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Abstract of Plenary Talks

Speaker: Matthias Beck

Title: Combinatorial Reciprocity Theorems

Abstract: A common theme of enumerative combinatorics is formed by counting functions that are polynomials. For example, one proves in any introductory graph theory course that the number of proper k-colorings of a given graph G is a polynomial in k, the chromatic polynomial of G. Combinatorics is abundant with polynomials that count something when evaluated at positive integers, and many of these polynomials have a (completely different) interpretation when evaluated at negative integers: these instances go by the name of combinatorial reciprocity theorems. For example, when we evaluate the chromatic polynomial of G at -1, we obtain (up to a sign) the number of acyclic orientations of G, that is, those orientations of G that do not contain a coherently oriented cycle. Combinatorial reciprocity theorems appear all over combinatorics. This talk will attempt to show some of the charm (and usefulness!) these theorems exhibit. Our goal is to weave a unifying thread through various combinatorial reciprocity theorems, by looking at them through the lens of geometry.

Speaker: Matthew Kahle

Title: Configuration spaces of disks in a strip

Abstract: This is work in progress with Bob MacPherson. We study configuration spaces of disks in an infinite strip. These spaces naturally generalize the well-studied configuration spaces of points in the plane, but giving the points thickness also has clear physical meaning: this is the energy landscape of a hard spheres gas. We are interested in the topology of these spaces, and we find qualitatively different ``regimes" of behavior: solid (where homology is trivial), liquid (where homology is unstable), and gas (where homology is stable). I will emphasize the combinatorial aspects of our methods in the talk.

Speaker: Caroline Klivans

Title: On the connectivity of three-dimensional tilings

Abstract: In this talk, I will discuss domino tilings of three dimensional manifolds. In particular, I will focus on the connected components of the space of tilings of such regions under local moves. Using topological techniques we introduce two parameters of tilings: the flux and the twist. Our main result characterizes when two tilings are connected by local moves in terms of these two parameters. (I will not assume any familiarity with the theory of tilings for the talk.)

Speaker: Kyungyong Lee

Title: The dimension of the frieze variety

Abstract: Conway and Coxeter introduced frieze patterns in 1973. These are arrays of positive integers satisfying a certain local rule. With the development of cluster algebras, more generalized frieze patterns have been defined and studied in the last decade. We define an algebraic variety associated to each frieze pattern, and show that the dimension of this variety is a new numerical invariant which exhibits the trichotomy: finite type, affine type, and wild type. This is based on a joint work with Matt Mills and Alexandra Seceleanu, and another joint work with Li Li and Ralf Schiffler.

Speaker: Isabella Novik

Title: Face numbers of centrally symmetric polytopes

Abstract: Many objects around us are symmetric. Yet, at present, we know much more about the face numbers of general (simplicial) polytopes than about those of centrally symmetric ones. This talk will survey several recent results as well as many remaining mysteries in the study of face numbers of centrally symmetric polytopes.

Speaker: Pavlo Pylyavskyy

Title: Zamolodchikov periodicity and integrability

Abstract: T-systems are certain discrete dynamical systems associated with quivers. They appear in several different contexts: quantum affine algebras and Yangians, commuting transfer matrices of vertex models, character theory of quantum groups, analytic Bethe ansatz, Wronskian-Casoratian duality in ODE, gauge/string theories, etc. Periodicity of certain T-systems was the main conjecture in the area until it was proven by Keller in 2013 using cluster categories. In this work we completely classify periodic T-systems, which turn out to consist of 5 infinite families and 4 exceptional cases, only one of the infinite families being known previously. We then proceed to classify T-systems that exhibit two forms of integrability: linearization and zero algebraic entropy. All three classifications rely on reduction of the problem to study of commuting Cartan matrices, either of finite or affine types. The finite type classification was obtained by Stembridge in his study of Kazhdan-Lusztig theory for dihedral groups, the other two classifications are new. This is joint work with Pavel Galashin.