Forthcoming Seminars › Seminar Series

Topology Seminar

Mon, 23/04/2012
15:45 		Lukasz Grabowksi (Imperial) Topology Seminar Add to calendar L3
On the decidability of the zero divisor problem
Let G be a finitely generated group generated by g_1,..., g_n. Consider the alphabet A(G) consisting of the symbols g_1,..., g_n and the symbols '+' and '-'. The words in this alphabet represent elements of the integral group ring Z[G]. In the talk we will investigate the computational problem of deciding whether a word in the alphabet A(G) determines a zero-divisor in Z[G]. We will see that a version of the Atiyah conjecture (together with some natural assumptions) imply decidability of the zero-divisor problem; however, we'll also see that in the group (Z/2 \wr Z)^4 the zero-divisor problem is not decidable. The technique which allows one to see the last statement involves "embedding" a Turing machine into a group ring.
Mon, 30/04/2012
15:45 		Martin Palmer (Oxford) Topology Seminar Add to calendar L3
Configuration spaces and homological stability
For a fixed background manifold $ M $ and parameter-space $ X $, the associated configuration space is the space of $ n $-point subsets of $ M $ with parameters drawn from $ X $ attached to each point of the subset, topologised in a natural way so that points cannot collide. One can either remember or forget the ordering of the n points in the configuration, so there are ordered and unordered versions of each configuration space. It is a classical result that the sequence of unordered configuration spaces, as $ n $ increases, is homologically stable: for each $ k $ the degree-$ k $ homology is eventually independent of $ n $. However, a simple counterexample shows that this result fails for ordered configuration spaces. So one could ask whether it's possible to remember part of the ordering information and still have homological stability. The goal of this talk is to explain the ideas behind a positive answer to this question, using 'oriented configuration spaces', in which configurations are equipped with an ordering - up to even permutations - of their points. I will also explain how this case differs from the unordered case: for example the 'rate' at which the homology stabilises is strictly slower for oriented configurations. If time permits, I will also say something about homological stability with twisted coefficients.
Mon, 14/05/2012
15:45 		Frederic Haglund (Orsay) Topology Seminar Add to calendar L3
Special actions on CAT(0) cube complexes
Mon, 21/05/2012
15:45 		Cornelia Drutu (Oxford) Topology Seminar Add to calendar L3
Equivalent notions of rank for manifolds of non-positive curvature and for mapping class groups of surfaces
In Riemannian geometry there are several notions of rank defined for non-positively curved manifolds and with natural extensions for groups acting on non-positively curved spaces. The talk shall explain how various notions of rank behave for mapping class groups of surfaces. This is joint work with J. Behrstock.
Mon, 28/05/2012
15:45 		Marc Lackenby (Oxford) Topology Seminar Add to calendar L3
Links with splitting number one
The unknotting number of a knot is an incredibly difficult invariant to compute. In fact, there are many knots which are conjectured to have unknotting number 2 but for which no proof of this is currently available. It therefore remains an unsolved problem to find an algorithm that determines whether a knot has unknotting number one. In my talk, I will show that an analogous problem for links is soluble. We say that a link has splitting number one if some crossing change turns it into a split link. I will give an algorithm that determines whether a link has splitting number one. (In the case where the link has two components, we must make a hypothesis on their linking number.) The proof that the algorithm works uses sutured manifolds and normal surfaces.
Mon, 28/05/2012
15:45 		Marc Lackenby (Oxford) Topology Seminar Add to calendar L3
Links with splitting number one

 The unknotting number of a knot is an incredibly difficult invariant to compute.
In fact, there are many knots which are conjectured to have unknotting number 2 but for
which no proof of this is currently available. It therefore remains an unsolved problem to find an
algorithm that determines whether a knot has unknotting number one. In my talk, I will
show that an analogous problem for links is soluble. We say that a link has splitting number
one if some crossing change turns it into a split link. I will give an algorithm that
determines whether a link has splitting number one. (In the case where the link has
two components, we must make a hypothesis on their linking number.) The proof
that the algorithm works uses sutured manifolds and normal surfaces.
Mon, 11/06/2012
15:45 		Piotr Przytycki (Warsaw) Topology Seminar Add to calendar L3
Mixed 3-manifolds are virtually special
This is joint work with Dani Wise and builds on his earlier work. Let M be a compact oriented irreducible 3-manifold which is neither a graph manifold nor a hyperbolic manifold. We prove that the fundamental group of M is virtually special. This means that it virtually embeds in a right angled Artin group, and is in particular linear over Z.

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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