Dynamics Seminar 20212022
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@ggroups.wisc.edu. For more information, contact Caglar Uyanik or Chenxi Wu.
Spring 2022
date  speaker  title  host(s)


Feb. 28  Rohini Ramadas (Warwick)  Coming soon  Dymarz 
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 subFinsler 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 subFinsler 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 lowdimensional 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 3manifolds
Hongbin Sun showed that a closed hyperbolic 3manifold 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 3manifolds which are not graphmanifolds.
Osama Khalil (Utah)
Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation
Khintchine's Theorem provides a zeroone 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 selfsimilar 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 padic 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 higherrank generalization of convex cocompact subgroups of rankone Lie groups. They are only defined for wordhyperbolic groups, but recently KapovichLeeb and Zhu have suggested possible definitions for an Anosov representation of a relatively hyperbolic group  aiming to give a higherrank 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)
Parametrizing transversals to horocycle flow
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)
The Hubbard tree is a combinatorial object that encodes the dynamic of a post critically finite polynomial map, and its topological entropy is called the core entropy. I will talk about an upcoming paper with Kathryn Lindsey and Giulio Tiozzo where we provide geometric constrains to the Galois conjugates of exponents of core entropy, which gives a necessary condition for a number to be the core entropy for a super attracting parameter.
Jack Burkart (UW Madison)
Geometry and Topology of Wandering Domains in Complex Dynamics
Let f: C > C be an entire function. In complex dynamics, the main objects of study are the Fatou set, the points where f and its iterates locally form a normal family, and the Julia set, which is the complement of the Fatou set and often has a fractal structure. The Fatou set is open, and connected components of the Fatou set map to each other. Connected components that are not periodic are called wandering domains.
In this talk, I give a biased survey of what we know about the existence of wandering domains in complex dynamics and their geometry and topology, highlighting both classical and recent results and some open problems.
Jayadev Athreya (UW Seattle)
Stable Random Fields, PattersonSullivan Measures, and Extremal Cocycle Growth
We study extreme values of groupindexed 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 PattersonSullivan 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 nonvanishing is equivalent to finiteness of the BowenMargulis measure for the associated unit tangent bundle U(X/G) provided X/G has nonarithmetic length spectrum. This is joint work with Mahan MJ and Parthanil Roy.
Funda Gültepe (U Toledo)
A universal CannonThurston map for the surviving curve complex Using the Birman exact sequence for pure mapping class groups, we construct a universal CannonThurston map onto the boundary of a curve complex for a surface with punctures we call surviving curve complex. In this talk, I will give some background and motivation for the work and I will explain the ingredients of the construction and then give a sketch of proof of the main theorem. Joint work with Chris Leininger and Witsarut Phoon.
Jonah Gaster (UWM)
The Markov ordering of the rationals
A rational number p/q determines a simple closed curve on a oncepunctured torus, which then has a welldefined length when the torus is endowed with a complete hyperbolic metric. When the metric is chosen so that the torus is “modular” (that is, when its holonomy group is conjugate into PSL(2,Z)), the lengths of the curves have special arithmetic significance, with connections to Diophantine approximation and number theory. Taking inspiration from McShane’s elegant proof of Aigner’s conjectures, concerning the (partial) ordering of the rationals induced by hyperbolic length on the modular torus, I will describe how hyperbolic geometry can be used to characterize monotonicity of the order so obtained along lines of varying slope in the (q,p)plane.
Chloe Avery (U Chicago)
Stable Torsion Length
The stable torsion length in a group is the stable word length with respect to the set of all torsion elements. We show that the stable torsion length vanishes in crystallographic groups. We then give a linear programming algorithm to compute a lower bound for stable torsion length in free products of groups. Moreover, we obtain an algorithm that exactly computes stable torsion length in free products of finite abelian groups. The nature of the algorithm shows that stable torsion length is rational in this case. As applications, we give the first exact computations of stable torsion length for nontrivial examples. This is joint work with Lvzhou Chen.
Daniel Levitin (UW Madison)
Metric Spaces of Arbitrary FinitelyGenerated Scaling Group
A quasiisometry between uniformly discrete spaces metric spaces of bounded geometry is scaling if it is coarsely kto1 for some positive real k, up to some error that takes into account the geometry of the space. The collection of k for which scaling selfmaps exist is a multiplicative group by composing maps. Scaling maps and the scaling group have been used to prove a variety of quasiisometric rigidity theorems for groups and spaces. In this talk, I construct a space with any finitelygenerated scaling group.
Archive of past Dynamics seminars
20202021 Dynamics_Seminar_20202021