Speaker: Steve Cantrell
Title: Rigidity of geometries: how to use dynamics
Abstract: We'll discuss how to use tools from dynamical systems and ergodic theory to study rigidity problems in geometry.
Speaker: Ujan Chakraborty
Title: Ergodic Operator Inequalities for Unimodular Amenable Groups through Random Walks
Abstract: Existing proofs of pointwise ergodic theorems for amenable groups are quite involved. This is principally due to the difficulty in obtaining the maximal inequality. Assuming unimodularity, we provide a new and transparent proof of these theorems using random walks that deals with the classical and noncommutative cases on an equal footing. We show that one may dominate the average on the group by that on the integers (a Markov operator). Maximal inequalities and pointwise convergence follow as corollaries.
Speaker: Xin Li
Title: An introduction to topological full groups
Abstract: This talk is about groups of dynamical origin which are called topological full groups. Roughly speaking, these groups describe symmetries of orbit structures arising from topological dynamics. The construction of topological full groups has recently attracted attention because it has led to new examples of infinite simple groups with interesting properties, answering several outstanding open questions in group theory.
My first goal is to give an introduction to topological full groups from the perspective of topological dynamics. My second goal is to explain connections to topics such as topological groupoids, operator algebras, and K-theory, which are areas I am currently working on.
Speaker: Giorgio Mangioni
Title: Random quotients preserve non-positive curvature
Abstract: A random walk on a finitely generated group G is an infinite sequence of group elements, each of which differs from the previous by a random generator. One can then define a random quotient by modding out the n-th steps of a collection of independent random walks, and study the behaviour of such quotient as the step n goes to infinity. In this talk we illustrate how several notions of "non-positive curvature" for groups survive under random quotients. As a case study, we show that random quotients of hyperbolic groups are again hyperbolic, providing a new proof of a theorem of Delzant (1996). This is joint work with Carolyn Abbott, Daniel Berlyne, Thomas Ng, and Alex Rasmussen.
Speaker: Saeed Shaabanian
Title: Covering time for Countable Markov Shifts
Abstract: Suppose we have a countable Markov shift equipped with a Gibbs measure. Since the orbit of almost every point in this system is dense, a natural question arises regarding the rate at which these orbits become dense. In this talk, we aim to provide an estimate of this rate, which, as it turns out, is related to the Minkowski dimension of the system.
Speaker: Alessandro Sisto
Title: Markov chains and large-scale geometry
Abstract: Random walks on groups have been studied extensively for several decades, but from the perspective of large-scale geometry it is also natural to study more general Markov chains. Indeed, random walks cannot be pushed-forward via so-called quasi-isometries, a very natural class of maps in geometric group theory, while more general Markov chains can. I will mostly discuss the general motivation, and time-permitting get to recent results.
Speaker: Mike Todd
Title: Almost sure orbits closeness
Abstract: Given a dynamical system $(X, f)$ with initial points $x, y\in X$ and a sequence $(r_n)_n$, we consider the set $E_n$ where there is some pair $f^i(x), f^j(y)$ with $1\leq i, j\le n$ such that the distance between these points is less than $r_n$. If $(r_n)_n$ shrinks sufficiently slowly, almost every pair $x, y$ will meet this condition infinitely often ($(\mu\times \mu)(\liminf_n E_n)=1$) or even have a pair which always satisfies this for all large enough $n$ ($(\mu\times \mu)(\limsup_n E_n)=1$). On the other hand, if $(r_n)_n$ shrinks too fast then the measures of these sets is zero.
In the case of the doubling map with Lebesgue we prove a dichotomy on the measure of $\limsup_n E_n$ being 0 or 1, depending on the summability of $n r_n$ or not. For more general systems, we get close to such a dichotomy, depending on the properties of the system. I will outline these and related results, joint work with Kirsebom, Kunde and Persson.
Speaker: Mike Whittaker
Title: Space filling curves, monotiles, and the half-hex
Abstract: I aim to relate the three notions in the title by studying a space filling curve through the half-hex substitution. We'll see that we naturally obtain the R1-rule of the Taylor-Socolar tile along with another tile that Jamie Walton and I studied. I also plan to mention a new direction.
