Gonzalez D'Leon's abstract: Choe, Oxley, Sokal and Wagner proved that the support of a homogeneous, multiaffine polynomial with the half-plane property is the set of bases of a matroid. This gives a new way of representing matroids. It is a challenging problem to determine whether a matroid is representable in this way or not. We develop a method to decide representability of matroids by polynomials with the half-plane property. The method uses the Tutte group of a matroid. We are able to prove that no projective geometry is representable and that a binary matroid is representable if and only if it is regular.

Norén's abstract:
Algebraic statistics is a new and interesting field of mathematics that combines statistics algebra and discrete geometry. The goal of algebraic statistics is to answer statistical questions with algebraic methods. A large part of that is to find useful Markov bases and other types of generating sets for toric ideals associated to hierarchical log-linear models. A huge problem in algebraic statistics has been the large number of variables in the polynomial rings where the toric ideals live. We introduce the graph homomorphism ideals to reduce the number of variables in a natural way. A graph homomorphism ideal is what arise when we take the ideal corresponding to certain hierarchical models and remove all the variables not corresponding to graph homomorphisms into some graph. We establish some basic properties of these new ideals by methods similar to those used for the hierarchical models by Diaconis, Sturmfels, Sullivant, et al. Then we study the case where the graph homomorphisms correspond to independent sets. The ideals arising here are especially nice and we present a new method to find Gröbner bases of degree two for ideals corresponding to bipartite graphs and graphs that are bipartite if one vertex is deleted. We also do computer calculations and find Markov bases for all graphs with eight or fewer vertices.