(2021.9.30) Mladen Bestvina (University of Utah)
Small and big mapping class groups
This was a special lecture as a part of the KAIX distinguished lecture series which consists of two colloquium style talks.
(2021.5.27) Tarik Aougab (Haverford College)
Simple length rigidity for covers
Suppose X and Y are finite covers of a fixed hyperbolic surface S. We first show that if for all closed curves gamma on S, gamma admits a simple lift to X if and only if it does to Y, then X and Y are equivalent covers. Using similar ideas, we address the question of when two covers of a fixed hyperbolic surface are isometric when their unmarked simple length spectra agree. We outline some sufficient criteria on the covers for this and generate families of examples. This represents joint work with Max Lahn, Marissa Loving, and Nick Miller.
[Lecture]
(2021.3) Sergio Fenley (IAS/Florida State University)
Partial hyperbolic dynamics in dimension 3.
This will be a 4 part minicourse. We will first cover some generalities on PH (partially hyperbolic diffeomorphisms). We then restrict to dimension 3 for the ambient manifold, and discuss Pujals' conjecture, and branching foliations, which in some sense are the main technical tool to analyze certain questions about PH in dimension 3. Then we will discuss PH homotopic to the identity. This is very involved and we obtain an enormous amount of structure for such in hyperbolic manifolds and in Seifert manifolds. Any homeomorphism in a closed hyperbolic 3-manifold has a finite power homotopic to the identity. We then study more PH in hyperbolic 3-manifolds and prove that if there is a PH in such a manifold, then there is also an Anosov flow in the manifold. If there is additional time, one possible topic to cover is collapsed Anosov flows.
Video recordings of the lectures are available via links below until March 30th.
[Lecture 1], [Lecture 2], [Lecture 3], [Lecture 4]
[Note 1], [Note 2], [Note 3], [Note 4]
(2021.1) Chenxi Wu (U of Wisconsin Madison)
Galois conjugate of entropies of interval maps
If a unimodal interval map sends the critical point to itself under finite iteration, the exponent of its entropy is an algebraic integer due to Perron-Frobenious theorem. When further assuming that the entropy is in a given interval, we are able to provide an algorithm characterizing the closure of all Galois conjugates of such algebraic integers. In an ongoing work we also generalized it to the setting of core entropy on quadratic Hubbard trees. This is a collaboration with Kathryn Lindsey, Diana Davis, Harrison Bray and Giulio Tiozzo.
[Lecture]
(2021.1) Yash Lodha (KIAS)
Amenability and paradoxical decompositions.
The notion of Amenability has its roots in the seminal work of von Neumann from the early 20th century. Since then, it has become a well studied subfield of modern group theory. In this mini course shall describe amenability from a rather algebraic point of view. In the first lecture we shall define amenability via the notion of Folner sets. The second lecture shall be focused on paradoxical decompositions and Tarski numbers. The third lecture shall focus on some recent results in the field, including new solutions to the von Neumann-Day problem.
[Lecture 1], [Lecture 2], [Lecture 3]
(2020.12) Yash Lodha (EPFL)
Spaces of countable groups
In this mini course I will describe two spaces of groups. The first is the Grigorchuk space of marked groups, and the second is the Polish space of enumerated groups. Both spaces provide a useful framework for the study of countable, discrete groups. After a gentle introduction, I shall describe some recent results in the field, such as the work of Minasyan, Osin and Witzel on quasi-isometric diversity, and the work of Elayavalli and Goldbring on the generic version of the von Neumann-Day problem. I shall also describe how some of my past and also some recent work (in part with coauthors) fits into the picture.