## 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: Autumn 2017
| Repeatable
for credit

## MATH 245A: Topics in Algebraic Geometry

Topics of contemporary interest in algebraic geometry. May be repeated for credit.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.

Last offered: Autumn 2019
| Repeatable
for credit

## MATH 245B: Topics in Algebraic Geometry

May be repeated for credit.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.

Last offered: Winter 2020
| Repeatable
for credit

## MATH 245C: Topics in Algebraic Geometry

May be repeated for credit.

Terms: Spr
| Units: 3
| Repeatable
for credit

Instructors:
Vakil, R. (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.

Last offered: Spring 2016
| Repeatable
for credit

## 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.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.

Terms: Aut
| Units: 3
| Repeatable
for credit

Instructors:
Soundararajan, K. (PI)

## MATH 249B: Topics in Number Theory

NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.

Terms: Win
| Units: 3
| Repeatable
for credit

Instructors:
Taylor, R. (PI)

## MATH 249C: Topics in Number Theory

NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.

Last offered: Spring 2020
| 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.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.

Terms: Spr
| Units: 3

Instructors:
Luk, J. (PI)

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