Month: January 2017

Seminars for Spring 2017

Here is the schedule of talks for Spring semester 2017. All talks are on Thursdays in MW154 at 3.00pm-4.00pm (unless otherwise indicated).

Jan 19: Dave Constantine (Wesleyan)

Feb 2: Florian Richter (Ohio State)

Feb 9: Harrison Bray (Michigan)

Feb 23: Donald Roberson (Utah)

Feb 23: COLLOQUIUM: Federico Rodriguez Hertz (Penn State)

Mar 2: Felipe Ramirez (Wesleyan)

Mar 3: [Friday 1.50-2.50pm]: Sebastian Van Strien (Warwick, UK)

Apr 6: Daniel Glasscock (Ohio State)

Apr 13: May Mei (Denison)

Apr 20: Osama Khalil (Ohio State)

May 25: Ayse Sahin (Wright State)

Jun 1: Joel Moreira (Northwestern)

Colloquium 11.3.16 DeMarco

Title: Complex dynamics and elliptic curves

Speaker: Laura DeMarco (Northwestern University)

Abstract: In this talk, I will present some connections between recent research in dynamical systems and the classical theory of elliptic curves and rational points. The main goal is to explain the role of dynamical stability and bifurcations in deducing arithmetic finiteness statements. I will focus on three examples: (1) the theorem of Mordell and Weil from the 1920s, presented from a dynamical point of view; (2) a recent result of Masser and Zannier about torsion points on elliptic curves, and (3) features of the Mandelbrot set.

Seminar 11.10.16 Pavlov

Title: Weakened specification properties and intrinsic ergodicity for subshifts

Speaker: Ronnie Pavlov (University of Denver)

Abstract: The specification property has been fundamental in the study of topological dynamical systems since its introduction by Bowen, and among other properties, implies intrinsic ergodicity, i.e. uniqueness of the measure of maximal entropy. For subshifts, specification amounts to the ability to concatenate arbitrary pairs of n-letter words given a uniform gap (i.e. independent of n).

In this setting, several natural weakenings of specification can be defined. For instance, one can allow non-uniform gaps, or allow a small number of “edits” to the words rather than including a gap. In both cases, one can parametrize by an auxiliary function f(n) controlling gap size/number of edits.

I will summarize some recent results in this area, beginning with examples showing that several weakened specification properties do not imply intrinsic ergodicity even for slowly growing f(n), and finishing with recent joint work with Vaughn Climenhaga, in which we defined a new weakened specification property which does imply intrinsic ergodicity for small enough f(n).

Seminar 10.20.16 Chen

Title: Two types of KAM-nondegenerate nearly integrable systems with positive metric entropy

Speaker: Dong Chen (Penn State)

Abstract: The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still have a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens outside KAM tori draws lots of attention. In this talk I will present two types of C^\infty small Lagrangian perturbation of the geodesic flow on a flat torus. Both resulting flows have positive metric entropy, from which we get positive metric entropy outside some KAM tori. What is special in the second type is that positive metric entropy comes from an arbitrarily small tubular neighborhood of one trajectory. This is a joint work with D. Burago and S. Ivanov.

Seminar 10.6.16 Sun

Title: On pointwise multiple convergence problems

Speaker: Wenbo Sun (The Ohio State University)

Abstract: Multiple convergence problems is a topic that has been widely studied in ergodic theory. These questions usually have interesting applications in combinatorics. While the L2L2 multiple convergence problems is well-understood by now, little is known for the pointwise convergence problems. In this talk, I will discuss about some recent advances in the pointwise convergence problems. This is joint work with Sebastian Donoso.

Seminar 9.22.16 Finkelshtein

Title: Diophantine approximation problems for groups of toral automorphisms

Speaker: Vladimir Finkelshtein (University of Illinois at Chicago)

Abstract: I will present sharp rates for a shrinking target problem for the action of an arbitrary subgroup of SL(2,Z) on the 2-torus. This can also be viewed as a noncommutative Diophantine approximation problem. The methods require construction of spectrally optimal random walks on groups acting properly cocompactly on Gromov hyperbolic spaces. Additionally, I will explain how similar estimates for this problem in higher dimension can be obtained using harmonic analysis.

Seminar 9.15.16 Burns

Speaker: Keith Burns (Northwestern)

Title: Mixing properties of the Weil-Petersson geodesic flow

Abstract: I will talk about the geodesic flow for the Weil-Petersson metric on the moduli space of a surface that supports hyperbolic metrics. This is a Riemannian metric with negative sectional curvatures. However the classical results of Anosov do not apply because the metric is incomplete and the sectional curvatures and their derivatives are not uniformly bounded. It was not until the 21st century that this geodesic flow was shown to be mixing (and in fact Bernoulli).

I will give some ideas from the proof and also from more recent work directed towards showing that the flow is exponentially mixing in case of the torus with one puncture.

This is joint work with Howie Masur, Carlos Matheus, and Amie Wilkinson