It meets weekly online usually on Mondays at

Participants take turns presenting a paper, which hopefully admits a nice one hour presentation.

This seminar is flexible, so if another time is better or one wishes to give a pre-talk to make it more accessible that is great.

Talks from session 1 (with videos!)

Title:Equidistribution of geodesic flow pushes via exponential mixing.

Abstract: Pick a point at random in a finite volume hyperbolic surface and simultaneously flow in all directions from it. For the typical starting point these expanding circles will equidistribute and this talk will present a (more general) argument of Margulis establishing this fact.

Speaker: Jon Chaika

Slides Note corrections are in red

Video

Title: A recurrence lemma and its applications and extensions

Abstract: I will discuss a lemma which is usually attributed to Atkinson, is very useful and apparently has been rediscovered multiple times. The lemma says that if (X, B, mu, T) is an ergodic system and f is in L^1(mu) with mean zero, then the Birkhoff sums of f do not diverge almost surely. Moreover the sums switch signs in a wide sense, infinitely many times. I will give the proof and discuss some applications of this result. Then I will discuss higher dimensional analogues, where the situation is significantly more complicated.

Speaker: Barak Weiss

Video link

Title: Measure rigidity of Cartan actions

Abstract: We'll take an introductory peek into the measure rigidity program for higher-rank abelian actions by looking at the simplest case, Anosov Z^k actions on (k+1)-dimensional tori. The main structures and ideas appearing in the theory will be explained, as well as how the situation becomes more complicated under fewer assumptions.

Video link

Speaker: Kurt Vinhage

Notes

Title: An invitation to "Entropy in Dimension One"

Abstract: Which real numbers arise as the entropies of continuous, multimodal, postcritically finite self-maps of real intervals? This is the "one-dimensional" analogue of a more famous open question: which real numbers arise as the dilatations of pseudo-Anosov surface diffeomorphisms? In "Entropy in Dimension One," W. Thurston answers this one-dimensional version of the question. We'll discuss a small subset of the many beautiful ideas and questions in this paper.

Speaker: Kathryn Lindsey

Notes

Video

Title: What is a Foster-Lyapunov-Margulis function?

Abstract: We'll show how a simple idea from probability theory on the recurrence of random walks can be used in many important dynamical and geometric situations, building on work of Eskin-Margulis and others. No prior knowledge of probability theory, random walks, or geometry is required. If time permits, as an unrelated "dessert" of sorts, we'll give a brief proof of the Hopf ratio ergodic theorem using the Birkhoff ergodic theorem for flows.

Video

Notes

Links mentioned: Meyn-Tweedie's Markov chains and stochastic stability, Eskin-Margulis' Recurrence Properties of Random Walks on Finite Volume Homogeneous Manifolds, Eskin-Margulis-Mozes' Upper Bounds and Asymptotics in a Quantitative Version of the Oppenheim Conjecture, Eskin-Mirzakhani-Rafi's Counting closed geodesics in strata, Athreya's Quantitative Recurrence and Large Deviations for Teichmuller Geodesic Flow and Kleinbock-Mirzadeh's Dimension estimates for the set of points with non-dense orbit in homogeneous spaces

Speaker: Jayadev Athreya

Title: The Smooth Realization Problem and Conjugation by Approximation

Abstract: Going back to the foundation work of von Neuman there is a question of whether there are smooth models of the models of classical ergodic theory. When both measure and map are required to be smooth there is only one known obstruction but essentially no general results. Within the class of zero entropy transformation we have a method called conjugation by approximation that can be used to realize many interesting properties. I will describe the method and some of the classical and modern consequences of this.

Video

Slides

Speaker: Alistair Windsor

Title: Accessibility for partially hyperbolic systems

Abstract: Accessibility is a fundamental tool when working with partially hyperbolic systems. For instance, in the 1970s it was used as a tool to show certain systems were transitive, and in the 1990s it was used to establish stable ergodicity. We will review the general notion and how it applies in these settings. We will also review the result from 2003 by Dolgopyat and Wilkinson on the C^1 density of stably transitive systems.

Speaker: Todd Fisher

Video

Slides

No seminar, thanksgiving

Title: Mixing from hyperbolicity following Babillot

Abstract: Following Martine Babillot’s “On the mixing property for hyperbolic systems” we will prove that a particular hyperbolic toral automorphism is mixing. The argument is much more general than this.

Babillot's paper

Slides

Speaker: Jon Chaika

Title: Uniqueness of clusters in percolation

Abstract: Suppose mu is a probability measure which is shift invariant on {0,1}^{Z^d} and we know that for almost every configuration x in {0,1}^{Z^d} there are connected components of 1s which are infinite. In this talk, we will follow a paper by Burton and Keane (generalising results by Aizenman, Kesten and Newman) to give an elegant proof of the fact that, under fairly general conditions (say full support), the number of connected components of infinite cardinality is at exactly one.

Speaker: Nishant Chandgotia