Past Junior Topology and Group Theory Seminar

6 February 2013
16:00
Montserrat Casals-Ruiz
Abstract
<p><span style="font-size: x-small;"><span style="font-size: 10pt;">The theory of equations over groups goes back to the very beginning of group theory and is linked to many deep problems in mathematics, such as the Diophantine problem over rationals. In this talk, we shall survey some of the key results on equations over groups, give an outline of the Makanin-Razborov process (an algorithm for solving equations over free groups) and its connections to other results in group theory and low-dimensional topology. <br /></span></span></p>
  • Junior Topology and Group Theory Seminar
30 January 2013
16:00
David Hume; Robert Kropholler; Martin Palmer and Alessandro Sisto
Abstract
<p><span style="font-size: x-small;"><span style="font-size: 10pt;">We will discuss (very) recent work by Hensel; Przytycki and Webb, who describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces. </span></span></p>
  • Junior Topology and Group Theory Seminar
30 January 2013
12:00
Lukasz Grabowski
Abstract
<p><span style="font-size: x-small;"><span style="font-size: 10pt;">The eponymous result is due to Bridson and Vogtmann, and was proven in their paper "Automorphisms of Automorphism Groups of Free Groups" (Journal of Algebra 229). While I'll remind you all the basic definitions, it would be very helpful to be already somewhat familiar with the outer space. </span></span></p>
  • Junior Topology and Group Theory Seminar
3 December 2012
(All day)
Diana Davis
Abstract

We will start with the square torus, move on to all regular polygons, and then look at a large family of flat surfaces called Bouw-Möller surfaces, made by gluing together many polygons. On each surface, we will consider the action of a certain shearing action on geodesic paths on the surface, and a certain corresponding sequence.

  • Junior Topology and Group Theory Seminar
28 November 2012
16:00
Will Cavendish
Abstract

A subgroup $H$ of a group $G$ is said to be engulfed if there is a
finite-index subgroup $K$ other than $G$ itself such that $H<K$, or
equivalently if $H$ is not dense in the profinite topology on $G$.  In
this talk I will present a variety of methods for showing that a
subgroup of a discrete group is engulfed, and demonstrate how these
methods can be used to study finite-sheeted covering spaces of
topological spaces.

  • Junior Topology and Group Theory Seminar
21 November 2012
16:00
Andrew Sale
Abstract
<p><span>Let F be a free group, and N a normal subgroup of F with derived subgroup N'. The Magnus embedding gives a way of seeing F/N' as a subgroup of a wreath product of a free abelian group over over F/N. The aim is to show that the Magnus embedding is a quasi-isometric embedding (hence "Q.I." in the title). For this I will use an alternative geometric definition of the embedding (hence "picture"), which I will show is equivalent to the definition which uses Fox calculus. Please note that we will assume no prior knowledge of calculus.</span></p>
  • Junior Topology and Group Theory Seminar
15 November 2012
16:30
Søren Fuglede Jørgensen
Abstract
In St John's College <p><span style="background-color: white;">In this part, I will redefine the quantum representations for $G = SU(2)$ making no mention of flat connections at all, instead appealing to a purely combinatorial construction using the knot theory of the Jones polynomial.<br /> <br /> Using these, I will discuss some of the properties of the representations, their strengths and their shortcomings. One of their main properties, conjectured by Vladimir Turaev and proved by Jørgen Ellegaard Andersen, is that the collection of the representations forms an infinite-dimensional faithful representation. As it is still an open question whether or not mapping class groups admit faithful finite-dimensional representations, it becomes natural to consider the kernels of the individual representations. Furthermore, I will hopefully discuss Andersen's proof that mapping class groups of closed surfaces do not have Kazhdan's Property (T), which makes essential use of quantum representations.</span></p>
  • Junior Topology and Group Theory Seminar
31 October 2012
16:00
Jason Semeraro
Abstract
<p><span style="font-size: x-small;"><span style="font-size: 10pt;">Saturated fusion systems are a next generation approach to the theory of finite groups- one major motivation being the opportunity to borrow techniques from homotopy theory. Extending work of Broto, Levi and Oliver, we introduce a new object - a 'tree of fusion systems' and give conditions (in terms of the orbit graph) for the completion to be saturated. We also demonstrate that these conditions are 'best possible' by producing appropriate counterexamples. Finally, we explain why these constructions provide a powerful way of building infinite families of fusion systems which are exotic (i.e. not realisable as the fusion system of a finite group) and give some concrete examples.</span></span></p>
  • Junior Topology and Group Theory Seminar

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