## MATH 259: mirror symmetry

| Repeatable for credit

## MATH 261A: Functional Analysis

Geometry of linear topological spaces. Linear operators and functionals. Spectral theory. Calculus for vector-valued functions. Operational calculus. Banach algebras. Special topics in functional analysis. May be repeated for credit.

| Repeatable for credit

## MATH 262: Applied Fourier Analysis and Elements of Modern Signal Processing (CME 372)

Introduction to the mathematics of the Fourier transform and how it arises in a number of imaging problems. Mathematical topics include the Fourier transform, the Plancherel theorem, Fourier series, the Shannon sampling theorem, the discrete Fourier transform, and the spectral representation of stationary stochastic processes. Computational topics include fast Fourier transforms (FFT) and nonuniform FFTs. Applications include Fourier imaging (the theory of diffraction, computed tomography, and magnetic resonance imaging) and the theory of compressive sensing.

## MATH 263A: Infinite-dimensional Lie Algebras

Basics of Kac-Moody Lie algebras, which include both finite dimensional semisimple Lie algebras and their infinite-dimensional analogs, up to the Kac-Weyl character formula and Macdonald identities, and the Boson-Fermion correspondence. May be repeated for credit. Prerequisite: 210 or equivalent.

| Repeatable for credit

## MATH 263B: Modular Representation Theory

Modular representation theory, a field largely created by Brauer, is the representation theory of finite groups over a field of characteristic p. It was a key tool in the classification of finite simple groups. Key features are the important roles played by projective modules, and the subtle relationship with the characteristic zero theory. Modular representation theory has strong similarities to the theory of groups of Lie type, with normalizers of p-subgroups playing the role of parabolics. Brauer and Green found deep relationships between the modular representation theory of the group and the simpler representation theory of such subgroups. In addition to such classical topics, we will look at some more recent developments.

| Repeatable for credit

Instructors:
Bump, D. (PI)

## MATH 264: Infinite Dimensional Lie Algebra

| Repeatable for credit

## MATH 266: Computational Signal Processing and Wavelets

Theoretical and computational aspects of signal processing. Topics: time-frequency transforms; wavelet bases and wavelet packets; linear and nonlinear multiresolution approximations; estimation and restoration of signals; signal compression. May be repeated for credit.

## MATH 269: Topics in symplectic geometry

May be repeated for credit.

| Repeatable for credit

Instructors:
Eliashberg, Y. (PI)

## MATH 270: Geometry and Topology of Complex Manifolds

Complex manifolds, Kahler manifolds, curvature, Hodge theory, Lefschetz theorem, Kahler-Einstein equation, Hermitian-Einstein equations, deformation of complex structures. May be repeated for credit.

| Repeatable for credit

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