Join us for Math Club Game Night! Enjoy an evening of Uno, Chess, card games, and of course, free pizza! Whether you're a math whiz or just looking for fun, bring your game face and join the excitement!

Vasily Dolgushev, Temple University

GT-shadows are morphisms of a groupoid GTSh who objects are finite index $B_3$-invariant subgroups of the free group on two generators. They may be thought of as approximations of elements of the Grothendieck-Teichmueller group GT. After a brief reminder of the groupoid GTSh, I will introduce the dihedral poset as a concrete subposet of the poset of objects of GTSh. For every element of the dihedral poset, we will describe its connected component in the groupoid GTSh. We will use connected components of certain objects of the dihedral poset to produce the first examples of finite non-abelian quotients of GT. My talk is based on the joint paper with Ivan Bortnovskyi, Borys Holikov and Vadym Pashkovskyi.

I will discuss joint work with Lafont, Minemyer, Sorcar, and Wells that provides an iterative procedure for constructing hyperbolic right-angled Coxeter groups that virtually algebraically fiber (that is, have a homomorphism onto Z with finitely generated kernel). One novel aspect of this procedure is that it produces examples in every cohomological dimension n > 1. This procedure combines work of Jankiewicz, Norin, and Wise with a generalization of a construction due to Osajda involving a `simplicial thickening' process.

Ino Loukidou (University of Chicago)

If M is a hyperbolic 3-manifold fibering over the circle, the fundamental group of M acts faithfully by homeomorphisms on a circle (the circle at infinity of the universal cover of the fiber), preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures (eg taut foliations, quasigeodesic or pseudo-Anosov flows) are known to give rise to universal circles - i.e. a circle with a faithful pi_1(M) action preserving a pair of invariant laminations - and these play a key role in relating the dynamical structure to the geometry of M. In this talk we introduce the idea of *Zippers*, which give a new and direct way to construct universal circles, streamlining the known construction in some cases, and giving a host of new constructions in others. This is joint work with Danny Calegari.

Let w be a word in a free group. A few years ago, Magee and I, relying on a work of Calegari, discovered that the stable commutator length of w, which is a well-studied topological invariant, can also be defined in terms of certain Fourier coefficients of w-random unitary matrices. But there are very natural ways to tweak the random-matrix side of this story: one may consider, for example, w-random permutations or w-random orthogonal matrices, and apply the same definition to obtain other "stable" invariants of w. Are these invariants interesting? Do they have, too, alternative topological/combinatorial definitions? In a joint work with Yotam Shomroni, we present concrete conjectures and begin to answer some of them. No background is assumed — I will define all notions.

Abstract: Symmetries of a group $G$ are encoded in the automorphism group $Aut(G)$. "Hidden symmetries" are encoded in the abstract commensurator $Comm(G)$. While many classes of finitely generated groups have reasonably well-understood commensurator -- for example, when $G$ is an arithmetic group, $Comm(G)$ is typically a group of matrices with rational entries -- the abstract commensurator of a free group, $Comm(F_2)$, is still somewhat mysterious. I will explain how Edgar A. Bering IV and I fleshed out a topological perspective of commensurations that allowed us to show that every countable locally finite group is a subgroup of $Comm(F_2)$.

Abstract: In this talk, I will describe a new geometric and combinatorial structure for the space of complex polynomials with a fixed number of roots. In particular, I will define a metric on the space of monic polynomials with d distinct centered roots, and I will introduce a finite cell structure for the metric completion. Each cell in this complex is a product of two Euclidean simplices, and the combinatorial structure comes from the dual presentation for the d-strand braid group. In particular, this provides a concrete connection between two classifying spaces for the braid group. This is joint work with Jon McCammond.

Abstract: We discuss what a "typical" short curve on a random large genus hyperbolic surface looks like. In particular, for each $L$, there are finitely many curves of length at most $L$. We find length scales at which such a curve chosen at random is highly likely to be non-simple, or fill the whole surface. It is known that, with respect to many commonly studied random models, a typical surface will be expander. That is, it will be "highly connected," in the sense that we get effective mixing of the geodesic flow. We will give results that hold for all expander surfaces, and hence for random surfaces with respect to many different random models. This is joint work with Ben Dozier.