Past Junior Topology and Group Theory Seminar

3 December 2014
16:00
Charles Cox
Abstract

Deciding whether or not two elements of a group are conjugate might seem like a trivial problem. However, there exist finitely presented groups where this problem is undecidable: there is no algorithm to output yes or no for any two elements chosen. In this talk Houghton groups (a family of groups all having solvable conjugacy problem) will be introduced as will the idea of twisted conjugacy: a generalisation of the conjugacy problem where an automorphism is also given. This will be our main tool in answering whether finite extensions and finite index subgroups of any Houghton group have solvable conjugacy problem.

  • Junior Topology and Group Theory Seminar
19 November 2014
16:00
Federico Vigolo
Abstract

In 1983 Kerckhoff settled a long standing conjecture by Nielsen proving that every finite subgroup of the mapping class group of a compact surface can be realized as a group of diffeomorphisms. An important consequence of this theorem is that one can now try to study subgroups of the mapping class group taking the quotient of the surface by these groups of diffeomorphisms. In this talk we will study quotients of surfaces under the action of a finite group to find bounds on the cardinality of such a group.

  • Junior Topology and Group Theory Seminar
12 November 2014
16:00
Giles Gardam
Abstract

The Dehn function of a group measures the complexity of the group's word problem, being the upper bound on the number of relations from a group presentation required to prove that a word in the generators represents the identity element. The Filling Theorem which was first stated by Gromov connects this to the isoperimetric functions of Riemannian manifolds. In this talk, we will see the classification of hyperbolic groups as those with a linear Dehn function, and give Bowditch's proof that a subquadratic isoperimetric inequality implies a linear one (which gives the only gap in the "isoperimetric spectrum" of exponents of polynomial Dehn functions).

  • Junior Topology and Group Theory Seminar
5 November 2014
16:00
Alexander Margolis
Abstract

We will give an outline of the proof by Kahn and Markovic who showed that a closed hyperbolic 3-manifold $\textbf{M}$ contains a closed $\pi_1$-injective surface. This is done using exponential mixing to find many pairs of pants in $\textbf{M}$, which can then be glued together to form a suitable surface. This answers a long standing conjecture of Waldhausen and is a key ingredient in the proof of the Virtual Haken Theorem.

  • Junior Topology and Group Theory Seminar
29 October 2014
16:00
Gareth Wilkes
Abstract

Dinits, Karzanov and Lomonosov showed that the minimal edge cuts of a finite graph have the structure of a cactus, a tree-like graph constructed from cycles. Evangelidou and Papasoglu extended this to minimal cuts separating the ends of an infinite graph. In this talk we will discuss a similar structure theorem for minimal vertex cuts separating the ends of a graph; these can be encoded by a succulent, a mild generalization of a cactus that is still tree-like.

  • Junior Topology and Group Theory Seminar
18 June 2014
17:00
Henry Bradford
Abstract


I shall outline a general method for finding upper bounds on the diameters of finite groups, based on the Solovay-Kitaev procedure from quantum computation. This method may be fruitfully applied to groups arising as quotients of many familiar pro-p groups. Time permitting, I will indicate a connection with weak spectral gap, and give some applications.

  • Junior Topology and Group Theory Seminar
18 June 2014
16:00
Simon Gritschacher
Abstract

Waldhausen defined higher K-groups for categories with certain extra structure. In this talk I will define categories with cofibrations and weak equivalences, outline Waldhausen's construction of the associated K-Theory space, mention a few important theorems and give some examples. If time permits I will discuss the infinite loop space structure on the K-Theory space.

  • Junior Topology and Group Theory Seminar
28 May 2014
16:00
Thomas Wasserman
Abstract
A one hour introduction to topological K-theory, that nifty generalised cohomology theory that is built starting from the semi-ring of vector bundles over a space. As I'll need it on Thursday I'll also explain a model for K-theory in terms of difference bundles, and, if time permits, its connection with Clifford algebras.
  • Junior Topology and Group Theory Seminar

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