David Cohen - On strongly aperiodic subshifts of finite type.

A subshift of finite type on a group G is a type of dynamical system which generalizes the idea of a regular language. We discuss how the geometry of G constrains the dynamical properties of subshifts of finite type on G.

Alireza Salehi Golsefidy - Local spectral gap
Let G be a nice topologolical group and Gamma be a dense subgroup of G. One can ask: how dense is Gamma? For instance, when G is compact, we get the best understanding answer if the action of Gamma on G (by left multiplication) has spectral gap. In a joint work with Boutonnet and Ioana, we define the notion of local spectral gap which in many ways plays the role of spectral gap for locally compact groups.

I will talk about the new results on the existence of spectral gap and local spectral gap. And then some of their applications will be mentioned, e.g. orbit equivalence rigidity, Banach-Ruziewicz problem, largeness of certain Galois representations.

Asaf Hadari - The image of mapping classes under homological representations
Mapping class groups have a very rich representation theory. The most well understood family of representations of these groups is the collection of homological representations. A great deal of work has been put into understanding these representations, but many basic questions remain open. We will address one such question by showing that an infinite order element of a mapping class group of a punctured surface has infinite order in the image of some homological representation.

Logan Hoehn - A complete classification of homogeneous plane continua
Coauthor: Lex G. Oversteegen
A space X is homogeneous if for every pair of points in $X$, there is a homeomorphism of $X$ onto itself taking one point to the other. Kuratowski and Knaster asked in 1920 whether the circle is the only homogeneous compact connected space (continuum) in the plane $\mathbb{R}^2$. Explorations of this problem fueled a significant amount of research in continuum theory, and among other things, led to the discovery of two new homogeneous continua in the plane: the pseudo-arc and the circle of pseudo-arcs. I will describe our recent result which implies that there are no more undiscovered homogeneous compact spaces in the plane.

Camille Horbez - Growth under random products of automorphisms of a free group
Given a nontrivial conjugacy class g in a free group F_N, what can be said about the typical growth of g under application of a random product of automorphisms of F_N? I will present a law of large numbers, a central limit theorem and a spectral theorem in this context. Similar results also hold for the growth of simple closed curves on a closed hyperbolic surface, under application of a random product of mapping classes of the surface. This is partly joint work with François Dahmani.

Priyam Patel - Separability Properties of Right-Angled Artin Groups
Right-Angled Artin groups (RAAGs) and their separability properties played an important role in the recent resolutions of some outstanding conjectures in low-dimensional topology and geometry. We begin this talk by defining two separability properties of RAAGs, residual finiteness and subgroup separability, and provide a topological reformulation of each. We then discuss joint work with K. Bou-Rabee and M.F. Hagen regarding quantifications of these properties for RAAGs and the implications of our results for the class of virtually special groups.

Felipe Ramirez - Shrinking targets on fractals
In a "shrinking targets problem'' we have some dynamical system on a space, and a nested sequence of subsets of that space, and we study the set of points of the space whose orbits under the dynamics enter these sets---the "targets''---infinitely often. This idea is only a simple generalization of the basic notion of recurrence, but it arises naturally in many types of problems, especially those connected to Diophantine approximation. I will discuss recent work with Henna Koivusalo, where we investigate a shrinking targets problem on self-affine fractals. We calculate the Hausdorff dimension of the set of points that recur to shrinking targets on our systems.

Anush Tserunyan - Probability groups
I will introduce a class of groups equipped with an invariant probability measure that respects the group structure in an appropriate sense; call such groups probability groups. This class contains all compact groups and is closed under taking ultraproducts with the induced Loeb measure. I will discuss the use of probability groups as a potential alternative to Furstenberg's correspondence principle. As an example, I will define a notion of mixing for probability groups and mention a double recurrence result for mixing probability groups that generalizes a theorem of Bergelson-Tao proved for ultra quasirandom groups, nevertheless having a considerably shorter proof.

Dave Witte-Morris - Arithmetic subgroups whose representations all map into GL(n,Z)
Suppose Gamma is an arithmetic subgroup of a semisimple Lie group G. For any homomorphism rho from G to GL(n,R), a classical paper of J.Tits determines whether rho(Gamma) is conjugate to a subgroup of GL(n,Z). Combining this with a well-known surjectivity result in Galois cohomology provides a short proof of the known fact that every G has an arithmetic subgroup Gamma, such that the containment is true for EVERY homomorphism rho. We will not assume the audience is acquainted with arithmetic groups, Galois cohomology, or the theorem of Tits.