## 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.

Terms: Win
| Units: 3
| Repeatable for credit

Instructors:
Weinstein, M. (PI)

## 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.

Last offered: Autumn 2016
| Repeatable for credit

## MATH 257B: Symplectic Geometry and Topology

Continuation of 257A. May be repeated for credit.

Terms: Aut
| Units: 3
| Repeatable for credit

Instructors:
Ionel, E. (PI)

## MATH 257C: Symplectic Geometry and Topology

Continuation of 257B. May be repeated for credit.

Terms: Spr
| Units: 3

Instructors:
Latschev, J. (PI)

## MATH 258: Topics in Geometric Analysis

May be repeated for credit.

Last offered: Winter 2017
| 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.

Last offered: Winter 2016

## MATH 263A: Algebraic Combinatorics and Symmetric Functions

Symmetric function theory unifies large parts of combinatorics. Theorems about permutations, partitions, and graphs now follow in a unified way. Topics: The usual bases (monomial, elementary, complete, and power sums). Schur functions. Representation theory of the symmetric group. Littlewood-Richardson rule, quasi-symmetric functions, combinatorial Hopf algebras, introduction to Macdonald polynomials. Throughout, emphasis is placed on applications (e.g. to card shuffling and random matrix theory). Prerequisite: 210A and 210B, or equivalent.

Last offered: Autumn 2016
| 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.

Last offered: Winter 2016
| Repeatable for credit

## MATH 263C: Topics in Representation Theory

May be repeated for credit.

Terms: Spr
| Units: 3
| Repeatable for credit

Instructors:
Bump, D. (PI)

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