## 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.

Last offered: Winter 2013
| Repeatable for credit

## MATH 233: Topics in Combinatorics: Non-constructive methods in combinatorics

Methods in combinatorics that prove the existence of certain objects without constructing them explicitly: The probabilistic method (concentration of measure, Lovasz local lemma), topological methods (Sperner's lemma, Brouwer's fixed-point and Borsuk-Ulam theorems), and algebraic methods (Nullstellensatz, the polynomial method and interlacing polynomials). We will also discuss the computational question of constructing the respective objects efficiently.

Terms: Spr
| Units: 3
| Repeatable for credit

Instructors:
Vondrak, J. (PI)

## 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.

Last offered: Winter 2013

## MATH 235A: Topics in combinatorics

This advanced course in extremal combinatorics covers several major themes in the area. These include extremal combinatorics and Ramsey theory, the graph regularity method, and algebraic methods.

Terms: Aut
| Units: 3
| Repeatable for credit

Instructors:
Fox, J. (PI)

## MATH 235B: Modern Markov Chain Theory

This is a graduate-level course on the use and analysis of Markov chains. Emphasis is placed on explicit rates of convergence for chains used in applications to physics, biology, and statistics. Topics covered: basic constructions (metropolis, Gibbs sampler, data augmentation, hybrid Monte Carlo); spectral techniques (explicit diagonalization, Poincaré, and Cheeger bounds); functional inequalities (Nash, Sobolev, Log Sobolev); probabilistic techniques (coupling, stationary times, Harris recurrence). A variety of card shuffling processes will be studies. Central Limit and concentration.

Terms: Win
| Units: 3
| Repeatable for credit

Instructors:
Diaconis, P. (PI)

## MATH 235C: Topics in Markov Chains

Classical functional inequalities (Nash, Faber-Krahn, log-Sobolev inequalities), comparison of Dirichlet forms. Random walks and isoperimetry of amenable groups (with a focus on solvable groups). Entropy, harmonic functions, and Poisson boundary (following Kaimanovich-Vershik theory).

Terms: Spr
| Units: 3
| Repeatable for credit

Instructors:
Zheng, T. (PI)

## MATH 236: Introduction to Stochastic Differential Equations

Brownian motion, stochastic integrals, and diffusions as solutions of stochastic differential equations. Functionals of diffusions and their connection with partial differential equations. Random walk approximation of diffusions. Prerequisite: 136 or equivalent and differential equations.

Terms: Win
| Units: 3

Instructors:
Papanicolaou, G. (PI)

## 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.

Last offered: Spring 2015

## MATH 238: Mathematical Finance (STATS 250)

Stochastic models of financial markets. Forward and futures contracts. European options and equivalent martingale measures. Hedging strategies and management of risk. Term structure models and interest rate derivatives. Optimal stopping and American options. Corequisites:
MATH 236 and 227 or equivalent.

Terms: Win
| Units: 3

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.

Terms: Spr
| Units: 3

Instructors:
Gu, Y. (PI)

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