Past Junior Topology and Group Theory Seminar

17 June 2021
10:00
Mireille Soergel
Abstract

We introduce the notion of systolic complexes and give conditions on presentations to construct such complexes using Cayley graphs.

We consider Garside groups to find examples of groups admitting such a presentation.
 

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  • Junior Topology and Group Theory Seminar
10 June 2021
10:00
Thibault Decoppet
Abstract

The goal of this talk is to present some elementary examples of fusion 2-categories whilst doing as little higher category theory as possible. More precisely, it turns out that up to a canonical completion operation, certain higher fusion categories are entirely described by their maximal subspaces. I will briefly motivate this completion operation in the 1-categorical case, and go on to explain why working with spaces is good enough in this particular case. Then, we will review some fact about $E_n$-algebras, and why they come into the picture. Finally, we will have a look at some small examples arising from finite groups.

  • Junior Topology and Group Theory Seminar
3 June 2021
17:00
Jonathan Fruchter
Abstract

Line patterns in free groups are collections of lines in the Cayley graph of a non-abelian free group F, which correspond to finite sets of words in F. Following Cashen and Macura, we will discuss line patterns by looking at the topology of Decomposition Spaces, which are quotients of the boundary of F that correspond to the different line patterns. Given a line pattern, we will also construct a cube complex whose isometry group is isomorphic to the group of quasi isometries of F which (coarsely) preserve the line pattern. This is a useful tool for studying the quasi isometric rigidity of related groups.

  • Junior Topology and Group Theory Seminar
20 May 2021
10:00
Abstract

For a group G and a finite dimensional linear representation σ : G → GLn(D) over a skew field (division ring) D, the agrarian invariants with respect to σ are the homological invariants of G with coefficient module Dn. In this talk I will discuss the relationship between agrarian invariants, L 2 -invariants, Thurston norm and twisted Alexander polynomials. I will also discuss an ongoing work with Dawid Kielak.

  • Junior Topology and Group Theory Seminar
13 May 2021
10:00
Daniel Woodhouse
Abstract

The conformal dimension of a hyperbolic group is a powerful numeric quasi-isometry invariant associated to its boundary.

As an invariant it is finer than the topological dimension and allows us to distinguish between groups with homeomorphic boundaries.

I will start by talking about what conformal geometry even is, before discussing how this connects to studying the boundaries of hyperbolic groups.

I will probably end by saying how jolly hard it is to compute.

 

  • Junior Topology and Group Theory Seminar
6 May 2021
10:00
Abstract

In this talk I will introduce the study of lattices in locally compact groups through their actions CAT(0) spaces. This is an extremely rich class of groups including S-arithmetic groups acting on products of symmetric spaces and buildings, right angled Artin and Coxeter groups acting on polyhedral complexes, Burger-Mozes simple groups acting on products of trees, and the recent CAT(0) but non biautomatic groups of Leary and Minasyan. If time permits I will discuss some of my recent work related to the Leary-Minasyan groups.

The join button will be published on the right (Above the view all button) 30 minutes before the seminar starts (login required).

  • Junior Topology and Group Theory Seminar
28 April 2021
10:00
Filippos Sytilidis
Abstract

The graph complex is a remarkable object with very rich structure and many, sometimes mysterious, connections to topology. To illustrate one such connection, I will attempt to construct a “self-linking” invariant of knots and expand on the ideas behind it.

The join button will be published on the right (Above the view all button) 30 minutes before the seminar starts (login required).

  • Junior Topology and Group Theory Seminar
10 March 2021
10:00
David Sheard
Abstract

In the world of finitely generated groups, presentations are a blessing and a curse. They are versatile and compact, but in general tell you very little about the group. Tietze transformations offer much (but deliver little) in terms of understanding the possible presentations of a group. I will introduce a different way of transforming presentations of a group called a Nielsen transformation, and show how topological methods can be used to study Nielsen transformations.

  • Junior Topology and Group Theory Seminar
3 March 2021
10:00
Marco Barberis
Abstract

Since its introduction in 1978 the curve complex has become one of the most important objects to study surfaces and their homeomorphisms. The curve complex is defined only using data about curves and their disjointness: a stunning feature of it is the fact that this information is enough to give it a rigid structure, that is every symplicial automorphism is induced topologically. Ivanov conjectured that this rigidity is a feature of most objects naturally associated to surfaces, if their structure is rich enough.

During the talk we will introduce the curve complex, then we will focus on its rigidity, giving a sketch of the topological constructions behind the proof. At last we will talk about generalisations of the curve complex, and highlight some rigidity results which are clues that Ivanov's Metaconjecture, even if it is more of a philosophical statement than a mathematical one, could be "true".

  • Junior Topology and Group Theory Seminar

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