# Past Junior Topology and Group Theory Seminar

13 February 2013
16:00
Martin Palmer
Abstract
<p><span>First of all, I will give an overview of what the phenomenon of homological stability is and why it's useful, with plenty of examples. I will then introduce configuration spaces -- of various different kinds -- and give an overview of what is known about their homological stability properties. A "configuration" here can be more than just a finite collection of points in a background space: in particular, the points may be equipped with a certain non-local structure (an "orientation"), or one can consider unlinked embedded copies of a fixed manifold instead of just points. If by some miracle time permits, I may also say something about homological stability with local coefficients, in general and in particular for configuration spaces. <br /></span></p>
• 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
23 January 2013
16:00
John Mackay
Abstract
<p><span style="font-size: x-small;"><span style="font-size: 10pt;">I'll discuss work of Wise and Ollivier-Wise that gives cubulations of certain small cancellation and random groups, which in turn shows that they do not have property (T).</span></span></p>
• Junior Topology and Group Theory Seminar
16 January 2013
16:00
Robert Kropholler
Abstract
<p>I will be looking at some conjectures and theorems closely related to the h-cobordism theorem and will try to show some connections between them and some group theoretic conjectures. </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