Lecture 1 

Lecturer

Matteo Migliorini

Title

Hyperbolic manifolds and fibrations

Abstract

In these lectures, we will study the interplay between hyperbolic manifolds and fibrations (i.e. fiber bundles) over the circle. We will see how on one hand fibering for a hyperbolic manifold is a very weird phenomenon, while on the other hand it is something very common, at least in dimension 3.

Indeed, by a theorem of Agol and Wise, all 3-manifolds virtually fiber over the circle. In higher dimensions the situation is much more mysterious, and the problem is that we lack all the tools we have available in dimension 3 to understand hyperbolic manifolds. The aim of these lectures is to introduce some combinatorial tools that allow to construct hyperbolic manifolds M equipped with a map f : M → S¹ that we are able to study.

We start by giving an overview of the world of hyperbolic manifolds, with a particular focus of their relationship with fibrations. We will see how the existence of a hyperbolic manifold of dimension n > 3 implies the existence of a hyperbolic group G with a "weird" subgroup H, i.e. that is of finite type but not hyperbolic.

Then we introduce Bestvina-Brady Morse theory, which is a piecewise-linear analogue of the more famous smooth version. This was originally introduced to study the finiteness properties of the kernel of certain epimorphisms  ϕ : G → Z; it can be seen as an algebraic analogue of fibrations.

In the appropriate setting one may promote a Bestvina-Brady Brady Morse function to a smooth one. So the problem of constructing a fibration can be reduced to an entirely combinatoric one.

Therefore we introduce Coxeter polytopes, that one can use to construct hyperbolic manifolds equipped with a cell complex structure. Using these, we can construct many different Bestvina-Brady Morse functions that we can investigate with some combinatorial techniques, introduced by Jankiewicz, Norin, and Wise; if some conditions are satisfied, the smoothing will produce a fibration.

Lecture 2

Lecturer

Florent Schaffhauser

Title

Vector bundles on Riemann surfaces

Abstract

Vector bundles on compact Riemann surfaces are of interest for at least two reasons:
1. They arise naturally in the study of analytic differential equations.
2. They provide non-Abelian analogues of the classical Jacobian variety in complex geometry.

In these lectures, we provide an introduction to these objects, focusing on their classification and the topology of their moduli spaces.

The first three lectures will focus on the general theory of holomorphic vector bundles on a compact Riemann surface, while the last three will present the gauge-theoretic approach developed by Atiyah, Bott and Donaldson in the 1980s.

More information: https://matematiflo.github.io/RTG/Lectures2024.html