Sep 7

Ludwik Dabrowski (SISSA): The weak HilbertSmith conjecture from the BorsukUlam type conjecture
Abstract: We show that a conjecture of Ageev follows from the BorsukUlamtype
conjecture of Baum, Dabrowski and Hajac. Then we explain how the Ageev
conjecture implies the weak version of the HilbertSmith conjecture which
states that no infinite compact zerodimensional group can act freely on a
manifold so that the orbit space is finite dimensional. The HilbertSmith
conjecture originates from the already settled Hilbert's fifth problem
concerning a characterization of Lie groups.

Sep 14

Camille Horbez (CNRS/Fields Institute): Growth under automorphisms of hyperbolic groups
Abstract: Let $G$ be a finitely generated group, let $S$ be a finite generating set of $G$, and let $f$ be an automorphism of $G$. A natural question is the following: what are the possible asymptotic behaviors for the length of $f^n(g)$, written as a word in the generating set $S$, as $n$ goes to infinity, and as $g$ varies in the group $G$?
We investigate this question in the case where $G$ is a torsionfree Gromov hyperbolic group. Growth was completely described by Thurston when $G$ is the fundamental group of a hyperbolic surface, and can be understood from BestvinaHandel’s work on traintracks when $G$ is a free group. We address the case of a general torsionfree hyperbolic group. We show in particular that every element $g$ has a welldefined exponential growth rate under iteration of $f$, and that only finitely many exponential growth rates arise as $g$ varies in $G$.
This is a joint work with Rémi Coulon, Arnaud Hilion and Gilbert Levitt.

Oct 5

Kiyoshi Igusa (Brandeis): Equivariant Hatcher construction
Abstract: This is a joint project with Tom Goodwillie extending earlier joint work with Goodwillie and Ohrt. The purpose of this project is to construct all exotic smooth structures on all smooth manifold bundles with a fiberwise group action in a stable range of dimensions. I will start by defining and enumerating the exotic smooth structures. By ``enumerate'' I mean compute the dimension of the vector space of exotic structures. To do this, we use a simplified version of Mackey functors.
We give a simple construction of these exotic structures using the irreducible real representations of a finite group $G$. We call it the ``equivariant Hatcher construction'' since it generalizes a classical construction due to Hatcher. We use higher Reidemeister torsion with coefficients in a Mackey functor to demonstrate that our construction (with all possible inputs) spans the vector space of all stable exotic structures on a fixed $G$bundle.

Oct 12

Mahan Mj (Tata Institute/Fields Institute): BowenMargulis measures and Extremal Cocycle Growth
Abstract: We establish a connection between extreme values of stable random fields arising in probability and groups $G$ acting geometrically on CAT(1) spaces $X$. The connection is mediated by the action of the group on its limit set equipped with the PattersonSullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth and show that its nonvanishing is equivalent to finiteness of the BowenMargulis measure for the associated unit tangent bundle $U(X/G)$ provided $X$ is not a tree whose edges are (up to scale) integers. We also establish an analogous statement for normal subgroups of free groups. This is joint work with Parthanil Roy.

Oct 19

Jing Tao (University of Oklahoma/Fields Institute): Big Torelli groups
Abstract: A surface $S$ is of finitetype if its fundamental group is finitely generated; otherwise, it is of infinite type. The mapping class group MCG($S$) of $S$ is the group of isotopy classes of orientationpreserving homeomorphisms of $S$. This is a wellstudied group when $S$ has finite type, but big mapping class groups, i.e. MCG($S$) of infinitetype surfaces, remain quite mysterious. But big mapping class groups arise naturally in various areas of mathematics and recently there has been a surge of interests in studying them. In this talk, I will discuss some recent results about the Torelli subgroup of MCG($S$). This is joint with Aramayona, Ghaswala, Kent, McLeay, and Winarski.

Nov 16

Bulent Tosun (Alabama): Contact surgeries, symplectic fillings and Lagrangian discs
Abstract: It is well known that all contact 3manifolds can be obtained from the standard contact structure on the 3sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, we will discuss some new results about positive contact surgeries and in particular completely characterize when contact ($r$)surgery is symplectically/Stein fillable for $r\in(0,1]$. This is joint work with James Conway and John Etnyre.

Nov 30

Jacob Russell (CUNY): Convexity in Hierarchically Hyperbolic Spaces
Abstract: Convexity is a fundamental notion across a variety of flavors of geometry. In the study of the course geometry of metric spaces, it is natural to study quasiconvexity i.e. convexity with respect to quasigeodesics. We study quasiconvexity in the class of hierarchically hyperbolic spaces; a generalization of Gromov hyperbolic spaces which contains the mapping class group, rightangled Artin and Coxeter groups, and many 3manifold groups. Inspired by the rich theory of quasiconvexity in hyperbolic spaces, we show that quasiconvex subsets of hierarchcially hyperbolic spaces mimic the behavior of quasiconvex subsets in hyperbolic spaces.
