## MATH 248: Introduction to Ergodic Theory

Topics may include 1) subadditive and multiplicative ergodic theorems, 2) notions of mixing, weak mixing, spectral theory, 3) metric and topological entropy of dynamical systems, 4) measures of maximal entropy. Prerequisites: Solid background in "Measure and Integration" (
Math 205A) and some functional analysis, including Riesz representation theorem and Hahn-Banach theorem (
Math 205B).

| Repeatable for credit

Instructors:
Mirzakhani, M. (PI)

## MATH 248A: Algebraic Number Theory

Structure theory and Galois theory of local and global fields, finiteness theorems for class numbers and units, adelic techniques. Prerequisites:
MATH 210A,B.

| Repeatable for credit

## MATH 249A: Topics in number theory

| Repeatable for credit

## 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.

## MATH 256B: Partial Differential Equations

Continuation of 256A.

| Repeatable for credit

Instructors:
Ryzhik, L. (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.

| Repeatable for credit

## MATH 257B: Symplectic Geometry and Topology

Continuation of 257A. May be repeated for credit.

| Repeatable for credit

## MATH 258: Topics in Geometric Analysis

May be repeated for credit.

| Repeatable for credit

Instructors:
Mazzeo, R. (PI)
;
White, B. (PI)

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