Courses

Matthias Beck (San Francisco State University, USA and Freie Universität Berlin, Germany)

Teaching assistant: Max Hlavacek (University of California, Berkeley)

Classifying combinatorial polynomials

Abstract: A common theme of enumerative combinatorics (often inspired or helped by commutative algebra) is formed by counting functions that are polynomials. Mirroring Herb Wilf's much-cherished and still-wide-open question which polynomials are chromatic?, we attempt to (partially) classify families of polynomials that arise from certain combinatorial situations. For example, a (not very deep) necessariy condition for a polynomial to be chromatic is that it is monic and has zero constant term. (We will prove much stronger conditions in our lectures.)

Our methodology is rooted in discrete geometry. The Ehrhart quasipolynomial of a rational polytope P (i.e., the convex hull of finitely many points in Q^d) encodes fundamental arithmetic data of P, namely, the number of integer lattice points in positive integral dilates of P. Because a polytope is given by linear constraints, we can model several combinatorial situations (e.g., graph coloring) using polyhedral concepts, and then count via Ehrhart quasipolynomials. The classification of Ehrhart polynomials (which stem from the integer-point count of lattice polytopes) is open already for degree 3, but we will discuss several general necessary conditions for Ehrhart (quasi-)polynomials, as well as classifications of Ehrhart polynomials of special families of polytopes. We will give several examples of problems in enumerative combinatorics that can be partially solved via structural results for Ehrhart (quasi-)polynomials.

Martina Juhnke-Kubitzke (Universität Osnabrück, Germany) and Uwe Nagel (University of Kentucky, USA)

Teaching assistant: Aida Maraj (Max Planck Institute for Mathematics in the Sciences (Leipzig), Germany)

Symmetries in algebra and combinatorics in finite and infinite settings

Abstract: Symmetries are abundant in algebra and combinatorics and have been studied intensely in the last years. In these lectures, we want to investigate symmetries in infinite and finite situation. With respect to the finite situation, we will consider simplicial complexes (or, more generally, monomial ideals) that are fixed under the action of a certain group. As a special case, we will treat centrally symmetric simplicial complexes and survey recent results that are known for those, including bounds for the face numbers of centrally symmetric polytopes. For symmetric simplicial complexes that are invariant under the action of other groups much less is known, and it is already of interest to construct infinite families of such. A possible direction to pursue is the study of edge ideals of graphs with a large automorphism group. With respect to the infinite situation, we have in mind ideals in polynomial rings in infinitely many variables and simplicial complexes on infinite vertex sets. This corresponds to considering chains of increasing symmetric ideals or symmetric simplicial complexes. Here symmetry is meant with respect to a symmetric group. We will discuss some known results concerning the asymptotic behavior of several algebraic and homological invariants along such chains, including their Hilbert series, codimension,projective dimension, Castelnuovo-Mumford regularity and Betti tables. As those chains of ideals have only attracted attention in the last few years, there are a lot of open problems and we will present several of these. In order to make the lectures accessible to a broad audience, we will introduce and survey results for arbitrary ideals/simplicial complexes.

Mateusz Michałek (Universität Konstanz, Germany)

Teaching assistant: Tim Seynnaeve (Universität Bern, Switzerland)

Applications of enumerative algebraic geometry: complete quadrics in statistics

Abstract: Enumerative algebraic geometry is a very active area of research reaching from old classical questions, like: "How many lines intersect four general lines in a 3-space?", to modern applications in string theory. Our first aim of the lectures is to introduce basic tools to make such computations - this is known as Schubert calculus. We will learn how to translate an enumerative problem to an intersection problem on a variety (like a Grassmannian) and how to solve this problem using combinatorics (equivalently algebraic computations on symmetric polynomials, equivalently computations in the cohomology ring).

The final aim is to develop theory to address new, important challenges coming from algebraic statistics. The very general, motivating question is how to find a probability distribution that best explains the observed data. In special cases (related to multivariate Gaussian distributions) such a best distribution is one of the solutions to a system of equations. These equations define a fiber of a finite algebraic map. The degree of the map is the measure of the complexity of the statistical model, known as the maximum likelihood degree. To compute it we will use the intersection theory on the variety of complete quadrics. Various open problems will be presented relating to algebra and algebraic geometry.

The statistical part will be only sketched, with the focus on objects like vector bundles, Chow ring and symmetric functions.