Dynamics Seminar: Difference between revisions
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===Vaibhav Gadre=== | ===Vaibhav Gadre=== | ||
A quadratic differential on a Riemann surface is equivalent to a half-translation structure on the surface by complex charts with half-translation transitions. The SL(2,R)-action on the complex plane takes half-translations to half-translations and so descends to moduli spaces of quadratic differentials. The diagonal part of the action is the Teichmuller flow. | |||
Apart from its intrinsic interest, the dynamics of Teichmuller flow is central to many applications in geometry, topology and dynamics. The Konstevich—Zorich cocycle which records the action of the flow on the absolute homology of the surface, plays a key role. | |||
In this talk, I will explain how the flow detects the topology of moduli spaces. Specifically, we will show that the flow group, namely the subgroup generated by almost flow loops, has finite index in the fundamental group. As a corollary, we will prove that the minus and plus (modular) Rauzy—Veech groups have finite index in the fundamental group, answering a question by Yoccoz. | |||
Using this, and Filip’s results on algebraic hulls and Zariski closures of modular monodromies, we prove that the Konstevich—Zurich cocycle (separately minus and plus pieces) have a simple Lyapunov spectrum, extending the work of Forni from 2002 and Avila—Viana from 2007. | |||
===Becky Eastham=== | ===Becky Eastham=== | ||
Revision as of 03:15, 31 August 2023
During the Fall 2023 semester, RTG / Group Actions and Dynamics seminar meets in room Van Vleck B235 on Mondays from 2:25pm - 3:15pm. 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, Chenxi Wu or Andy Zimmer.
Fall 2023
| date | speaker | title | host(s) | |
|---|---|---|---|---|
| September 11 | Vaibhav Gadre (Glasgow) | Teichmuller flow detects the fundamental group | Apisa | |
| September 18 | Becky Eastham (UW Madison) | TBA | local | |
| September 25 | Brandis Whitfield (Temple) | TBA | Loving | |
| October 2 | Hanh Vo (Arizona State) | TBA | Dymarz | |
| October 9 | Yandi Wu (UW Madison) | TBA | local | |
| October 16 | Sanghoon Kwak (Utah) | Mapping class groups of Infinite graphs — “Big Out(Fn)” | Loving | |
| October 23 | Sara Maloni (UVA) | TBA | Uyanik | |
| October 30 | Giulio Tiozzo (Toronto) | TBA | Uyanik | |
| November 6 | Emily Stark (Wesleyan) | TBA | Uyanik | |
| November 13 | Hongming Nie (Stony Brook) | TBA | Wu | |
| November 20 | Rose Morris-Wright (Middlebury) | TBA | Dymarz | |
| November 27 | Luke Jeffreys (UW Madison) | TBA | local | |
| December 4 | ||||
| December 11 |
Fall Abstracts
Vaibhav Gadre
A quadratic differential on a Riemann surface is equivalent to a half-translation structure on the surface by complex charts with half-translation transitions. The SL(2,R)-action on the complex plane takes half-translations to half-translations and so descends to moduli spaces of quadratic differentials. The diagonal part of the action is the Teichmuller flow.
Apart from its intrinsic interest, the dynamics of Teichmuller flow is central to many applications in geometry, topology and dynamics. The Konstevich—Zorich cocycle which records the action of the flow on the absolute homology of the surface, plays a key role.
In this talk, I will explain how the flow detects the topology of moduli spaces. Specifically, we will show that the flow group, namely the subgroup generated by almost flow loops, has finite index in the fundamental group. As a corollary, we will prove that the minus and plus (modular) Rauzy—Veech groups have finite index in the fundamental group, answering a question by Yoccoz.
Using this, and Filip’s results on algebraic hulls and Zariski closures of modular monodromies, we prove that the Konstevich—Zurich cocycle (separately minus and plus pieces) have a simple Lyapunov spectrum, extending the work of Forni from 2002 and Avila—Viana from 2007.
Becky Eastham
Brandis Whitfield
Hanh Vo
Yandi Wu
Sanghoon Kwak
Surfaces and graphs are closely related; there are many parallels between the mapping class groups of finite-type surfaces and finite graphs, where the mapping class group of a finite graph is the outer automorphism group of a free group of (finite) rank. A recent surge of interest in infinite-type surfaces and their mapping class groups begs a natural question: What is the mapping class group of an “infinite” graph? In this talk, I will explain the answer given by Algom-Kfir and Bestvina and present recent work, joint with George Domat (Rice University), and Hannah Hoganson (University of Maryland), on the coarse geometry of such groups.
Sara Maloni
Giulio Tiozzo
Emily Stark
Hongming Nie
Rose Morris-Wright
Luke Jeffreys
Archive of past Dynamics seminars
2022-2023 Dynamics_Seminar_2022-2023
2021-2022 Dynamics_Seminar_2021-2022
2020-2021 Dynamics_Seminar_2020-2021