The talk will be based on work done by Furstenberg, taken mainly from his paper "Randon Walks and Discrete Subgroups of Lie Groups". We will present the idea of a boundary attached to a random walk on a group, and explain intuitively how it can be applied to prove that SL2(R) and SLn(R) - for n greater than 2 - do not have isomorphic lattices. Then we focus on a key step in that proof: Constructing a random walk on a lattice in SLn(R) that has the same boundary as a "spherical" random walk on SLn(R) itself.
Abstract: Cut and project point sets are defined by identifying a strip of a fixed n-dimensional lattice (the "cut"), and projecting the lattice points in that strip to a d-dimensional subspace (the "project"). Such sets have a rich history in the study of mathematical models of quasicrystals, and include well known examples such as the Fibonacci chain and vertex sets of Penrose tilings.
This will be a research talk. The abstract is below:
A group G is stable in permutations if every almost-action of G on a finite set is close to some actual action. Part of the interest in this notion comes from the observation that a non-residually finite stable group cannot be sofic.
I will show that surface groups are stable in a flexible sense, that is if one is allowed to "add a few extra points" to the action. This is the first non-trivial stability result for a non-amenable group.
Abstract: In this talk we will discuss some recent work by Guihéneuf and Lefeuvre who prove that shadowing is generic for Lebesgue measure preserving transformations of the 2-torus. We will spend most of our time motivating the problem, discussing the history of such questions- specifically touching upon earlier work of Oxtoby, Ulam, Alpern and Prasad and some general techniques used in the area. Time permitting, we will discuss the recent proof given by Guihéneuf and Lefeuvre.