## MATH 232: Topics in Probability: Percolation Theory

An introduction to first passage percolation and related general tools and models. Topics include early results on shape theorems and fluctuations, more modern development using hyper-contractivity, recent breakthrough regarding scaling exponents, and providing exposure to some fundamental long-standing open problems. Course prerequisite: graduate-level probability.

Terms: Aut
| Units: 3
| Repeatable for credit

Instructors:
Basu, R. (PI)

## MATH 233A: Topics in Combinatorics

| Repeatable for credit

## MATH 233B: Topics in Combinatorics: Polyhedral Techniques in Optimization

LP duality and min-max formulas; matchings, spanning trees, matroids, matroid union and intersection; packing of trees and arborescences; submodular functions, continuous extensions and optimization.

Terms: Win
| Units: 3
| Repeatable for credit

Instructors:
Vondrak, J. (PI)

## MATH 233C: Topics in Combinatorics

Terms: Spr
| Units: 3
| Repeatable for credit

Instructors:
Fox, 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.
http://statweb.stanford.edu/~adembo/large-deviations/

Terms: Spr
| Units: 3

Instructors:
Dembo, A. (PI)

## 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.

Last offered: Autumn 2015
| Repeatable for credit

## 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.

Last offered: Winter 2016
| Repeatable for credit

## 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).

Last offered: Spring 2016
| Repeatable for credit

## 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)
;
Chen, L. (TA)

## 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

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