MATH202A
Overview
Subject: Math
Course Number: 202A
Department: Mathematics
Course Level: Graduate
Course Title
Introduction to Topology and Analysis
Course Description
Metamathematics of predicate logic. Completeness and compactness theorems. Interpolation theorem,definability,theory of models. Metamathematics of number theory, recursive functions, applications to truth and provability. Undecidable theories. Sequence begins fall.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH202B
Overview
Subject: Math
Course Number: 202B
Department: Mathematics
Course Level: Graduate
Course Title
Introduction to Topology and Analysis
Course Description
Measure and integration. Product measures and Fubini-type theorems. Signed measures; Hahn and Jordan decompositions. Radon-Nikodym theorem. Integration on the line and in Rn. Differentiation of the integral. Hausdorff measures. Fourier transform. Introduction to linear topological spaces, Banach spaces and Hilbert spaces. Banach-Steinhaus theorem; closed graph theorem. Hahn-Banach theorem. Duality; the dual of LP. Measures on locally compact spaces; the dual of C(X). Weak and weak-* topologies; Banach-Alaoglu theorem. Convexity and the Krein-Milman theorem. Additional topics chosen may include compact operators, spectral theory of compact operators, and applications to integral equations.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH204
Overview
Subject: Math
Course Number: 204
Department: Mathematics
Course Level: Graduate
Course Title
Ordinary Differential Equations
Course Description
Rigorous theory of ordinary differential equations. Fundamental existence theorems for initial and boundary value problems, variational equilibria, periodic coefficients and Floquet Theory, Green's functions, eigenvalue problems, Sturm-Liouville theory, phase plane analysis, Poincare-Bendixon Theorem, bifurcation, chaos.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH205
Overview
Subject: Math
Course Number: 205
Department: Mathematics
Course Level: Graduate
Course Title
Theory of Functions of a Complex Variable
Course Description
Normal families. Riemann Mapping Theorem. Picard's theorem and related theorems. Multiple-valued analytic functions and Riemann surfaces. Further topics selected by the instructor may include: harmonic functions, elliptic and algebraic functions, boundary behavior of analytic functions and HP spaces, the Riemann zeta functions, prime number theorem.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH206
Overview
Subject: Math
Course Number: 206
Department: Mathematics
Course Level: Graduate
Course Title
Functional Analysis
Course Description
Spectrum of an operator. Analytic functional calculus. Compact operators. Hilbert-Schmidt operators. Spectral theorem for bounded self-adjoint and normal operators. Unbounded self-adjoint operators. Banach algebras. Commutative Gelfand-Naimark theorem. Selected additional topics such as Fredholm operators and Fredholm index, Calkin algebra, Toeplitz operators, semigroups of operators, interpolation spaces, group algebras.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH207
Overview
Subject: Math
Course Number: 207
Department: Mathematics
Course Level: Graduate
Course Title
Unbounded Operators
Course Description
Unbounded self-adjoint operators. Stone's Theorem, Friedrichs extensions. Examples and applications, including differential operators. Perturbation theory. Further topics may include: unbounded operators in quantum mechanics, Stone-Von Neumann Theorem. Operator semigroups and evolution equations, some non-linear operators. Weyl theory of defect indices for ordinary differential operators.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH208
Overview
Subject: Math
Course Number: 208
Department: Mathematics
Course Level: Graduate
Course Title
C*-algebras
Course Description
Basic theory of C*-algebras. Positivity, spectrum, GNS construction. Group C*-algebras and connection with group representations. Additional topics, for example, C*-dynamical systems, K-theory.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH209
Overview
Subject: Math
Course Number: 209
Department: Mathematics
Course Level: Graduate
Course Title
Von Neumann Algebras
Course Description
Basic theory of von Neumann algebras. Density theorems, topologies and normal maps, traces, comparison of projections, type classification, examples of factors. Additional topics, for example, Tomita Takasaki theory, subfactors, group actions, and noncommutative probability.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH212
Overview
Subject: Math
Course Number: 212
Department: Mathematics
Course Level: Graduate
Course Title
Several Complex Variables
Course Description
Power series developments, domains of holomorphy, Hartogs' phenomenon, pseudo convexity and plurisubharmonicity. The remainder of the course may treat either sheaf cohomology and Stein manifolds, or the theory of analytic subvarieties and spaces.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH219
Overview
Subject: Math
Course Number: 219
Department: Mathematics
Course Level: Graduate
Course Title
Dynamical Systems
Course Description
Diffeomorphisms and flows on manifolds. Ergodic theory. Stable manifolds, generic properties, structural stability. Additional topics selected by the instructor.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH222A
Overview
Subject: Math
Course Number: 222A
Department: Mathematics
Course Level: Graduate
Course Title
Partial Differential Equations
Course Description
The theory of boundary value and initial value problems for partial differential equations, with emphasis on nonlinear equations. Laplace's equation, heat equation, wave equation, nonlinear first-order equations, conservation laws, Hamilton-Jacobi equations, Fourier transform, Sobolev spaces.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH222B
Overview
Subject: Math
Course Number: 222B
Department: Mathematics
Course Level: Graduate
Course Title
Partial Differential Equations
Course Description
The theory of boundary value and initial value problems for partial differential equations, with emphasis on nonlinear equations. Second-order elliptic equations, parabolic and hyperbolic equations, calculus of variations methods, additional topics selected by instructor.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH224A
Overview
Subject: Math
Course Number: 224A
Department: Mathematics
Course Level: Graduate
Course Title
Mathematical Methods for the Physical Sciences
Course Description
Introduction to the theory of distributions. Fourier and Laplace transforms. Partial differential equations. Green's function. Operator theory, with applications to eigenfunction expansions, perturbation theory and linear and non-linear waves. Sequence begins fall.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH224B
Overview
Subject: Math
Course Number: 224B
Department: Mathematics
Course Level: Graduate
Course Title
Mathematical Methods for the Physical Sciences
Course Description
Introduction to the theory of distributions. Fourier and Laplace transforms. Partial differential equations. Green's function. Operator theory, with applications to eigenfunction expansions, perturbation theory and linear and non-linear waves. Sequence begins fall.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH258
Overview
Subject: Math
Course Number: 258
Department: Mathematics
Course Level: Graduate
Course Title
Harmonic Analysis
Course Description
Basic properties of Fourier series, convergence and summability, conjugate functions, Hardy spaces, boundary behavior of analytic and harmonic functions. Additional topics at the discretion of the instructor.
Units
Minimum Units: 4
Maximum Units: 4
Prior Terms Offered
MATH278
Overview
Subject: Math
Course Number: 278
Department: Mathematics
Course Level: Graduate
Course Title
Topics in Analysis
Course Description
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.
Units
Minimum Units: 4
Maximum Units: 4