Talks – Midwest dynamics and group actions seminar

01/11: Thomas Hille

Distribution of Values of Irrational Forms at Integral Points and Spherical Averages

Abstract: Let $Q$ be a non-degenerate indefinite quadratic form in $n$ variables. In the mid 80’s, Margulis proved the Oppenheim conjecture, which states that if $n \geq 3$ and $Q$ is not proportional to a rational form, then the set of values of $Q$ at integral points is dense in $\mathbb{R}$ . In some cases, homogeneous forms of higher degree exhibit the same behavior if the number of variables is large enough in terms of the degree and if the group preserving the form is large enough, then the set of values at integral points can be studied from the point of view of homogeneous dynamics.

In this talk we will discuss the problem of effective and quantitative distribution of values of certain forms at integral points. A central and recurrent theme revolves around (asymptotic) estimates of certain spherical averages going back to the work of Eskin, Margulis and Mozes.

This talk is based on the one hand on joint work with P. Buterus, F. Götze and G. Margulis and on the other hand with E. Fromm and H. Oh.

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01/25: Lewis Bowen

Title: A New Infinite-Dimensional Multiplicative Ergodic Theorem

Abstract: In 1960, Furstenberg and Kesten introduced the problem of describing the asymptotic behavior of products of random matrices as the number of factors tends to infinity. Oseledets’ proved that such products, after normalization, converge almost surely. This theorem has wide-ranging applications to smooth ergodic theory and rigidity theory. It has been generalized to products of random operators on Banach spaces by Ruelle and others. I will explain a new infinite-dimensional generalization based on von Neumann algebra theory which accommodates continuous Lyapunov distribution. This will be a gentle introductory-style talk; no knowledge of von Neumann algebras will be assumed. This is joint work with Ben Hayes (U. Virginia) and Yuqing Frank Lin (Ben-Gurion U.).

Slides

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11/09: Andrew Zimmer

Convex co-compact representations of 3-manifold groups

Abstract: A representation of a finitely generated group into the projective linear group is called convex co-compact if it has finite kernel and its image acts convex co-compactly on a properly convex domain in real projective space. In this talk I will discuss the case of 3-manifold groups and prove that the fundamental group of a closed irreducible orientable 3-manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic, or Euclidean × Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. This extends a result of Benoist about convex real projective structures on closed 3-manifolds. In each case, I will also describe what these representations look like. This is joint work with Mitul Islam (a graduate student at the University of Michigan).

Slides

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11/02: Minju Lee

Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends

Abstract: This is joint work with Hee Oh. We establish an analogue of Ratner’s orbit closure theorem for any connected closed subgroup generated by unipotent elements in $\mathrm{SO}(d,1)$ acting on the space $\Gamma\backslash\mathrm{SO}(d,1)$ , assuming that the associated hyperbolic manifold $M=\Gamma\backslash\mathbb{H}^d$ is a convex cocompact manifold with Fuchsian ends. For $d = 3$ , this was proved earlier by McMullen, Mohammadi and Oh. In a higher dimensional case, the possibility of accumulation on closed orbits of intermediate subgroups causes serious issues, but in the end, all orbit closures of unipotent flows are relatively homogeneous. Our results imply the following: for any $k\geq 1$ ,
(1) the closure of any $k$ -horosphere in $M$ is a properly immersed submanifold;
(2) the closure of any geodesic $(k+1)$ -plane in $M$ is a properly immersed submanifold;
(3) an infinite sequence of maximal properly immersed geodesic $(k+1)$ -planes intersecting $\mathrm{core} M$ becomes dense in $M$ .

Slides

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10/26: Homin Lee

Global rigidity theorems for actions of higher rank lattices

Abstract: We will talk about rigidity theorems for smooth actions of a higher rank lattice $\Gamma$ on compact manifolds following the philosophy of the Zimmer program. Previously many global rigidity phenomena are known due to the presence of “higher rank” with “property (T)” on the lattice side, and “Anosov” on the dynamics side. In this talk, we will discuss two global rigidity theorems that relax each condition. The main ingredients are cocycle superrigidity and its generalizations.

Slides

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10/19: Florian Richter

Additive and geometric transversality of fractal sets in the reals and integers

Abstract: Using the language of fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the 1960s that explore the relationship between digit expansions of real numbers in distinct prime bases. While his famous x2 x3 conjecture remains open, recent solutions to some of his “transversality conjectures” have shed new light on old problems. In this talk we explore analogues of results surrounding Furstenberg’s conjectures in the discrete setting of the integers, with the aim of understanding the independence of sets of integers that are structured with respect to different prime bases. This is based on joint work with Daniel Glasscock and Joel Moreira.

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10/12: Ben Lowe

Minimal Surfaces in Negatively Curved 3-Manifolds and Dynamics

Abstract: The Grassmann bundle of tangent 2-planes over a closed hyperbolic 3-manifold M has a natural foliation by (lifts of) immersed totally geodesic planes in M. I am going to talk about work I’ve done on constructing foliations whose leaves are (lifts of) minimal surfaces in a metric on M of negative sectional curvature, which are deformations of the totally geodesic foliation described above. The foliations we construct make it possible to use homogeneous dynamics to study how closed minimal surfaces in variable negative curvature are distributed in the ambient 3-manifold. Many of the ideas here come from recent work of Calegari-Marques-Neves. I was able to prove some preliminary results on the dynamics of these foliations, but much remains to be understood.

Slides

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10/05: Davi Obata

Open sets of partially hyperbolic systems having a unique SRB measure

Abstract: For a dynamical system, a physical measure is an ergodic invariant measure that captures the asymptotic statistical behavior of the orbits of a set with positive Lebesgue measure. A natural question in the theory is to know when such measures exist.

It is expected that a “typical” system with enough hyperbolicity (such as partial hyperbolicity) should have such measures. A special type of physical measure is the so-called hyperbolic SRB (Sinai-Ruelle-Bowen) measure. Since the 70`s the study of SRB measures has been a very active topic of research.

In this talk, we will see some new examples of open sets of partially hyperbolic systems with two dimensional centers having a unique SRB measure. One of the key features for these examples is a rigidity result for a special type of measure (the so-called u-Gibbs measure) which allows us to conclude the existence of the SRB measures.

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09/28: Alireza Salehi Golsefidy

Two new concepts for compact groups: Spectral independence and local randomness

Abstract: I will explain two new concepts for compact groups mentioned in the title. Their basic properties and their connections with the FAb property, quasi-randomness, and super-approximation will be outlined. I will present how these ideas help us show that a Borel probability measure m on the product of compact open subgroups of two locally non-isomorphic simple analytic groups has spectral gap when its projection to each factor has. (Joint work with Keivan Mallahi-Karai and Amir Mohammadi)

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09/21: Wouter van Limbeek

Commensurators and arithmeticity of thin groups

Abstract: The commensurator of a discrete, Zariski-dense subgroup $\Gamma$ of a simple Lie group $G$ contains information on the arithmetic nature of $\Gamma$ : For example, in 1974, Margulis proved that if $\Gamma$ is a lattice and its commensurator is dense, then $\Gamma$ is arithmetic. In 2011, Shalom asked if the same is true only assuming $\Gamma$ is Zariski-dense in $G$ . I will report on recent progress on this question that uses ideas from infinite ergodic theory, profinite actions, representation theory, Brownian motion, random walks and the structure of locally compact groups. I will show how these combine to give information on commensurators and applications to problems in group theory, hyperbolic 3-manifolds, and structure of arithmetic lattices in Lie groups. This is joint work with D. Fisher and M. Mj.

Slides