Algebraic statistics and contingency tables

In this talk I will give an overview of the role of algebraic statistics in the statistical analysis of contingency tables. I will survey major areas in which algebraic methods proved to be crucial and provided a fertile ground for novel research directions: computation of sharp integer bounds for cell entries, existence of maximum likelihood estimates, simulation from probability distributions on spaces of tables, Markov bases, high-dimensional sparse tables with structural zeros, log-linear model selection. I will give examples that illustrate this methodology and talk about open problems.