Forthcoming events in this series
Homogeneous Einstein metrics and the graph theorem.
Abstract
First of all, we are going to recall some basic facts and definitions about homogeneous Riemannian manifolds. Then we are going to talk about existence and non-existence of invariant Einstein metrics on compact homogeneous manifolds. In this context, we have that it is possible to associate to every homogeneous space a graph. Then, the graph theorem of Bohm, Wang and Ziller gives an existence result of invariant Einstein metrics on a compact homogeneous space, based on properties of its graph. We are going to discuss this theorem and sketch its proof.
Optimal embeddings of groups into Hilbert spaces
Abstract
We begin by showing the underlying ideas Bourgain used to prove that the Cayley graph of the free group of finite rank can be embedded into a Hilbert space with logarithmic distortion. Equipped with these ideas we then tackle the same problem for other metric spaces. Time permitting these will be: amalgamated products and HNN extensions over finite groups, uniformly discrete hyperbolic spaces with bounded geometry and Cayley graphs of cyclic extensions of small cancellation groups.
3-manifolds and their fundamental groups
Abstract
We'll discuss 2 ways to decompose a 3-manifold, namely the Heegaard
splitting and the celebrated geometric decomposition. We'll then see
that being hyperbolic, and more in general having (relatively)
hyperbolic fundamental group, is a very common feature for a 3-manifold.
Normal Forms, Factorability and Cohomology of HV-groups
Abstract
A factorability structure on a group G is a specification of normal forms
of group elements as words over a fixed generating set. There is a chain
complex computing the (co)homology of G. In contrast to the well-known bar
resolution, there are much less generators in each dimension of the chain
complex. Although it is often difficult to understand the differential,
there are examples where the differential is particularly simple, allowing
computations by hand. This leads to the cohomology ring of hv-groups,
which I define at the end of the talk in terms of so called "horizontal"
and "vertical" generators.
Analysis on boundaries of hyperbolic groups
Abstract
We'll survey some of the ways that hyperbolic groups have been studied
using analysis on their boundaries at infinity.
A brief introduction to the Steenrod squares and their applications
Abstract
I will give a brief introduction to the Steenrod squares and move on to show some applications of them in Topology and Geometry.
Slow Ultrafilters and asymptotic cones of proper metric spaces
Abstract
The construction of the asymptotic cone of a metric space which allows one to capture the "large scale geometry" of that space has been introduced by Gromov and refined by van den Dries and Wilkie in the 1980's. Since then asymptotic cones have mainly been used as important invariants for finitely generated groups, regarded as metric spaces using the word metric.
However since the construction of the cone requires non-principal ultrafilters, in many cases the cone itself is very hard to compute and seemingly basic questions about this construction have been open quite some time and only relatively recently been answered.
In this talk I want to review the definition of the cone as well as considering iterated cones of metric spaces. I will show that every proper metric space can arise as asymptotic cone of some other proper space and I will answer a question of Drutu and Sapir regarding slow ultrafilters.
Nonstandard methods in geometry: asymptotic cones
Abstract
After a quick-and-dirty introduction to nonstandard analysis, we will
define the asymptotic cones of a metric space and we will play around
with nonstandard tools to show some results about them.
For example, we will hopefully prove that any separable asymptotic cone
is proper and we will classify real trees appearing as asymptotic cones
of groups.
Conjugacy problem and conjugator length
Abstract
A brief survey of the above.
Geoghegan stacks and finiteness properties of groups
Abstract
Geoghegan's stack construction is a tool for analysing groups
that act on simply connected CW complexes, by providing a topological
description in terms of cell stabilisers and the quotient complex,
similar to what Bass-Serre theory does for group actions on trees. We
will introduce this construction and see how it can be used to give
results on finiteness properties of groups.
Random Walks: Getting from A to A.
Abstract
This talk introduces the topic of random walks on a finitely generated group and asks what properties of such a group can be detected through knowledge of such walks.
16:00
Bounded cohomology and quasi-homomorphisms
Abstract
Starting from a definition of the cohomology of a group, we will define the bounded cohomology of a group. We will then show how quasi-homomorphisms lead to cocycles in the second bounded cohomology group, and use this to look at the second bounded cohomology of some of our favourite groups. If time permits we will end with some applications.