## MATH 243: Functions of Several Complex Variables

Holomorphic functions in several variables, Hartogs phenomenon, d-bar complex, Cousin problem. Domains of holomorphy. Plurisubharmonic functions and pseudo-convexity. Stein manifolds. Coherent sheaves, Cartan Theorems A&B. Levi problem and its solution. Grauert¿s Oka principle. nPrerequisites:
MATH 215A and experience with manifolds.

Last offered: Winter 2011
| Repeatable
for credit

## MATH 244: Riemann Surfaces

Riemann surfaces and holomorphic maps, algebraic curves, maps to projective spaces. Calculus on Riemann surfaces. Elliptic functions and integrals. Riemann-Hurwitz formula. Riemann-Roch theorem, Abel-Jacobi map. Uniformization theorem. Hyperbolic surfaces. (Suitable for advanced undergraduates.) Prerequisites:
MATH 106 or
MATH 116, and familiarity with surfaces equivalent to
MATH 143,
MATH 146, or
MATH 147.

Last offered: Spring 2009
| Repeatable
for credit

## MATH 245A: Topics in Algebraic Geometry

Topics of contemporary interest in algebraic geometry. May be repeated for credit.

| Repeatable
3 times
(up to 9 units total)

## MATH 245B: Topics in Algebraic Geometry

May be repeated for credit.

Terms: Win
| Units: 3
| Repeatable
3 times
(up to 9 units total)

Instructors:
Li, J. (PI)

## MATH 245C: Topics in Algebraic Geometry

May be repeated for credit.

Terms: Spr
| Units: 3
| Repeatable
for credit

Instructors:
Venkatesh, A. (PI)

## MATH 246: Topics in number theory: L-functions

The Riemann Zeta function and Dirichlet L-functions, zero-free regions and vertical distribution of the zeros, primes in arithmetic progressions, the class number problem, Hecke L-functions and Tate's thesis, Artin L-functions and the Chebotarev density theorem, Modular forms and Maass forms.nnPrerequisites: Algebraic Number Theory.

Terms: Spr
| Units: 3
| Repeatable
for credit

Instructors:
Entin, A. (PI)

## 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).

Last offered: Autumn 2014
| Repeatable
for credit

## MATH 249A: Topics in number theory

Topics of contemporary interest in number theory. May be repeated for credit.

Last offered: Autumn 2013
| Repeatable
3 times
(up to 9 units total)

## MATH 249B: Topics in Number Theory

Terms: Win
| Units: 3
| Repeatable
3 times
(up to 9 units total)

Instructors:
Conrad, B. (PI)

## MATH 249C: Topics in Number Theory

Terms: Spr
| Units: 3
| Repeatable
for credit

Instructors:
Soundararajan, K. (PI)

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