This is a hybrid seminar, featuring both online and local in-person speakers. The seminar broadly focuses on the ergodic theory of smooth dynamical systems, dynamics of group actions, homogeneous dynamics, and their applications in geometry. It is designed for a general mathematical audience, and graduate students and postdocs are especially encouraged to attend. If you are interested in participating, please feel free to reach out at mishrap@wfu.edu or veconid@wfu.edu.
Time: Monday, 3:30 AM Eastern Time.
Venue: Manchester 125, if you would to attend virtually please send an email to mishrap@wfu.edu or veconid@wfu.edu
Contact: mishrap@wfu.edu or veconid@wfu.edu
Organizers: Pratyush Mishra and Dominic Veconi.
Abstract: An Anosov manifold with boundary is a compact smooth Riemannian manifold M with strictly convex boundary, hyperbolic trapped set (possibly empty), and no conjugate points. We will discuss when M admits a codimension zero isometric embedding into a closed Riemannian manifold with Anosov geodesic flow and demonstrate how to use such embeddings to prove the marked boundary distance rigidity conjecture for Anosov surfaces with boundary. This talk is based on joint work with Dong Chen and Andrey Gogolev as well as joint work with Thibault Lefeuvre.
Abstract: The purpose of this talk is to discuss pointwise ergodic theorems. We focus on the noncommutative (nilpotent) variant of the Furstenberg-Bergelson-Leibman conjecture, which states that the ergodic averages converge pointwise almost everywhere, provided that the underlying measure preserving transformations generate a nilpotent group.
Abstract: The classical Livšic Theorem for smooth, transitive Anosov flows states that any Hölder continuous function whose integral over any periodic orbit vanishes must be a derivative. In this talk, we will discuss a generalization of the Livšic Theorem to the setting of geodesic flows on locally CAT(–1) spaces, which include negatively curved Riemannian manifolds but are much more general. Time permitting, we will also discuss applications to the volume rigidity problem for locally CAT(-1) spaces. This is joint work with Dave Constantine and Elvin Shrestha.
Abstract: SRB measures, being physical measures, are of prime importance in partially hyperbolic systems. Their existence is an open problem - in general. Nevertheless, a related, more general class of measures - known as u-Gibbs states, were known to exist by a theorem of Pesin-Sinai. I will explain how one can adapt the factorization technique, pioneered by Eskin-Mirzakhani, to the setting of smooth dynamics and prove that for quantitatively non-integrable systems a (generalized) u-Gibbs state must be an SRB measure. If time permits, I will try to describe some of the key ideas and constructions of the Eskin-Mirzakhani technique.
Abstract: Given a vector field on a compact manifold, we will construct a countable partition with finite entropy for any invariant probability measure. The partition will be compatible with Liao's tubular neighborhoods, in which the holonomies along the flow lines are well-defined. For "good" vector fields, we will show that the partition is entropy-generating.
Abstract: We discuss the thermodynamic and ergodic properties of a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the pseudo-Anosov map is uniformly hyperbolic outside of a neighborhood of a set of singularities, and the trajectories are slowed down so the differential is the identity at the singularities. Using Young towers, we prove existence and uniqueness of equilibrium measures for the family of observables known as the geometric $t$-potentials. This set of equilibrium measures includes a unique smooth Sinai-Ruelle-Bowen physical measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the Central Limit Theorem.
Abstract: We discuss the thermodynamic and ergodic properties of a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the pseudo-Anosov map is uniformly hyperbolic outside of a neighborhood of a set of singularities, and the trajectories are slowed down so the differential is the identity at the singularities. Using Young towers, we prove existence and uniqueness of equilibrium measures for the family of observables known as the geometric $t$-potentials. This set of equilibrium measures includes a unique smooth Sinai-Ruelle-Bowen physical measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the Central Limit Theorem.
Abstract: Consider a countable discrete group G and its subgroup space Sub(G), the collection of all subgroups of G. Sub(G) is a compact metrizable space with respect to the Chabauty topology (the topology induced from the product topology on {0,1}^G). The normal subgroups of G are the fixed points of (Sub(G),G). Furthermore, the G-invariant probability measures of this dynamical system are known as invariant random subgroups (IRSs).
Recently, Glasner and Lederle have introduced the notion of Boomerang subgroups. Among many other remarkable results, they strengthen the well-known Margulis's normal subgroup Theorem.
To a countable discrete group G, we can also associate an algebraic object L(G), called the group von Neumann algebra. More recently, in a joint work with Hartman and Oppelmayer, we introduced the notion of Invariant Random Algebra (IRA), an invariant probability measure on the collection of subalgebras of L(G).
Motivated by the works of Glasner and Lederle, in ongoing joint work with Yair Glasner, Yair Hartman, and Yongle Jiang, we introduce the notion of Boomerang subalgebras in the context of L(G). In this talk, we shall show that every Boomerang subalgebra of a torsion-free non-elementary hyperbolic group (an example of this is a Free group generated by two elements a and b) comes from a Boomerang subgroup. We shall also discuss its connection to understanding IRAs in such groups.