## 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)

## 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 237A: Topics in Financial Math: Market microstructure and trading algorithms

Introduction to market microstructure theory, including optimal limit order and market trading models. Random matrix theory covariance models and their application to portfolio theory. Statistical arbitrage algorithms.

Terms: Spr
| Units: 3
| Repeatable for credit

Instructors:
Papanicolaou, G. (PI)

## 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.

Last offered: Spring 2016

## 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: Autumn 2017
| Repeatable for credit

## MATH 245A: Topics in Algebraic Geometry

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

Terms: Aut
| Units: 3
| Repeatable for credit

Instructors:
Vakil, R. (PI)

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