Commutative Algebra and Algebraic Geometry
Organizer: David Eisenbud
Date: Tuesday, November 20
3:45: Seth Sullivant, North Carolina State U: Algebraic Statistics
Algebraic statistics advocates polynomial algebra as a tool for addressing problems in statistics and its applications. This connection is based on the fact that most statistical models are defined either parametrically or implicitly via polynomial equations. The idea is summarized by the phrase "Statistical models are semialgebraic sets". I will try to illustrate this idea with two examples, the first coming from the analysis of contingency tables, and the second arising in computational biology. I will keep the algebraic and statistical prerequisites to a minimum and keep the talk accessible to a broad audience.
5:00: Bernd Sturmfels, UC Berkeley: Commutative Algebra of Statistical Ranking
A model for statistical ranking is a family of probability distributions whose states are orderings of a fixed finite set of items. We represent the orderings as maximal chains in a graded poset. The most widely used ranking models are parameterized by rational function in the model parameters, so they define algebraic varieties. We study these varieties from the perspective of combinatorial commutative algebra. One of our models, the Plackett-Luce model, is non-toric. Five others are toric: the Birkhoff model, the ascending model, the Csiszar model, the inversion model, and the Bradley-Terry model. For these models we examine the toric algebra, its lattice polytope, and its Markov basis. This is joint work with Volkmar Welker.