Speaker: Joel Moreira (OSU)

Title: On $\{x + y, xy\}$ patterns in large sets of countable fields

Abstract:

Department of Mathematics

Speaker: Joel Moreira (OSU)

Title: On $\{x + y, xy\}$ patterns in large sets of countable fields

Abstract:

In a previous joint work with V. Bergelson we showed that a large subset of a countable field K contains a non-trivial pair {x+y,xy} with x and y in K. Recently we obtained an alternative proof of this result. This new approach uses similar ergodic theoretical techniques together with limits along ultrafilters, and it gives some improvements on our previous work. In particular we show that a large subset of countable field with characteristic 0 contains a non-trivial pair {n+x,nx} where n is an integer and x is in K.

Speaker: Donald Robertson (Ohio State)

Title: The Corners Theorem for Quasirandom Groups

Abstract: The corners theorem, proved by Ajtai and Szemeredi in 1974, states that for any prescribed density, any dense subset of a large enough n-by-n grid contains the vertices of a right triangle. In this talk I will describe recent joint work with V. Bergelson and P. Zorin-Kranich on a version of the corners theorem for quasirandom groups (finite groups without low-dimensional complex representations) via ergodic theory.

Speaker: Joe Rosenblatt (IUPUI)

Title: Coboundaries

Abstract: A coboundary is a difference $H – H \circ \tau$. The function $H$ is the transfer function. It is well-known that for ergodic transformations, a mean-zero function can be approximated by coboundaries. How well one can do this is constrained mostly by bounds on the norm of the transfer function. Similarly, even ergodic maps have almost invariant functions, with the primary constraint being the degree of almost invariance and how long the almost invariance should last. These two issues are related to the ongoing effort to construct (many) pairs of maps without common coboundaries

**Organised in collaboration with the OSU Logic seminar**

Speaker: Anush Tserunyan (UIUC)

Title: Probability groups as an alternative to Furstenberg’s correspondence

Abstract: Multiple recurrence results for amenable groups are most commonly proven via the Furstenberg correspondence principle, which allows for switching from the hard-to-work-with translation action of \Gamma on (\Gamma, d), where d is an upper density function, to a more friendly measure-preserving action of \Gamma on a (genuine) probability space (X, \mu). I suggest an alternative correspondence principle, where we switch the group itself to a more infinitary group G, which, nevertheless, is equipped with a translation-invariant (genuine) probability measure \mu, and the corresponding action is simply the translation action of G on itself. The object (G, \mu) is an example of, what I call, a probability group, the class of which includes all compact groups and is closed under ultraproducts. One of the advantages of studying measure-preserving actions of probability groups over those of amenable groups is that we can integrate over the group, which makes many statements, such as the mean ergodic theorem, boil down to Fubini’s theorem. As an example, we will give a short proof of a triple recurrence result proved by Bergelson–Tao for quasirandom groups.

Speaker: Yair Hartman (Weizmann Institute, Israel)

Title: The Furstenberg entropy realization problem

Speaker: Russell Ricks (Michigan)

Title: Flat strips in rank one CAT(0) spaces

Abstract: Let X be a proper, geodesically complete CAT(0) space under a geometric (that is, properly discontinuous, cocompact, and isometric) group action on X; further assume X admits a rank one axis. Using the Patterson-Sullivan measure on the boundary, we construct a generalized Bowen-Margulis measure on the space of geodesics in X. However, in order to construct this measure, we must prove a couple structural results about the original CAT(0) space X. First, with respect to the Patterson-Sullivan measure, almost every point in the boundary of X is isolated in the Tits metric. Second, under the Bowen-Margulis measure, almost no geodesic bounds a flat strip of any positive width. Then, with the generalized Bowen-Margulis measure, we can characterize when the length spectrum of X is arithmetic (that is, the set of translation lengths is contained in a discrete subgroup of the reals). In this talk, we will discuss the constructions and some of the issues involved.

Speaker: Lei Yang (Yale)

Title: Equidistribution of evolution of curves in homogeneous space under diagonal flow

Abstract: In this talk, we consider a compact analytic curve φ : I → H = SO(n, 1) and embed it into some homogeneous space G/Λ where H ⊂ G and HΛ is dense in G. Fix a maximal R-split Cartan subgroup A = {at : t ∈ R}, we wonder under which condition the expanded curves {atφ(I) : t > 0} tend to be equidistributed. It turns out that it is true if the image of φ(I)is not contained in any proper totally geodesic submanifold of H. It answers a question of Nimish Shah and extends his previous result. And then we will talk the main idea to prove the same result if we replace the analytic curve by only smooth curve, this project is ongoing and joint with Nimish Shah.

Speaker: Ilya Vinogradov (Bristol, UK)

Title: Effective Ratner Theorem for ASL(2, R) and the gaps of the sequence \sqrt n modulo 1

Abstract: Let G=SL(2,\R)\ltimes R^2 and Gamma=SL(2,Z)\ltimes Z^2. Building on recent work of Strombergsson we prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of Gamma\G, which projects to a closed horocycle in the unit tangent bundle to the modular surface. We use this to answer a question of Elkies and McMullen by making effective the convergence of the gap distribution of sqrt n mod 1.

Here is our complete program for Fall 2014:

August 21: Vaughn Climenhaga (Houston)

August 28: Ian Melbourne (Warwick, UK)

Sept 11: Ilya Vinogradov (Bristol, UK)

Sept 18: Lei Yang (Yale)

Sept 25: Russell Ricks (Michigan) (joint seminar with Geometric Group Theory)

Oct 23: Yair Hartman (Weizmann Institute, Israel)

Oct 30: Anush Tserunyan (UIUC, joint seminar with Logic)

Nov 6: Joe Rosenblatt (IUPUI)

Nov 13: Donald Robertson (OSU)

Nov 20: Joel Moreira (OSU)

Speaker: Ian Melbourne (Warwick)

Title: Mixing for dynamical systems with infinite measure

Abstract: We describe results on mixing for a large class of dynamical systems (both discrete and continuous time) preserving an infinite ergodic invariant measure. The method is based on operator renewal theory and an extension of the renewal-theoretic techniques of Garsia & Lamperti from probability theory.

This is joint work with Dalia Terhesiu.

Title: “Tower constructions from specification properties”

Speaker: Vaughn Climenhaga (Houston)

Abstract: Given a dynamical system with some hyperbolicity, the equilibrium states associated to sufficiently regular potentials often display stochastic behaviour. Two important tools for studying these equilibrium states are specification properties and tower constructions. I will describe how both uniform and non-uniform specification properties can be used to deduce existence of a tower with exponential tails, and hence to establish various statistical properties.