Title: Quantitative multiple recurrence in ergodic theory

Speaker: Sebastian Donoso (Universidad de O’Higgins)

Abstract: In this talk I will survey recent development of the multiple recurrence problem in ergodic theory. For a probability space (X,,μ) and measure preserving transformations T1,…,Td, the problem is to study the largeness of the set of n∈ℕ such that

where a1,…,ad take integer values on the integers and F is a suitable function. I will mention key results and comment on the problem for commuting transformations, linear and polynomial functions ai. I plan to provide some proofs that rely on combinatorial constructions.

Title: Smooth Local Rigidity of Algebraic Actions

Speaker: Zhenqi Wang (Michigan State University)

Abstract: At first, we will introduce the background of algebraic actions and give some interesting examples. Next, we will review of various smooth rigidity results for higher-rank algebraic actions and recent progress. Finally, we will talk about the new progress on smooth rigidity of rank-one partially hyperbolic actions and future directions.

Title: Subsequences of multiple correlations

Speaker: Anh Le (Northwestern University)

Abstract: The results of Bergelson-Host-Kra and of Leibman say that a multiple correlation sequence can be decomposed as a sum of a nilsequence and a null sequence. Inspired by these results, Frantzikinakis asks the following question: Let rnrn be the sequence of primes, or [nc], or 2n. For a multiple correlation a(n), is it true that there exists a nilsequence b(n) and null sequence e(n) such that a(rn)=b(rn)+e(n)?

In this talk, I’ll briefly discuss why the answer is affirmative for the sequences of primes and [nc]. However, our main focus will be about 2n. The answer for this sequence is also yes and surprisingly related to the notion of sets of Bohr recurrence.

Title: Primitive rational points on expanding horospheres

Speaker: Manuel Luethi (ETH-Zurich)

Abstract: An observation by Marklof implies that the primitive rational points of denominator n along the stable horocycle orbits of large volume determined by n equidistribute within a proper submanifold of the unit tangent bundle to the modular surface. We examine the general behavior of primitive rational points along expanding horospheres and prove joint equidistribution in products of the torus and the unit tangent bundle to the modular surface using effective mixing for congruence quotients.

Title: Rigidity of higher rank smooth abelian action with hyperbolicity

Speaker: Disheng Xu (University of Chicago)

Abstract: In this talk we will show our recent results (some are working in progress) with Damjanovic and Damjanovic, Wilkinson on rigidity phenomena of higher rank abelian actions with certain hyperbolicity. In addition, we will introduce some “final conjectures” in this area.

Title: Quantitative disjointness of nilflows and horospherical flows

Speaker: Asaf Katz (University of Chicago)

Abstract: In his influential disjointness paper, H. Furstenberg proved that weakly-mixing systems are disjoint from irrational rotations (and in general, Kronecker systems), a result that inspired much of the modern research in dynamics. Recently, A. Venkatesh managed to prove a quantitative version of this disjointness theorem for the case of the horocyclic flow on a compact Riemann surface. I will discuss Venkatesh’s disjointness result and present a generalization of this result to more general actions of nilpotent groups, utilizing structural results about nilflows proven by Green-Tao-Ziegler. If time permits, I will discuss certain applications of such theorems to sparse equidistribution problems and number theory.

Title: Random perturbations of predominantly expanding multimodal circle maps

Speaker: Yun Yang (CUNY)

Abstract: In this talk we will study the effects of IID random perturbations of amplitude ϵ>0 on the asymptotic dynamics of one-parameter families {fa:S1→S1,a∈[0,1]} of smooth multimodal maps which “predominantly expanding”, i.e., |f′a|≫1 away from small neighborhoods of the critical set {f′a=0}. We will obtain, for any ϵ>0, a checkable, finite-time criterion on the parameter aa for random perturbations of the map fafa to exhibit (i) a unique stationary measure, and (ii) a positive Lyapunov exponent comparable to ∫S1log|f′a|dx.

Title: Ergodicity of Iwasawa continued fractions

Speaker: Joseph Vandehey (The Ohio State University)

Abstract: The connections between the behavior of geodesic flow on the modular surface and the Gauss map for the regular continued fraction expansion date back to Artin. Recently, Anton Lukyanenko and myself were able to extend this connection to higher dimensional hyperbolic spaces and higher dimensional continued fraction algorithms. This allows us to give proofs of ergodicity for a variety of new continued fraction expansions, including quaternionic, octonionic, and Heisenberg continued fractions. We will discuss this result and some of the odd consequences of it.

Here is the list of seminars for Fall 2018.

August 30: Joe Vandehey (OSU)

September 6: Yun Yang (CUNY)

September 13: Asaf Katz (University of Chicago)

September 27: Disheng Xu (University of Chicago)

October 9: Manuel Luethi (ETH Zurich)

October 18: COLLOQUIUM: Amir Mohammadi (University of California San Diego)

November 1: Anh Le (Northwestern)

November 9: Zhenqi Wang (Michigan State)

December 6: Sebastian Donoso (University of O’Higgins, Chile)