printer friendly pageMATH 231: Mathematics and Statistics of Gambling (STATS 334)
Probability and statistics are founded on the study of games of chance. Nowadays, gambling (in casinos, sports and the Internet) is a huge business. This course addresses practical and theoretical aspects. Topics covered: mathematics of basic random phenomena (physics of coin tossing and roulette, analysis of various methods of shuffling cards), odds in popular games, card counting, optimal tournament play, practical problems of random number generation. Prerequisites: Statistics 116 and 200.
Terms: Spr
| Units: 3
MATH 231A: An Introduction to Random Matrix Theory (STATS 351A)
Patterns in the eigenvalue distribution of typical large matrices, which also show up in physics (energy distribution in scattering experiments), combinatorics (length of longest increasing subsequence), first passage percolation and number theory (zeros of the zeta function). Classical compact ensembles (random orthogonal matrices). The tools of determinental point processes.
Last offered: Autumn 2008
MATH 231C: Free Probability
Background from operator theory, addition and multiplication theorems for operators, spectral properties of infinite-dimensional operators, the free additive and multiplicative convolutions of probability measures and their classical counterparts, asymptotic freeness of large random matrices, and free entropy and free dimension. Prerequisite:
STATS 310B or equivalent.
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.
Last offered: Autumn 2016
| Repeatable
for credit
MATH 233A: Topics in Combinatorics
Geometry of polynomials and non-constructive proofs in combinatorics: The independence polynomial, the Lovasz Local Lemma and Shearer's Lemma. Real-rooted polynomials, stable polynomials, Ramanujan graphs and the Kadison-Singer problem. Strongly Rayleigh measures and negative dependence. Applications in algorithms.
Terms: Win
| Units: 3
| Repeatable
for credit
Instructors:
Vondrak, J. (PI)
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.
Last offered: Winter 2017
| Repeatable
for credit
MATH 233C: Topics in Combinatorics
Last offered: Spring 2017
| 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.
http://statweb.stanford.edu/~adembo/large-deviations/
Last offered: Spring 2017
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
(up to 99 units total)
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
(up to 99 units total)
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