## MATH 232: Topics in Probability: Percolation Theory

An introduction to some of the most important theorems and open problems in percolation theory. Topics include some of the difficult early breakthroughs of Kesten, Menshikov, Aizenman and others, and recent fields-medal winning works of Schramm, Lawler, Werner and Smirnov. Prerequisites: graduate-level probability. Offered every 1-2 years.

| Repeatable for credit

## MATH 234: Large Deviations Theory (STATS 374)

Combinatorial estimates and the method of types. Large deviation probabilities for partial sums and for empirical distributions, Cramer's and Sanov's theorems and their Markov extensions. Applications in statistics, information theory, and statistical mechanics. Prerequisite:
MATH 230A or
STATS 310. Offered every 2-3 years.

## MATH 237: Default and Systemic Risk

Introduction to mathematical models of complex static and dynamic stochastic systems that undergo sudden regime change in response to small changes in parameters. Examples from materials science (phase transitions), power grid models, financial and banking systems. Special emphasis on mean field models and their large deviations, including computational issues. Dynamic network models of financial systems and their stability.

Instructors:
Papanicolaou, G. (PI)

## MATH 239: Computation and Simulation in Finance

Monte Carlo, finite difference, tree, and transform methods for the numerical solution of partial differential equations in finance. Emphasis is on derivative security pricing. Prerequisite: 238 or equivalent.

Instructors:
Menz, G. (PI)

## MATH 243: Functions of Several Complex Variables

| Repeatable for credit

## MATH 244: Riemann Surfaces

Compact Riemann surfaces and algebraic curves; cohomology of sheaves; Serre duality; Riemann-Roch theorem and application; Jacobians; Abel's theorem. May be repeated for credit.

| Repeatable for credit

## MATH 245A: Topics in Algebraic Geometry: Moduli Theory

Topics in the study of moduli spaces: Basic of algebraic surfaces, Hodge structure of surfaces, moduli of K3 surfaces, cycles and rational curves in K3 surfaces, Torelli for K3 surfaces.

| Repeatable for credit

## MATH 245B: Topics in Algebraic Geometry

May be repeated for credit.

| Repeatable for credit

Instructors:
Vakil, R. (PI)

## MATH 247: Topics in Group Theory

Topics include the Burnside basis theorem, classification of p-groups, regular and powerful groups, Sylow theorems, the Frattini argument, nilpotent groups, solvable groups, theorems of P. Hall, group cohomology, and the Schur-Zassenhaus theorem. The classical groups and introduction to the classification of finite simple groups and its applications. May be repeated for credit.

| Repeatable for credit

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