Graduate Computational Algebraic Geometry Seminar
Sonja Petrovic
UIC
Markov degrees of hierarchical models arising from Betti numbers of Stanley-Reisner ideals
Abstract: There are two seemingly unrelated classical objects associated to a
simplicial complex: a hierarchical model and a Stanley-Reisner ring.
A hierarchical model gives rise to a toric ideal, a relationship that
is a staple of algebraic statistics. The degrees of generators of this
ideal are dubbed "Markov degrees" and encode the complexity of the
model.
In turn, a Stanley-Reisner ideal is a monomial ideal whose algebraic
properties are encoded by the combinatorial properties of the complex.
Betti numbers encode ranks of free modules in a minimal free
resolution of the Stanley-Reisner ring, a central object in
commutative algebra.
In this talk, I will introduce all of these concepts, and present a
recent result which explores a first connection between Markov degrees
of the model and Betti numbers of the Stanley-Reisner ideal.
As an application of the main theorem, we recover a result of Froberg
which classifies simplicial complexes with linear resolutions.
This talk is based on joint work with Erik Stokes, preprint available
at arXiv:0910.1610v1
Thursday November 12, 2009 at 11:00 AM in SEO 612