September 30: Dylan Thurston (Indiana)
Title: Conformal embeddings of graphs
Abstract: Given a topological branched covering f: (S^2,P) \to (S^2, P) of the sphere by itself, with branch values contained in P, can f be realised as a rational map? William Thurston gave a criterion in 1982: If f has a hyperbolic orbifold, it can be realised as a rational map iff there is not an invariant multi-curve satisfying certain conditions. This condition is hard to apply in practice, since it involves checking infinitely many multi-curves, but is nevertheless useful. We give a complete positive combinatorial condition, using a novel notion of \emph{conformal embedding}, a stricter condition than being 1-Lipshitz. A topological branched covering can be realised as a rational map iff a sufficiently large inverse iterate of a metric spine conformally embeds inside itself. This is joint-work-in-progress with Kevin Pilgrim.
October 7: Zhiren Wang (Yale)
Title: Global Rigidity of Anosov Z^r Actions on Tori and Nilmanifolds
Abstract: As part of a more general conjecture by Katok and Spatzier, it was asked if all smooth Anosov Z^r-actions on tori, nilmanifolds and infranilmanifolds without rank-1 factor actions are, up to smooth conjugacy, actions by automorphisms. In this talk, we will discuss a recent joint work with Federico Rodriguez Hertz that affirmatively answers this question.
October 14: Aaron Brown (Penn State)
Title: Topological, smooth, and measure rigidity for higher-rank lattice actions
Abstract: Given an Anosov diffeomorphism of a torus, J. Franks proved there is a continuous conjugacy between the nonlinear action and the toral automorphism induced by the action on first homology. We ask under what conditions Franks's theorem generalizes to lattice actions. That is, given an irreducible lattice in a semisimple Lie group, and an action of the lattice on a torus, under what conditions is the nonlinear lattice action semi-conjugate to the algebraic action induced by the representation on first homology? Assuming the linear representation is Anosov and the lattice is higher-rank, we prove a semi-conjugacy exists under certain technical hypotheses on the induced linear representation. These conditions can be verified for lattices in many lie groups and many linear representations. Assuming the non-linear action contains an Anosov element, the semi-conjugacy can be promoted to a diffeomorphism. We note that we do not assume the existence of an invariant measure. We also consider lattice actions on arbitrary manifolds and ask under what conditions there is an invariant measure for the action. Assuming some non-resonance of Lyapunov exponent functionals, we prove the existence of an invariant measure. This is joint work with Federico Rodriguez-Hertz (Penn State) and Zhiren Wang (Yale)
October 21: Andrés Sambarino (University of Chicago)
Title: Higher rank counting problems
Abstract: Let Gamma be a discrete group of isometries of a symmetric space X. The orbital counting problem consists on finding an asymptotic for #{g in Gamma: d_X(o,g o)\leq T} as T goes to infinity, where o is a given point on X. The purpose of the lecture is to discuss this problem for a class of subgroups that contains Hitchin representations of surface groups. The novelty of this situation is that even when one restricts the problem to the 'convex core' the group still has infinite co-volume.
October 28: Todd Fisher (Brigham Young University)
Title: Entropy variation for smooth systems
Abstract: Topological entropy is locally constant among uniformly hyperbolic diffeomorphisms and certain partially hyperbolic maps. It is also constant on neighborhood of some diffeomorphisms that are not partially hyperbolic, but are isotopic to an Anosov map. We show that this phenomenon is not universal: there exists an open set of diffeomorphisms where the topological entropy is nowhere locally constant.
November 4: Ana Rechtman (IRMA, Université de Strasbourg)
Title: On the minimal set of Kuperberg's plug
Abstract: In 1993 K. Kuperberg constructed examples of smooth and real analytic flows without periodic orbits on any closed 3-manifold. These examples continue to be the only known examples with such properties and are constructed using plugs. After reviewing K. Kuperberg's construction, I will present part of a study of the minimal set in these plugs. Under some generic assumptions, the minimal set has topological dimension 2 and is stratified: the 1-dimensional strata has two connected components, both dense in the 2-dimensional strata. I will explain the main properties of this set.
November 11: Robert Niemeyer (University of New Mexico, Albuquerque)
Title: Current results and future directions regarding dynamics on fractal billiard tables
Abstract: In this talk, we will discuss recent results on two fractal billiard tables: the Koch snowflake and (for lack of a better name and none ever given in the literature) the T-fractal. Initially, an investigation of the flow on a fractal billiard table was made on the Koch snowflake. Results on the Koch snowflake motivated us to investigate the dynamics on the T-fractal billiard table. Substantial progress has been made in determining periodic orbits of the T-fractal billiard table. We detail some of the recent results concerning periodic orbits, determined in collaboration with M. L. Lapidus and R. L. Miller. Less has been done to determine what may constitute an equidistributed orbit of the T-fractal billiard. We provide substantial experimental and theoretical evidence in support of the existence of an orbit that is dense in the T-fractal billiard table but is not a space-filling curve. We briefly touch on a long-term goal of determining a Veech dichotomy for the flow on a fractal billiard table, namely that, in a fixed direction, the flow is either closed or uniquely ergodic. Parts of this talk will be suitable for an advanced undergraduate/beginning graduate student audience.
November 18: Rafael Potrie (Universidad de la Republica, Uruguay)
Title: Cocycles of diffeomorphisms over a hyperbolic basis
Abstract: The study of cocycles over hyperbolic dynamics has been studied since the 70's both to obtain rigidity results as well as understanding geometric properties of those dynamics. This started with the work of Livsic and now it is usual to call it "Livsic theory". Recently, some attention has been given to the study of cocycles taking values in topological groups other than the reals. Kalinin proved recently a Livsic type result for cocycles taking values in Lie groups. We give a Livsic type result for cocycles taking values in diffeomorphisms of the one-dimensional manifolds as well as some partial results for cocycles with values in any diffeomorphism group. This is joint work with A. Kocsard (UFF, Rio de Janeiro).
November 25: Alex Furman (UIC)
Title: On simplicity of the Lyapunov spectrum for some dynamical systems
Abstract: Let (X,m,T) be an ergodic probability measure preserving system, and F:X?SL_d(R) be a measurable integrable function. The associated Lyapunov exponents \lambda_1\ge \lambda_2\ge \dots\ge \lambda_d describe the asymptotic behavior of the products F_n(x)=F(T^{n?1}x)?F(Tx)F(x) ; namely the exponential expansion/contraction rates under F_n(x). An important problem is to determine whether the exponents are all distinct: \lambda_i>\lambda_{i+1}. Simplicity of the spectrum (in the above sense) for products of independent random variables was established by Guivarc'h-Raugi and Gol'dsheid-Margulis in 70s and 80s. More recently, simplicity of the spectrum for a particular system related to Teichmuller flow and Interval Exchange Transformations was conjectured by Kontsevich-Zorich and proved by Avila-Viana. Another case of simplicity of the spectrum appeared in the work of Eskin-Mirzakhani (proved in Eskin-Matheus). In the talk I will describe a joint work with Uri Bader in which we give a "soft" proof of simplicity of the Lyapunov spectrum for a class of systems, including the above mentioned situations.
December 2: Danny Calegari (University of Chicago)
Title: 3-Manifolds everywhere
Abstract: 3-Manifolds (and 3-manifold groups) are (almost) everywhere, including where you might not expect them. I will explain how to construct them (with some help from ergodic theory) and to recognize them. Some of this is joint work with Henry Wilton, building on previous joint work with Alden Walker.
December 9: Yiannis Konstantoulas (UIUC)
Title: Effective multiple mixing for Weyl chamber actions
Abstract: A famous theorem of Mozes implies that a mixing action of a semisimple group (with certain conditions) on a probability space $X$ is mixing of all orders. Prompted by a question of D. Dolgopyat, we study in this work the quantitative behaviour of multiple correlation integrals for the Weyl chamber action of such a group using the properties of semidirect products embedded in simple groups of higher rank. We will demonstrate how to get explicit rates of decay for spaces of well behaved functions and give examples for classical groups.