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)
Title: Unique Equilibrium States for Geodesic Flows on Surfaces without Focal Points
Speaker: Nyima Kao (University of Chicago)
Abstract: It is well-known that for compact uniformly hyperbolic systems Hölder potentials have unique equilibrium states. However, it is much less known for non-uniformly hyperbolic systems. In his seminal work, Knieper proved the uniqueness of the measure of maximal entropy for the geodesic flow on compact rank 1 non-positively curved manifolds. A recent breakthrough made by Burns, Climenhaga, Fisher, and Thompson which extended Knieper’s result and showed the uniqueness of the equilibrium states for a large class of non-zero potentials. This class includes scalar multiples of the geometric potential and Hölder potentials without carrying full pressure on the singular set. In this talk, I will discuss a further generalization of these uniqueness results, following the scheme of Burns-Climenhaga-Fisher-Thompson, to equilibrium states for the same class of potentials over geodesic flows on compact rank 1 surfaces without focal points. This work is an MRC project joint with Dong Chen, Kiho Park, Matthew Smith, and Régis Varão.
Title: The Erdos sumset conjecture
Speaker: Florian Richter (Ohio State University)
Abstract: A longstanding open conjecture of Erdos states that every subset of the integers with positive density contains a sum B+C of two infinite sets B and C. I will talk about recent joint work with Joel Moreira and Donald Robertson in which we resolve this conjecture. Our poof utilizes ideas and methods coming from Ergodic Theory, including Bergelson’s intersectivity lemma, various decomposition theorems of arithmetic functions into structured and pseudo-random components, and some borrowed techniques from Beiglboeck’s proof of Jin’s theorem.
Title: A classification of intermediate β-transformations
Speaker: Tony Samuel (Cal Poly San Luis Obispo)
Abstract: In this talk we consider transformations of the unit interval of the form $\beta x + \alpha \bmod{1}$ where $1 < \beta < 2$ and $0 \leq α \leq 2 – \beta$. These transformations are called intermediate β-transformations. We will discuss some old and new results concerning these transformations, for instance, their kneading sequences, their absolutely continuous invariant measures and dynamical properties such as topological transitivity and the sub-shift of finite type property. Moreover, we address how the kneading sequences and absolutely continuous invariant measures change as we let $(\beta, \alpha)$ converge to $(1, \theta)$, for some $\theta \in [0, 1]$. Finally, some open problems and applications of these results to one-dimensional Lorenz maps and quasicrystals will be alluded to.
Title: Limiting behavior of multiple polynomial averages
Speaker: Andreas Koutsogiannis (Ohio State University)
Abstract: The study of the norm limiting behavior of multiple ergodic averages has been of great importance in the area of ergodic theory. A central result, Szemerédi’s theorem (i.e., every subset of natural numbers of positive upper density contains arbitrary long arithmetic progressions) follows by a classical result on multiple ergodic averages due to Furstenberg. In this talk we will mainly deal with averages along integer part of special families of real polynomials, for a single transformation (recent joint work with D. Karageorgos) as well as for multiple commuting transformations; we will refer to results along other integer valued sequences mainly due to Bergelson, Knutson, Leibman, Chu, Frantzikinakis, Host and Kra. We will also sketch how to obtain by the aforementioned results, the corresponding results along prime numbers.
Title: Periodic point growth for C^2 maps of the two sphere
Speaker: Yun Yang (CUNY)
Abstract: There are two basic mechanisms producing periodic orbits in a dynamical systems: contraction (via the Contraction mapping theorem, Banach fixed point theorem) and degree (via topological methods such as the Lefschetz theorem).These two mechanisms play an important role in the joint work of Enrique Pujals, Michael Shub and myself with assistance from Sylvain Crovisier on Shub’s conjecture: Let f : M -> M be a C2 map of a compact manifold. Then the exponential growth rate in fixed points of fn bounded below by the growth rate of the Lefschetz numbers of f^n. In this talk, I will present the proof of this conjecture in the case where f: S2 -> S2 has positive entropy and reverses orientation in the direction of vanishing exponents.
Title: On the relationship between entropy and periodic orbits
Speaker: Bryce Weaver (Xavier University)
Abstract: Via Pesin theory, complexity, measured by entropy, is inexorably linked to structure, namely stable/unstable manifolds. Another such link is the relationship between entropy and growth of periodic orbits. One pioneer result in this area was by G. Margulis in the late ‘60s for geodesics flows on negatively curved manifolds. We explore an adaptation of the technique established by G. Margulis in the case of case of geodesic flows on manifolds with regions of positive curvature. We then close with some settings where there is potential for similar adaptations, in particular the Bunimovich stadium billiards.
Title: Lee-Yang zeros for the Cayley Tree and expanding maps of the circle
Speaker: Roland Roeder (IUPUI)
Abstract: I will explain how to use detailed properties of expanding maps of the circle (Shub-Sullivan rigidity, Ledrappier-Young formula, large deviations principle,…) to study the limiting distribution of Lee-Yang zeros for the Ising Model on the Cayley Tree. No background in mathematical physics is expected of the audience. This is joint work with Ivan Chio, Anthony Ji, and Caleb He.
Title: The automorphism group of a zero entropy symbolic system
Speaker: Van Cyr (Bucknell)
Abstract: The symmetries of a symbolic dynamical system X form an interesting and often complicated group called its automorphism group. Although this group is always countable, it is frequently extremely complex for positive entropy subshifts (containing free subgroups, the fundamental group of every 2-manifold, and every finite group). By contrast, the group of automorphisms of a zero entropy subshift is often considerably more tame and it has been possible to prove a number of strong algebraic results. In this talk I will discuss some of these results and open problems.
Title: Recurrence along some sequences involving primes
Speaker: Younghwan Son (Pohang University of Science and Technology, Korea)
Abstract: One of the important themes in ergodic theory is the phenomena of recurrence, which concerns how the initial state returns to the original state. The ergodic method introduced by Furstenberg has been used to deduce the combinatorial structures inherent in large subsets of integers by proving recurrence statements in dynamical systems.
In this talk, we will present some general results on uniform distribution involving primes to establish new recurrence statements, which refine the previous results obtained by Sarkozy and Furstenberg. This is a joint work with V. Bergelson and G. Kolesnik.
Title: Badly approximable points on manifolds and unipotent orbits in homogeneous spaces
Speaker: Lei Yang (Sichuan University, China)
Abstract: We will study n-dimensional badly approximable points on manifolds. Given an smooth non-degenerate submanifold in R^n, we will show that any countable intersection of the sets of weighted badly approximable points on the manifold has full Hausdorff dimension. This strengthens a previous result of Beresnevich by removing the condition on weights and weakening the analytic condition on manifolds to smooth condition. Compared with the work of Beresnevich, we study the problem through homogeneous dynamics. It turns out that the problem is closely related to the study of distribution of long pieces of unipotent orbits in homogeneous spaces.
Title: Generic properties of extensions
Speaker: Michael Schnurr (Max-Planck-Institut, Germany)
Abstract: Motivated by the classical results by Halmos and Rokhlin on the genericity of weakly but not strongly mixing transformations and the Furstenberg tower construction, we show that weakly but not strongly mixing extensions on a fixed product space with both measures non-atomic are generic. In particular, a generic extension does not have an intermediate nil-factor.
Title: Marked length rigidity for Fuchsian buildings
Speaker: Dave Constantine (Wesleyan University)
Abstract: Suppose we are given two metrics on a space and told that for every element of the fundamental group of the space, the length of the shortest curve representing it for each metric is the same. Must the two metrics be the same? This is the marked length spectrum (MLS) rigidity problem. Most famously, the answer is `yes’ for negatively-curved Riemannian metrics on closed surfaces, yet the problem remains wide open for negatively curved metrics in higher dimensions.
In this talk I’ll discuss joint work with Jean-Francois Lafont proving some MLS rigidity results for Fuchsian buildings. The proof uses the combinatorial structure of the buildings, as well as an extension of the technology developed to prove MLS rigidity for surfaces which we must carefully adapt to overcome the non-surface-like behavior of the buildings.
Title: Cohomology of smooth abelian group actions and applications of exotic topological groups
Speaker: Kurt Vinhage (University of Chicago)
Abstract: We will discuss the cohomology of homogeneous partially hyperbolic abelian group actions and recently (re)discovered tools which combine group homology, Lie criteria for topological groups, and some exotic topological groups, namely the free product of Lie groups. We will also discuss how these tools may be used to obtain smooth rigidity results, as well as other possible applications, including in the rank one setting.
Title: Khintchine recurrence for upper Banach density along filters
Speaker: John Johnson (Ohio State University)
Abstract: Khintchine’s recurrence theorem is a well-known extension of Poincaré’s recurrence theorem improving the “quality” of the recurrence in a measure-preserving transformation on a probability space: if $A$ is a subset of the probability space with positive measure, then the collection $\{ n \in \mathbb{N} : \mu(A \cap T^{-n}A) > 0 \}$ has bounded gaps. This recurrence theorem also has a combinatorial analog for sets with positive upper Banach density: if $A \subseteq \mathbb{N}$ has positive upper Banach density $d^*(A) > 0$, then the collection $\{ n \in \mathbb{N} : d^ *(A \cap -n + A) > 0 \}$ has bounded gaps. By iteration, this combinatorial theorem yields Szemerédi’s affine cube lemma as an easy corollary. (The affine cube lemma is one important component in the combinatorial proof of Roth’s theorem on three-term arithmetic progressions.)
In joint work with Florian Richter, we define a notion of upper Banach density, relative to filters on the positive integers, and, under some relatively mild conditions on the underlying filters, prove an analog of Khintchine’s recurrence theorem for this generalization of Banach density as well. As a combinatorial consequence, we can obtain a restricted version of Szemerédi’s affine cube lemma
Title: Disjointness properties of some parabolic flows
Speaker: Adam Kanigowski (Pennsylvania State University)
Abstract: We study disjointness properties of some parabolic flows such as smooth flows on surfaces, time changes of horocycle flows and Heisenberg nilflows. We show that in the above classes time p and q automorphisms are disjoint. This in particular implies Sarnak conjecture. The approach is based on a new version of Ratner’s property (non-uniform Ratner’s property).
Title: Divergent trajectories in arithmetic homogeneou spaces of rational rank two
Speaker: Nattalie Tamam (Tel Aviv University)
Abstract: In the theory of Diophantine approximations, singular points are ones for which Dirichlet’s theorem can be infinitely improved. It is easy to see that all rational points are singular. In the special case of dimension one, the only singular points are the rational ones. In higher dimensions, points lying on a rational hyperplane are also obviously singular. However, in this case there are additional singular points. In the dynamical setting the singular points are related to divergent trajectories. In the talk I will define obvious divergent trajectories and explain the relation to rational points. In addition, I will present the more general setting involving Q-algebraic groups. Lastly I will discuss results concerning classification of divergent trajectories in Q-algebraic groups.
Title: The dichotomy between structure and randomness in multiplicative number theory
Speaker: Florian Richter (Ohio State University)
Abstract: We will begin the talk by discussing a dichotomy theorem in multiplicative number theory which asserts that any multiplicative function (that satisfies certain minor regularity conditions) is either a (special kind of) almost periodic function or a pseudo-random function. Then we will explore how this phenomenon extends to other classical objects coming from multiplicative number theory. In particular, we will study the combinatorial and dynamical properties of level sets of multiplicative functions and I will present a structure theorem which says that for any level set E of an arbitrary multiplicative function there exists a highly structured superset R such that E is a pseudo-random subset of R.
Title: Distributional Lattices in Symmetric spaces.
Speaker: Elliot Paquette (Ohio State University)
Abstract: A Riemannian symmetric space X is a Riemannian manifold in which it is possible to reflect all geodesics through a point by an isometry of the space. A lattice in such a space can be considered as a discrete subgroup G of isometries so that a Borel fundamental domain of the quotient space G/X has finite Riemannian volume. Lattices mirror the structure of the ambient space in many ways: for example, X is amenable if and only if the the ambient space is amenable. We introduce the notion of a distributional lattice, generalizing the notion of lattice, by considering measures on discrete subsets of X having finite Voronoi cells and certain distributional invariance properties. Non-lattice distributional lattices exist in any Riemannian symmetric space: the Voronoi tessellation of a stationary Poisson point process is an example. With an appropriate notion of amenability, the amenability of a distributional lattice is equivalent to the amenability of the ambient space. We give some open problems related to these processes and some pretty pictures.
Title: Moebius disjointness for models of an ergodic system and beyond
Speaker: Mariusz Lemanczyk (Nicolaus Copernicus University, Toruń, Poland)
Abstract: In 2010, P. Sarnak formulated the following conjecture:
For each zero entropy topological system (X,T), we have
$$
\lim_{N\to\infty}\frac1N\sum_{n\leq N}f(T^nx)\mu(n)\to 0$$
for each $f\in C(X)$ and $x\in X$. Here $\mu$ stands for the arithmetic
Moebius function. The talk will concentrate on motivations for Sarnak’s conjecture
(relations with celebrated Chowla conjecture in number theory), role of ergodic theory in it and some recent progress.