## Virtual Seminar on

## Geometry and Topology

### https://visgat.cayley.kr

## How to join

**Zoom Meeting ID** 739-035-2844

**Password** Zero minus the Euler characteristic of a closed oriented surface of genus 4000

**Time **Wed 4 pm or Thursday 9 am (or so) in KST (Korea Standard Time)

## 2021

**January 7, 11, 13 (Th, M, W),**** ****4**** pm Korea**

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.

**January 14 (Th),**** ****10 a****m Korea**

Chenxi Wu (U of Wisconsin Madison)

*Canonical metric on surfaces and manifolds*

A classical theorem by Kazhdan says that the canonical metric on surfaces converges to hyperbolic metric when listed to a sequence of regular covers that converges to the universal cover. We generalize the result to other sequences of regular covers as well as higher dimension using Lück's L^2 techniques. This is a collaboration with Farbod Shokrieh and Hyungryul Baik.

**January 15 (F),**** ****10 a****m Korea**

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.

**January 20 (W), 2021**

4:30pm - 6:00pm in Korean time = 8:30am - 10:00am in France

Federica Fanoni (CNRS)

*Isospectral hyperbolic surfaces of infinite type*

Two hyperbolic surfaces are said to be (length) isospectral if they have the same length spectrum (i.e., the same collection of lengths of primitive closed geodesics, counted with multiplicity). For closed surfaces, there is an upper bound on the size of isospectral hyperbolic structures depending only on the topology. We will show that the situation is different for infinite-type surfaces, by constructing large families of isospectral hyperbolic structures on surfaces of infinite genus.