Winter Semester 2022/2023

Abstract:

Spectral sequences are a very helpful and powerful tool of homological algebra to compute some graded modules, most notably the (co)homology of (co)chain complexes. This minicourse will give a introduction to the theory of spectral sequences and will discuss some applications to bounded cohomology, in particular we will construct the Hochschild-Serre spectral sequence in bounded cohomology.

The first two talks of this course are intended as an elementary introduction to spectral sequences. We will see their definition, some methods to construct them and we will discuss some of the most best known examples of spectral sequences and possible applications of these examples.

Only in the third talk we turn our attention to bounded cohomology. We will straight ahead dive into the construction of the Hochschild-Serre spectral sequence in bounded cohomology. As an application we will discuss some aspects of the characterisation of boundedly acyclic morphisms by Moraschini and Raptis.

Minicourse notes available here
Recordings: Talk 1 Talk 2 Talk 3 

Abstract:

In the first two talks of this thematic block, we will present the notion of minimal volume introduced by Gromov, and after an overview of some of the main results in the literature, we will give a fairly detailed account of Gromov's proof of his "main inequality", which relates minimal volume and simplicial volume.

In the third talk, we present Braun-Sauer's proof of a generalization of Gromov's main inequality: the bound on the curvature of the manifold M is replaced by an upper bound on the volume of radius 1 balls in the universal cover of M. The proof shows actually that the integral foliated simplicial volume of M is a lower bound, up to a multiplicative constant, for its minimal volume.

Recordings: Talk 1 Talk 2 Talk 3 

Abstract:

Given an upper bound on the volumes of unit balls in the universal cover of a closed Riemannian n-manifold M, can the volume of M be an arbitrarily small fraction of the simplicial volume?  A theorem of Guth and Braun—Sauer says no.  We outline an alternative proof using Papasoglu’s method of area-minimizing separating sets.

Recording 

Abstract:

First talk: The first talk of this thematic block will be about the bounded Euler class for an action of a group on the circle. The bounded Euler class is a generalisation of Poincaré's rotation number that was introduced by Ghys to study the dynamics of circle actions. We will give several points of view on the bounded Euler class and explain how it can be used to classify circle actions up to semi-conjugacy.

Slides Recording 

Second talk: The Euler class of a sphere bundle is a cohomology class of the base space which, intuitively, measures how far the bundle is from admitting a global section. We will define the cocycle from which this class arises, state some of its main properties and see how we can define this class also for vector bundles. Finally, we will prove that, for the tangent bundle of a smooth closed oriented manifold, the Euler class is equal to the Euler characteristic.

Recording

Third talk: In this talk we will hopefully understand how the previous talks are related to each other. On first hand we have a notion of (bounded) Euler class for an action on the circle, and on the other one we have seen how to associated an Euler class to a sphere bundle.

The correspondence between flat circle bundles and actions on the circle hence provides a double notion of bounded Euler class in this framework. Luckily, they coincide and, in case of surfaces, their norm is bounded by the Euler characteristic. This fact, known as Milnor-Wood inequality, has interesting consequences in the study of geometric representations (Goldman) and can be obtained also in different settings. We will try to make an overview of the topic, following Chapters 10, 11 and 12 of Frigerio’s book. 

Recording

Fourth talk: A well-known conjecture by Chern states that the Euler characteristic of closed affine manifolds is always zero. During this talk, we will describe the problem and we will present a possible strategy using simplicial volume. Quite surprisingly, this problem is related with two classical questions about the vanishing of simplicial volume by Gromov and Lück, respectively.

More precisely, we will give an overview of all these problems and then we will concentrate on a result by Bucher, Connell and Lafont showing that Chern conjecture is true for aspherical affine manifolds satisfying technical conditions on their holonomy.

Recording