Dynamics Seminar 2022-2023: Difference between revisions
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|[https://sites.google.com/view/zhufeng-math/home Feng Zhu] (UW Madison) | |[https://sites.google.com/view/zhufeng-math/home Feng Zhu] (UW Madison) | ||
|[[#Feng Zhu| '' | |[[#Feng Zhu| ''Relatively Anosov representations: a dynamical notion of geometric finiteness'']] | ||
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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"). | 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. | |||
===Ethan Farber=== | ===Ethan Farber=== | ||
Revision as of 15:38, 24 October 2022
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
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.
Ethan Farber
Lukas Geyer
Harry Baik
MurphyKate Montee
Marissa Loving
Tina Torkaman
Spring 2023
| date | speaker | title | host(s) |
|---|---|---|---|
| January 30 | Pierre-Louis Blayac (Michigan) | TBA | Zhu and Zimmer |
| February 6 | Karen Butt (Michigan) | TBA | Zimmer |
| March 6 | Filippo Mazzoli (UVA) | TBA | Zhu |
| 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 | Apisa and Uyanik |
Spring Abstracts
Pierre-Louis Blayac
Karen Butt
Filippo Mazzoli
Rose Morris-Wright
Carolyn Abbott
Samantha Fairchild
Jon Chaika
Archive of past Dynamics seminars
2021-2022 Dynamics_Seminar_2021-2022
2020-2021 Dynamics_Seminar_2020-2021