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 kcolorings 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 wellstudied 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 threedimensional 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:
Tsystems 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, WronskianCasoratian duality in ODE, gauge/string theories, etc. Periodicity
of certain Tsystems 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 Tsystems,
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 Tsystems 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
KazhdanLusztig theory for dihedral groups, the other two classifications are new.
This is joint work with Pavel Galashin.

