Seminar 6.1.17 Moreira

Title: Quasi-disjointness

SpeakerJoel Moreira (Northwestern University)

Abstract: The Kronecker factor of a measure preserving system is the factor generated by all the almost periodic functions and has an explicit algebraic description. A surprising result of Berg essentially classifies the joinings of two ergodic measure preserving Z-systems in terms of the joinings of their Kronecker factors (which have an easy algebraic description), if one of the systems is (measurable) distal. Using Berg’s terminology, he showed that any ergodic Z-system is quasidisjoint. from any ergodic distal Z-system. In this talk I will explore the notion of quasidisjointness, and present an alternative definition of quasidisjointness which makes sense for measure preserving actions of any group and agrees with Berg’s definition for Z actions.

Seminar 5.25.17 Sahin

Title: Symbolic and Ergodic Actions of the discrete Heisenberg group

SpeakerAyşe Sahin (Wright State University)

Abstract: We describe two different actions of the Heisenberg group, Γ: odometer actions and aperiodic shifts of finite type. Both types of actions rely on the geometric structure of the Heisenberg group, but exploit different frameworks. We show in the first case that the odometer actions are indexed by a triple of data describing finite index subgroups of Γ, as described by a short exact sequence associated to the Heisenberg group. The aperiodic shift of finite type, on the other hand, is constructed using the Cayley graph structure of a semi-direct product group.

Seminar 4.20.17 Khalil

Title: Birkhoff’s Theorem, Sparse Equidistribution and Translates of Measures on Homogeneous Spaces

SpeakerOsama Khalil (Ohio State University)

Abstract: Many applications of homogeneous dynamics in number theory and geometry involve showing that the translates of a non-invariant measure by a sequence of transformations become equidistributed towards a distinguished ergodic invariant measure. We will discuss some of these results and the applications of their pointwise refinements to sparse equidistribution of unipotent flows and diophantine approximation.

Seminar 4.13.17 Mei

Title: Spectral Properties of Continuum Fibonacci Schrodinger Operators

SpeakerMay Mei (Denison University)

Abstract: The Nobel Prize-winning discovery of quasicrystals irrevocably changed the paradigm of crystalline structure. From the standpoint theoretical physics, questions about the time evolution of quantum particles lead to the study of a Schrodinger operator that depends on parameters that we will refer to as the potential, which corresponds to the potential energy of the system, and the coupling constant which encodes interaction strength. In particular, we are interested in potentials that are generated by ergodic transformations. Discrete Schrodinger operators with potentials generated by aperiodic subshifts over a finite alphabet have been studied since the mid 1980’s. More specifically, the Fibonacci Hamiltonian has been a particularly well-studied example. Several continuum analogues have also been considered. In this talk, we will offer a survey of discrete ergodic Schrodinger operators and discuss spectral properties of one particular continuum Fibonacci Schrodinger operator in which each letter of the subshift sequence is replaced with a function.

Seminar 4.6.17 Glasscock

TitleCombinatorial aspects of recurrence in Szemerédi’s theorem and applications addressing the interplay between additive and multiplicative largeness

SpeakerDaniel Glasscock (Ohio State University)

AbstractForty-two years ago, Endre Szemerédi proved that subsets of the natural numbers with positive upper density contain arbitrarily long arithmetic progressions.  To this day, Szemerédi’s theorem and its relatives continue to stimulate new research in many fields, including combinatorics, additive number theory, harmonic analysis, and dynamics.  In this talk, I will explain how some of the finer combinatorial aspects of recurrence in Szemerédi’s theorem can be derived from the (purely combinatorial) density Hales-Jewett theorem.  I will then demonstrate how this extra information is useful in two applications relating notions of additive and multiplicative largeness.

Seminar 3.3.17 Van Strien

Title: Heterogeneously coupled maps

Speaker: Sebastian Van Strien (Imperial College, London)

Abstract: Our main aim is to rigorously study dynamics of Heterogeneously Coupled Maps (HCM). Such systems are determined by a network with heterogeneous degrees. Some nodes, called hubs, are very well connected and most nodes interact with few others. The local dynamics on each node is chaotic, coupled with other nodes according to the network structure. Such systems are very hard to describe in full. Nevertheless we are able to describe the system over exponentially large time scales (in terms of the total number of nodes of the network). In particular, we show that the dynamics of hub nodes can be very well approximated by a low-dimensional system.

Seminar 3.2.17 Ramirez

Title: Some questions about inhomogeneous approximation

Speaker: Felipe Ramire (Wesleyan)

Abstract: Khintchine’s Theorem (1924) states that almost all (respectively, almost no) real numbers can be approximated by rationals at a given rate, provided that the rate is monotonic and corresponds to a divergent (resp. convergent) series. In 1941, Duffin and Schaeffer showed by way of example that the monotonicity condition cannot be removed. They formulated the famous and resistant Duffin—Schaeffer Conjecture in response to this example. I will discuss an analogue of this situation for inhomogeneous approximations. From the point of view of dynamics, this talk is about toral translations.

Colloquium 2.23.17 Rodriguez Hertz

Title: New developments in the theory of smooth actions

Speaker: Federico Rodriguez Hertz (Pennsylvania State University)

Abstract: In recent years several new advances in the theory of lattice actions have been made. In this talk I will present some of the key ingredients to these advances. I plan to keep the talk at an elementary level so only some basic notions of measure theory and differentiation on manifolds should be needed.

Seminar 2.23.17 Robertson

Title: Families of mild mixing interval exchange transformations

SpeakerDonald Robertson (University of Utah)

Abstract: Interval exchange transformations are piecewise linear isometries of the unit interval. Almost every interval exchange transformation is rigid. In this talk I will describe full Hausdorff dimension families of interval exchange transformations that are mild mixing.

Seminar Bray 2.9.17

Title: A measure of maximal entropy for geodesic flows of nonstrictly convex Hilbert geometries

SpeakerHarrison Bray (University of Michigan)

Abstract: Strictly convex Hilbert geometries naturally generalize constant negatively curved Riemannian geometries, and the geodesic flow on quotients has been well-studied by Benoist, Crampon, Marquis, and others. In contrast, nonstrictly convex Hilbert geometries in three dimensions have the feel of nonpositive curvature, but also have a fascinating geometric irregularity which forces the geodesic flow to avoid direct application of existing nonuniformly hyperbolic theory. In this talk, we present a geometric approach to studying the geodesic flow in this setting, culminating in a measure of maximal entropy which is ergodic for the geodesic flow.

Seminar 2.7.17 Richter

Title: Kronecker disjointness of dynamical systems and a spectral refinement of the BHK-decomposition.

Speaker: Florian Richter (Ohio State University)

Abstract: We investigate how spectral properties of measure preserving systems are reflected in ergodic averages and multiple ergodic averages (aka. non-conventional ergodic averages) arising from this system. This leads to natural strengthenings of the classical mean ergodic theorem and its various generalizations. In particular, we derive new weighted multiple ergodic theorems.

Seminar 1.19.17 Constantine

Title: The specification property and its consequences for CAT(-1) spaces

Speaker: Dave Constantine (Wesleyan)

Abstract: The specification property is a strong dynamical property which allows one to find orbits in a system which obey quite robust sets of constraints. The canonical example of a flow with specification is the geodesic flow on a compact, negatively curved manifold. The CAT(-1) property is an attempt to capture the essential features of negative curvature in the metric space setting, without assumptions on the regularity of the space.

In this talk, covering joint work with Jean-Francois Lafont and Dan Thompson, I’ll prove that a weak form of the specification property holds for geodesic flows on compact, locally CAT(-1) geodesic metric spaces. I’ll discuss some of the dynamical consequences that follow from this property, and try to indicate how the CAT(-1) geometry of the spaces plays an essential role in the proof.

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