Month: August 2014

Seminar 9.11.14 Vinogradov

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.

Seminar Fall 2014

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)

Seminar 8.28.14 Melbourne

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.

Seminar 8.21.14 Climenhaga

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.

Seminar 6.25.14 Adams

Title: A Case of Anything Goes in Infinite Ergodic Theory

Speaker: Terry Adams, US DoD

Seminar Type:  Ergodic Theory/Probability

Abstract: Dynamical systems are well studied in the finite measure preserving case. Many of the same principles do not apply for infinite measure preserving transformations. As an example, for an invertible finite measure preserving transformation, its Cartesian product is ergodic if and only if it is weak mixing. As a consequence, all products of all non-zero powers are ergodic. In the case of invertible infinite measure preserving transformations, the situation is quite different. We give a class of transformations that demonstrate just about any reasonable behavior when it comes to ergodicity and conservativity of products of powers. Also, we’ll provide background on “weak” mixing notions in infinite measure.

Seminar 5.14.14 Van Strien

Title: Dynamics on random networks

Speaker: Sebastian Van Strien, Imperial College, London

Seminar Type:  Ergodic Theory/Probability

Abstract: Networks in which some nodes are highly connected and others have low connectivity are ubiquitous (they are used to model the brain, the internet, cities etc). In this talk I will consider coupled dynamics on randomly selected networks of this type. The methods used for analyzing this are related to those used to show stochastic stability of certain dynamical systems. This work is joint with Tiago Pereira and Jeroen Lamb (Imperial College)

Colloquium 4.3.14 Eskin

Title: The SL(2,R) action on moduli space

Speaker: Alex Eskin, University of Chicago

Abstract: We prove some ergodic-theoretic rigidity properties of the action of SL(2; R) on the moduli space of compact Riemann surfaces. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2; R) is supported on an invariant a ffine submanifold. The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work. This is joint work with Maryam Mirzakhani and Amir Mohammadi.

Seminar 3.20.14 Burago

Title: Just so stories (R. Kipling)

Speaker: Dima Burago,  Penn State

Seminar Type:  Ergodic Theory and Probability

Abstract: This is not a usual type of a seminar talk, though I have given several talks with almost the same title lately. Still the talks are different. I made transparencies for more than 20 topics, two to three slides per topic. Necessary definitions, formulations of key results, hints towards proofs, open problems. I choose about 8 topics per talk. The choice depends on the audience, the weather, what I had for breakfast and such. The only thing that unites the topics is that they have been of interest to me in past number of years. They all are related to geometry, PDEs, dynamics, geometric group theory and such.

Colloquium 3.13.14 Fisher

Title: Entropy for smooth systems

Speaker: Todd Fisher, Brigham Young University

Abstract: Dynamical systems studies the long-term behavior of systems that evolve in time.  It is well known that given an initial state the future behavior of a system is unpredictable, even impossible to describe in many cases. The entropy of a system is a number that quantifies the complexity of the system.  In studying entropy, the nicest classes of smooth systems are ones that are structurally stable. Structurally stable systems, for example automorphisms of the torus, are those that do not undergo bifurcations for small perturbations.  In this case, the entropy remains constant under perturbation. Outside of the class of structurally stable systems, a perturbation of the original system may undergo bifurcation. However, this is a local phenomenon, and it is unclear when and how the local changes in the system lead to global changes in the complexity of the system.  We will state recent results describing how the entropy (complexity) of the system may change under perturbation for systems that are not structurally stable.

Seminar 3.6.14 Carnovale

Title: Gowers Norms and Multiple Recurrence of Sparse Measures on R^d

Speaker: Marc Carnovale, The Ohio State University

Seminar Type:  Ergodic Theory and Probability Seminar

Abstract: It is classical that any positive measure subset of the reals must contain scaled, translated images of any finite configuration of points. Does this still hold for natural classes of ”large” singular sets? A construction of Keleti says that Hausdorff dimension 1 is insufficient to guarantee such a result even for 3-term arithmetic progressions (3APs), while a result of Laba and Pramanik says that the stronger notion of Fourier dimension does yield a result for 3APs, but leaves the case of longer progressions open. Using the notion of intersections of measures from geometric measure theory as a guide, we study a quantity which can be thought of as measuring the multiple recurrence properties under the shift operator of a singular measure, rather than a set, in the torus, and show that it is positive when the measure under question satisfies a certain “higher order Fourier dimension” condition.

Seminar 2.13.14

Title: Finite Sum Sets and Minimally Almost Periodic Groups

Speaker: Donald Robertson, The Ohio State University

Seminar Type: Ergodic Theory and Probability Seminar

Abstract: A result due to Hindman states that, no matter how the positive integers are finitely partitioned, one cell of the partition contains a sequence and all its sums without repetition. Straus, answering a question of Erdos, later gave an example showing that a density version of Hindman’s result does not hold. He exhibited sets of positive integers with arbitrarily large density, each having the property that no shift contains a sums set of the above kind. In this talk I will present recent joint work with V. Bergelson, C. Christopherson and P. Zorin-Kranich in which we generalize Straus’s example to a class of locally compact, second countable, amenable groups and show, using ergodic theory techniques recently developed by Host and Austin, that positive density subsets of groups outside this class must contains sets with strong combinatorial properties. In particular, this allows us to give a combinatorial characterization of minimally almost periodic, amenable groups.