Dynamics Seminar 2022-2023

The Dynamics seminar meets in room B329 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 Paul Apisa, Marissa Loving, Caglar Uyanik, or Chenxi Wu. Contact Caglar Uyanik with your wisc email to get the zoom link for virtual talks.

Fall 2022

date	speaker	title	host(s)
September 12	Jing Tao (OU)	Genericity of pseudo-Anosov maps	Dymarz and Uyanik
September 19	Rebekah Palmer (Temple)(virtual)	Totally geodesic surfaces in knot complements	VIRTUAL
September 26	Beibei Liu (MIT)	The critical exponent: old and new	Dymarz
October 3	Grace Work (UW-Madison)	Discretely shrinking targets in moduli space	local
October 10	Jean Pierre Mutanguha (Princeton)	Canonical forms for free group automorphisms	Uyanik
October 17	Anthony Sanchez (UCSD)	Kontsevich-Zorich monodromy groups of translation covers of some platonic solids	Uyanik
October 24	Alena Erchenko (U Chicago)	postponed	Uyanik and Work
October 31	Feng Zhu (UW Madison)	Relatively Anosov representations: a dynamical notion of geometric finiteness	local
November 7	Ethan Farber (BC)	TBA	Loving
November 14	Lukas Geyer (Montana)	TBA	Burkart
November 21	Harry Hyungryul Baik (KAIST)	TBA	Wu
November 28	MurphyKate Montee (Carleton)	TBA	Dymarz
December 5	Marissa Loving (UW Madison)	TBA	local
December 12	Tina Torkaman (Harvard)	TBA	Uyanik

Fall Abstracts

Jing Tao

By Nielsen-Thurston classification, every homeomorphism of a surface is isotopic to one of three types: finite order, reducible, or pseudo-Anosov. While there are these three types, it is natural to wonder which type is more prevalent. In any reasonable way to sample matrices in SL(2,Z), irreducible matrices should be generic. One expects something similar for pseudo-Anosov maps. In joint work with Erlandsson and Souto, we define a notion of genericity and show that pseudo-Anosov maps are indeed generic. More precisely, we consider several "norms" on the mapping class group of the surface, and show that the proportion of pseudo-Anosov maps in a ball of radius r tends to 1 as r tends to infinity. The norms can be thought of as the natural analogues of matrix norms on SL(2,Z).

Rebekah Palmer

Studying totally geodesic surfaces has been essential in understanding the geometry and topology of hyperbolic 3-manifolds.  Recently, Bader--Fisher--Miller--Stover showed that containing infinitely many such surfaces compels a manifold to be arithmetic.  We are hence interested in counting totally geodesic surfaces in hyperbolic 3-manifolds in the finite (possibly zero) case.  In joint work with Khánh Lê, we expand an obstruction, due to Calegari, to the existence of these surfaces.  On the flipside, we prove the uniqueness of known totally geodesic surfaces by considering their behavior in the universal cover.  This talk will explore this progress for both the uniqueness and the absence.

Beibei Liu

The critical exponent is an important numerical invariant of discrete groups acting on negatively curved Hadamard manifolds, Gromov hyperbolic spaces, and higher-rank symmetric spaces. In this talk, I will focus on discrete groups acting on hyperbolic spaces (i.e., Kleinian groups), which is a family of important examples of these three types of spaces. In particular, I will review the classical result relating the critical exponent to the Hausdorff dimension using the Patterson-Sullivan theory and introduce new results about Kleinian groups with small or large critical exponents.

Grace Work

The shrinking target problem characterizes when there is a full measure set of points that hit a decreasing family of target sets under a given flow. This question is closely related to the Borel Cantilli lemma and also gives rise to logarithm laws. We will examine the discrete shrinking target problem in a general and then more specifically in the setting of Teichmuller flow on the moduli space of unit-area quadratic differentials.

Jean Pierre Mutanguha

The Nielsen–Thurston theory of surface homeomorphisms can be thought of as a surface analogue to the Jordan Canonical Form. I will discuss my progress in developing a similar canonical form for free group automorphisms. (Un)Fortunately, free group automorphisms can have arbitrarily complicated behaviour. This is a significant barrier to translating arguments that worked for surfaces into the free group setting; nevertheless, the overall ideas/strategies do translate!

Anthony Sanchez

Platonic solids have been studied for thousands of years. By unfolding a platonic solid we can associate to it a translation surface. Interesting information about the underlying platonic solid can be discovered in the cover where more (dynamical and geometric) structure is present. The translation covers we consider have a large group of symmetries that leave the global composition of the surface unchanged. However, the local structure of paths on the surface is often sensitive to these symmetries. The Kontsevich-Zorich mondromy group keeps track of this sensitivity.

In joint work with R. Gutiérrez-Romo and D. Lee, we study the monodromy groups of translation covers of some platonic solids and show that the Zariski closure is a power of SL(2,R). We prove our results by finding generators for the monodromy groups, using a theorem of Matheus–Yoccoz–Zmiaikou that provides constraints on the Zariski closure of the groups (to obtain an "upper bound"), and analyzing the dimension of the Lie algebra of the Zariski closure of the group (to obtain a "lower bound").

Feng Zhu

Putting hyperbolic metrics on a finite-type surface S gives us linear representations of the fundamental group of S into PSL(2,R) with many nice geometric and dynamical properties: for instance they are discrete and faithful, and in fact stably quasi-isometrically embedded.

In this talk, we will introduce (relatively) Anosov representations, which generalise this picture to higher-rank Lie groups such as PSL(d,R) for d>2, giving us a class of (relatively) hyperbolic subgroups there with similarly good geometric and dynamical properties.

This is mostly joint work with Andrew Zimmer.

Spring 2023

date	speaker	title	host(s)
January 30	Pierre-Louis Blayac (Michigan)	TBA	Zhu and Zimmer
February 6	Karen Butt (Michigan)	TBA	Zimmer
February 13	Elizabeth Field (Utah)	TBA	Loving
February 20	open	TBA
February 27	open	TBA
March 6	Filippo Mazzoli (UVA)	TBA	Zhu
March 13	Spring Break, No Seminar
March 20	Rose Morris-Wright (Middlebury)	TBA	Dymarz
March 27	Carolyn Abbott (Brandeis)	TBA	Dymarz and Uyanik
April 3	Samantha Fairchild (Osnabrück)	TBA	Apisa
April 10	Jon Chaika (Utah)	TBA	Uyanik
April 17	open	TBA
April 24	Tarik Aougab (Haverford)	TBA	Loving
May 1	open	TBA

Archive of past Dynamics seminars

2021-2022 Dynamics_Seminar_2021-2022

2020-2021 Dynamics_Seminar_2020-2021