## MATH 256A: Partial Differential Equations

The theory of linear and nonlinear partial differential equations, beginning with linear theory involving use of Fourier transform and Sobolev spaces. Topics: Schauder and L2 estimates for elliptic and parabolic equations; De Giorgi-Nash-Moser theory for elliptic equations; nonlinear equations such as the minimal surface equation, geometric flow problems, and nonlinear hyperbolic equations.

Last offered: Spring 2014

## MATH 256B: Partial Differential Equations

Continuation of 256A.

Last offered: Winter 2015
| Repeatable for credit

## MATH 257A: Symplectic Geometry and Topology

Linear symplectic geometry and linear Hamiltonian systems. Symplectic manifolds and their Lagrangian submanifolds, local properties. Symplectic geometry and mechanics. Contact geometry and contact manifolds. Relations between symplectic and contact manifolds. Hamiltonian systems with symmetries. Momentum map and its properties. May be repeated for credit.

Terms: Win
| Units: 3
| Repeatable for credit

Instructors:
Eliashberg, Y. (PI)

## MATH 257B: Symplectic Geometry and Topology

Continuation of 257A. May be repeated for credit.

Terms: Spr
| Units: 3
| Repeatable for credit

Instructors:
Ganatra, S. (PI)

## MATH 257C: Symplectic Geometry and Topology

Continuation of 257B. May be repeated for credit.

Last offered: Spring 2015

## MATH 258: Topics in Geometric Analysis

May be repeated for credit.

Terms: Win
| Units: 3
| Repeatable for credit

Instructors:
Mazzeo, R. (PI)

## 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.

Terms: Win
| Units: 3

Instructors:
Candes, E. (PI)
;
Bates, E. (TA)

## 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.

Last offered: Winter 2014
| Repeatable for credit

## MATH 263B: Crystal Bases: Representations and Combinatorics

Crystal Bases are combinatorial analogs of representation theorynof Lie groups. We will explore different aspects of thesenanalogies and develop rigorous purely combinatorial foundations.

Terms: Win
| Units: 3
| Repeatable for credit

Instructors:
Bump, D. (PI)

## MATH 263C: Topics in Representation Theory

May be repeated for credit.

Last offered: Spring 2015
| Repeatable for credit

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