Past 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
24 October 2012
16:00
David Hume
Abstract
<p><span style="color: black; font-family: Tahoma; font-size: x-small;"><span style="font-size: 10pt;">We give a brief overview of hyperbolic metric spaces and the relatively hyperbolic counterparts, with particular emphasis on the quasi-isometry class of trees. We then show that an understanding of the relative version of such spaces - quasi tree-graded spaces -&nbsp; has strong consequences for mapping class groups. In particular, they are shown to embed into a finite product of (possibly infinite valence) simplicial trees. This uses and extends the work of Bestvina, Bromberg and Fujiwara.</span></span></p>
  • Junior Topology and Group Theory Seminar
17 October 2012
16:00
Elisabeth Fink
Abstract
<p><span style="font-size: x-small;"><span style="font-size: 10pt;">I will explain a construction of a group acting on a rooted tree, related to the Grigorchuk group. Those groups have exponential growth, at least under certain circumstances. I will also show how it can be seen that any two elements fulfil a non-trivial relation, implying the absence of non-cyclic free subgroups.</span></span></p>
  • Junior Topology and Group Theory Seminar
10 October 2012
16:00
Henry Bradford
Abstract
The study of free groups and their automorphisms has a long pedigree, going back to the work of Nielsen and Dehn in the early 20th century, but in many ways the subject only truly reached maturity with the introduction of Outer Space by Culler and Vogtmann in the “Big Bang” of 1986. In this (non-expert) talk, I will walk us through the construction of Outer Space and some related complexes, and survey some group-theoretic applications.
  • Junior Topology and Group Theory Seminar
26 September 2012
16:00
Anitha Thillaisundaram
Abstract
<p>We use Schlage-Puchta's concept of p-deficiency and Lackenby's property of p-largeness to show that a group having a finite presentation with p-deficiency greater than 1 is large. What about when p-deficiency is exactly one? We also generalise a result of Grigorchuk on Coxeter groups to odd primes.</p>
  • Junior Topology and Group Theory Seminar
25 April 2012
16:00
Moritz Rodenhausen
Abstract

A construction by McCool gives rise to a finite presentation for the stabiliser of a finite set of conjugacy classes in a free group under the action of Aut(F_n) or Out(F_n). An important concept of my talk are rigid elements, which will allow to simplify these huge presentations. Finally I will sketch applications to centralisers in Aut(F_n).

  • Junior Topology and Group Theory Seminar

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