## 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)

## 2020

**August 5 (W) 4 - 5:30 pm,** Zoom 739-035-2844

Mahan Mj (Tata Institute of Fundamental Research)

*Percolation on Hyperbolic groups*

We study first passage percolation (FPP) in a Gromov-hyperbolic group G with boundary equipped with the Patterson-Sullivan measure. We associate an i.i.d. collection of random passage times to each edge of a Cayley graph of G, and investigate classical questions about asymptotics of first passage time as well as the geometry of geodesics in the FPP metric. Under suitable conditions on the passage time distribution, we show that the 'velocity' exists in almost every direction, and is almost surely constant by ergodicity of the G-action on the boundary.

For every point on the boundary, we also show almost sure coalescence of any two geodesic rays directed towards the point. Finally, we show that the variance of the first passage time grows linearly with word distance along word geodesic rays in every fixed boundary direction.

This is joint work with Riddhipratim Basu.

**August 12 (W) No Talk**

**August 19, 21, 24, 25 (W F M Tu, mini-course) 4 - 5:30 pm**, Zoom 739-035-2844

Bram Petri (Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université)

*Extremal problems and probabilistic methods in hyperbolic geometry*

Even if we know many things about hyperbolic manifolds, there are many open extremal problems on them. To name a few:

- How does the maximal systole among closed hyperbolic n-manifolds of volume at most V grow as a function of V?

- How does the minimal diameter among closed hyperbolic n-manifolds of volume at least V grow as a function of V?

- Are there closed hyperbolic n-manifolds of arbitrarily large volume whose spectral gap is larger than that of hyperbolic n-space?

Even for surfaces (i.e n=2), many of these extremal problems are open. In this case, answers to these questions also provide insight into the shape of deformation spaces of hyperbolic surfaces.

In these lectures, I will discuss some of these problems. I will talk about what is known about them and how random constructions of hyperbolic manifolds sometimes provide answers.

**August 26 (W) 10 - 11:30 am,** Zoom 739-035-2844

Andrew Putman (University of Notre Dame)

*TBA*

TBA

**September 2 (W) 4 - 5:30 pm,** Zoom 739-035-2844

Michal Ferov (The University of Newcastle)

*TBA*

TBA

**September 9 (W) 4 - 5:30 pm,** Zoom 739-035-2844

Tengren Zhang (National University of Singapore)

*TBA*

TBA

**September 23 (W) 4 - 5:30 pm,** Zoom 739-035-2844

TBA (TBA)

*TBA*

TBA

**October 14 (W) 9 - 10:30 am,** Zoom 739-035-2844

TBA (TBA)

*TBA*

TBA

**October 28 (W) & more (mini-course). 9 - 10:30 am,** Zoom 739-035-2844

Michael Landry (Washington University St. Louis)

*TBA*

TBA