MATH 235B: Modern Markov Chain Theory
This is a graduatelevel 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: not given this year, last offered Winter 2016

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
MATH 235C: Topics in Markov Chains
Classical functional inequalities (Nash, FaberKrahn, logSobolev 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 KaimanovichVershik theory).
Terms: not given this year, last offered Spring 2016

Units: 3

Repeatable for credit

Grading: Letter or Credit/No 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

Grading: Letter or Credit/No Credit
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.
Terms: not given this year, last offered Spring 2015

Units: 3

Grading: Letter or Credit/No Credit
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

Grading: Letter or Credit/No 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

Grading: Letter or Credit/No Credit
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: not given this year, last offered Spring 2016

Units: 3

Grading: Letter or Credit/No Credit
MATH 243: Functions of Several Complex Variables
Holomorphic functions in several variables, Hartogs phenomenon, dbar complex, Cousin problem. Domains of holomorphy. Plurisubharmonic functions and pseudoconvexity. Stein manifolds. Coherent sheaves, Cartan Theorems A&B. Levi problem and its solution. Grauert¿s Oka principle. nPrerequisites:
MATH 215A and experience with manifolds.
Terms: not given this year, last offered Winter 2011

Units: 3

Repeatable for credit

Grading: Letter or Credit/No Credit
MATH 244: Riemann Surfaces
Riemann surfaces and holomorphic maps, algebraic curves, maps to projective spaces. Calculus on Riemann surfaces. Elliptic functions and integrals. RiemannHurwitz formula. RiemannRoch theorem, AbelJacobi 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.
Terms: not given this year, last offered Autumn 2017

Units: 3

Repeatable for credit

Grading: Letter or Credit/No 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

Grading: Letter or Credit/No Credit
Instructors:
Vakil, R. (PI)
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