, normal processes, convariance, stationary processes, renewal equations, and Markov chains. Prereq.: MATH 241.

MATH 243. Dynamical System I. 3 crs. Systems of differential equations existence, uniqueness and continuity of solutions, linear systems, including constant coefficients, asymptotic behaviour, periodic coefficients; stability of linear and almost linear systems; the Poincaré-Bendix theorem; global stability (Lyapunov method); differential equations and dynamical systems - including closed orbits structural stability and 2-dimensional flow. Prereq.: Graduate status.

MATH 244. Dynamical Systems II. 3 crs. Introduction to Chaos; local bifurcations: - center manifolds, normal forms, equilibria; and periodic orbits; averaging and perturbation: - Poincaré maps, Hamiltonian systems and Melnikov's methods; hyperbolic sets, symbolic dynamics and strange attractors: - Smale Horseshoe, invariant sets, Markov partitions and statistical properties; global bifurcations: - Lorentz and Hopf bifurcations; Chaos in discrete dynamical system. Prereq.: MATH 243.