# Dynamics Seminar 2021-2022

The Dynamics seminar meets in room **901 of Van Vleck Hall** on **Mondays** from **2:30pm - 3:20pm**. To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu. For more information, contact Caglar Uyanik or Chenxi Wu.

## Fall 2021

## Abstracts

### Nate Fisher (UW Madison)

*Boundaries, random walks, and nilpotent groups*

In this talk, we will discuss boundaries and random walks in the Heisenberg group. We will discuss a class of sub-Finsler metrics on the Heisenberg group which arise as the asymptotic cones of word metrics on the integer Heisenberg group and describe new results on the boundaries of these polygonal sub-Finsler metrics. After that, we will explore experimental work to examine the asymptotic behavior of random walks in this group. Parts of this work are joint with Sebastiano Nicolussi Golo.

### Caglar Uyanik (UW Madison)

*Dynamics on currents and applications to free group automorphisms*

Currents are measure theoretic generalizations of conjugacy classes on free groups, and play an important role in various low-dimensional geometry questions. I will talk about the dynamics of certain "generic" elements of Out(F) on the space of currents, and explain how it reflects on the algebraic structure of the group.

### Michelle Chu (UIC)

*Prescribed virtual torsion in the homology of 3-manifolds*

Hongbin Sun showed that a closed hyperbolic 3-manifold virtually contains any prescribed torsion subgroup as a direct factor in homology. In this talk we will discuss joint work with Daniel Groves generalizing Sun’s result to irreducible 3-manifolds which are not graph-manifolds.

### Osama Khalil (Utah)

*Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation*

Khintchine's Theorem provides a zero-one law describing the approximability of typical points by rational points. In 1984, Mahler asked whether the same holds for Cantor’s middle thirds set. His question fits into a long studied line of research aiming at showing that Diophantine sets are highly random and are thus disjoint, in a suitable sense, from highly structured sets.

We will discuss the first complete analogue of Khintchine’s theorem for certain self-similar fractal measures, recently obtained in joint work with Manuel Luethi. The key ingredient in the proof is an effective equidistribution theorem for fractal measures on the space of unimodular lattices, generalizing a long history of similar results for smooth measures beginning with Sarnak’s work in the eighties. To prove the latter, we associate to such fractals certain p-adic Markov operators, reminiscent of the classical Hecke operators, and leverage their spectral properties. No background in homogeneous dynamics will be assumed.

### Theodore Weisman (UT Austin)

*Relative Anosov representations and convex projective structures*

Anosov representations are a higher-rank generalization of convex cocompact subgroups of rank-one Lie groups. They are only defined for word-hyperbolic groups, but recently Kapovich-Leeb and Zhu have suggested possible definitions for an Anosov representation of a relatively hyperbolic group - aiming to give a higher-rank generalization of geometrical finiteness.

In this talk, we will introduce a more general version of relative Anosov representation which also interacts well with the theory of convex projective structures. In particular, the definition includes projectively convex cocompact representations of relatively hyperbolic groups, and allows for deformations of cusped convex projective manifolds (including hyperbolic manifolds) in which the cusp groups change in nontrivial ways.

### Grace Work (UW Madison)

There are many interesting dynamical flows that arise in the context of translation surfaces, including the horocycle flow. One application of the horocycle flow is to compute the distribution of the gaps between slopes of saddle connections on a specific translation surface. This method was first developed by Athreya and Chueng in the case of the torus, where the question can be restated in terms of Farey fractions and was solved by R. R. Hall using methods from analytic number theory. An important step in this process is to find a good parametrization of a transversal to horocycle flow. We will show how to do this explicitly in the case of the octagon, how it generalizes to a specific class of translation surfaces, lattice surfaces, (both joint work with Caglar Uyanik), and examine how to parametrize the transversal for a generic surface in a given moduli space.

### Chenxi Wu (UW Madison)

"TBA"

### Jack Burkart (UW Madison)

"TBA"

### Jayadev Athreya (UW Seattle)

*Stable Random Fields, Patterson-Sullivan Measures, and Extremal Cocycle Growth*

We study extreme values of group-indexed stable random fields for discrete groups G acting geometrically on spaces X in the following cases: (1) G acts freely, properly discontinuously by isometries on a CAT(-1) space X, (2) G is a lattice in a higher rank Lie group, acting on a symmetric space X, (3) G is the mapping class group of a surface acting on its Teichmuller space. The connection between extreme values and the geometric action is mediated by the action of the group G on its limit set equipped with the Patterson-Sullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth which measures the distortion of measures on the boundary in comparison to the movement of points in the space X and show that its non-vanishing is equivalent to finiteness of the Bowen-Margulis measure for the associated unit tangent bundle U(X/G) provided X/G has non-arithmetic length spectrum. This is joint work with Mahan MJ and Parthanil Roy.

### Funda Gültepe (U Toledo)

"TBA"

### Jonah Gaster (UWM)

*TBA*

### Chloe Avery (U Chicago)

"TBA"

### Matt Clay (Arkansas)

"TBA"

## Archive of past Dynamics seminars

2020-2021 Dynamics_Seminar_2020-2021