Past Junior Topology and Group Theory Seminar

9 March 2016
16:00
Renee Hoekzema
Abstract

Orientable manifolds can only have an odd Euler characteristic in dimensions divisible by 4. I will prove the analogous result for spin and string manifolds, where the dimension can only be a multiple of 8 and 16 respectively. The talk will require very little background. I'll go over the definition of spin and string structures, discuss cohomology operations and Poincare duality.

  • Junior Topology and Group Theory Seminar
2 March 2016
16:00
Alex Margolis
Abstract

I will present a basic overview of finiteness conditions, group cohomology, and related quasi-isometry invariance results. In particular, I will show that if a group satisfies certain finiteness conditions, group cohomology with group ring coefficients encodes some structure of the `homology at infinity' of a group. This is seen for hyperbolic groups in the work of Bestvina-Mess, which relates the group cohomology to the Čech cohomology of the boundary.

  • Junior Topology and Group Theory Seminar
20 January 2016
16:00
Federico Vigolo
Abstract

I will illustrate how to build families of expanders out of 'very mixing' actions on measure spaces. I will then define the warped cones and show how these metric spaces are strictly related with those expanders.

  • Junior Topology and Group Theory Seminar
2 December 2015
16:00
Nicolaus Heuer
Abstract

Quasihomomorphisms (QHMs) are maps $f$ between groups such that the
homomorphic condition is boundedly satisfied. The case of QHMs with
abelian target is well studied and is useful for computing the second
bounded cohomology of groups. The case of target non-abelian has,
however, not been studied a lot.

We will see a technique for classifying QHMs $f: G \rightarrow H$ by Fujiwara and
Kapovich. We will give examples (sometimes with proofs!) for QHM in
various cases such as

  • the image $H$  hyperbolic groups,
  • the image $H$ discrete rank one isometries,
  • the preimage $G$ cyclic / free group, etc.

Furthermore, we point out a relation between QHM and extensions by short
exact sequences.

  • Junior Topology and Group Theory Seminar
25 November 2015
16:00
Federico Vigolo
Abstract

A family of expanders is a sequence of finite graphs which are both sparse and highly connected. Firstly defined in the 80s, they had huge applications in applied maths and computer science. Moreover, it soon turned out that they also had deep implications in pure maths. In this talk I will introduce the expander graphs and I will illustrate a way to construct them by approximating actions of groups on probability spaces.

  • Junior Topology and Group Theory Seminar

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