Bram Petri

Extremal problems and probabilistic methods in hyperbolic geometry

Where: Virtual Seminar on Geometry and Topology

Abstract

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.

Lecture notes

I will post my notes here after each lecture.
DISCLAIMER: These notes will probably be full of mistakes and typos, I'm happy to hear about any of them.

Notes