Past Seminars › Seminar Series

Topology Seminar (past)

Mon, 17/12/2012
16:30 		Michael Hopkins (Harvard University, USA) Topology Seminar Add to calendar
Astor Lecture: The homotopy groups of spheres
I will describe the history of the homotopy groups of spheres, and some of the many different roles they have come to play in mathematics.
Mon, 26/11/2012
15:45 		Marc Lackenby (Oxford) Topology Seminar Add to calendar L3
A polynomial upper bound on Reidemeister moves
Consider a diagram of the unknot with c crossings. There is a sequence of Reidemeister moves taking this to the trivial diagram. But how many moves are required? In my talk, I will give an overview of my recent proof that there is there is an upper bound on the number of moves, which is a polynomial function of c.
Mon, 19/11/2012
15:45 		Andrew Sale (Oxford) Topology Seminar Add to calendar L3
Finding Short Conjugators in Wreath Products and Free Solvable Groups

The question of estimating the length of short conjugators in between
elements in a group could be described as an effective version of the
conjugacy problem. Given a finitely generated group $G$ with word metric
$d$, one can ask whether there is a function $f$ such that two elements
$u,v$ in $G$ are conjugate if and only if there exists a conjugator $g$ such
that $d(1,g) \leq f(d(1,u)+d(1,v))$. We investigate this problem in free
solvable groups, showing that f may be cubic. To do this we use the Magnus
embedding, which allows us to see a free solvable group as a subgroup of a
particular wreath product. This makes it helpful to understand conjugacy
length in wreath products as well as metric properties of the Magnus
embedding.
Mon, 12/11/2012
15:45 		Andrew Stacey (Trondheim University and Oxford) Topology Seminar Add to calendar L3
That which we call a manifold ...
It's well known that the mapping space of two finite dimensional manifolds can be given the structure of an infinite dimensional manifold modelled on Frechet spaces (provided the source is compact). However, it is not that the charts on the original manifolds give the charts on the mapping space: it is a little bit more complicated than that. These complications become important when one extends this construction, either to spaces more general than manifolds or to properties other than being locally linear. In this talk, I shall show how to describe the type of property needed to transport local properties of a space to local properties of its mapping space. As an application, we shall show that applying the mapping construction to a regular map is again regular.
Mon, 05/11/2012
15:45 		Noah Snyder (MPI Bonn) Topology Seminar Add to calendar L3
Radford's theorem and the belt trick
Topological field theories give a connection between topology and algebra. This connection can be exploited in both directions: using algebra to construct topological invariants, or using topology to prove algebraic theorems. In this talk, I will explain an interesting example of the latter phenomena. Radford's theorem, as generalized by Etingof-Nikshych-Ostrik, says that in a finite tensor category the quadruple dual functor is easy to understand. It's somewhat mysterious that the double dual is hard to understand but the quadruple dual is easy. Using topological field theory, we show that Radford's theorem is exactly the consequence of the Dirac belt trick in topology. That is, the double dual corresponds to the generator of $ \pi_1(\mathrm{SO}(3)) $ and so the quadruple dual is trivial in an appropriate sense exactly because $ \pi_1(\mathrm{SO}(3)) \cong \mathbb{Z}/2 $. This is part of a large project, joint with Chris Douglas and Chris Schommer-Pries, to understand local field theories with values in the 3-category of tensor categories via the cobordism hypothesis.
Mon, 29/10/2012
15:45 		Oscar Randal-Williams (Cambridge University) Topology Seminar Add to calendar L3
Stable moduli spaces of high dimensional manifolds
I will discuss recent joint work with S. Galatius, in which we generalise the Madsen–Weiss theorem from the case of surfaces to the case of manifolds of higher even dimension (except 4). In the simplest case, we study the topological group $ \mathcal{D}_g $ of diffeomorphisms of the manifold $ \#^g S^n \times S^n $ which fix a disc. We have two main results: firstly, a homology stability theorem—analogous to Harer's stability theorem for the homology of mapping class groups—which says that the homology groups $ H_i(B\mathcal{D}_g) $ are independent of $ g $ for $ 2i \leq g-4 $. Secondly, an identification of the stable homology $ H_*(B\mathcal{D}_\infty) $ with the homology of a certain explicitly described infinite loop space—analogous to the Madsen–Weiss theorem. Together, these give an explicit calculation of the ring $ H^*(B\mathcal{D}_g;\mathbb{Q}) $ in the stable range, as a polynomial algebra on certain explicitly described generators.
Mon, 22/10/2012
15:45 		Shengkui Ye (Oxford) Topology Seminar Add to calendar L3
Matrix group actions on CAT(0) spaces and manifolds
I will talk about the fixed-point properties of matrix groups acting CAT(0) paces, spheres and acyclic manifolds. The matrix groups include general linear groups, sympletic groups, orthogonal groups and classical unitary groups over general rings. We will show that for lower dimensional CAT(0) spaces, the group action of a matrix group always has a global fixed point and that for lower dimensional spheres and acyclic manifolds, a group action by homeomorphisms is always trivial. These results give generalizations of results of Farb concerning Chevalley groups over commutative rings and those of Bridson-Vogtmann, Parwani and Zimmermann concerning the special linear groups SL_{n}(Z) and symplectic groups Sp_{2n}(Z).
Mon, 15/10/2012
15:45 		Tilman Bauer (Oxford and KTH) Topology Seminar Add to calendar L3
A chromatic approach to unstable homotopy
Mon, 08/10/2012
15:45 		Richard Schwartz (Brown University) Topology Seminar Add to calendar L3
Patterns of squares, polytope exchange transformations, and renormalization
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.
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, 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, 14/05/2012
15:45 		Frederic Haglund (Orsay) Topology Seminar Add to calendar L3
Special actions on CAT(0) cube complexes
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, 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, 05/03/2012
15:45 		Andy Tonks (London Metropolitan University) Topology Seminar Add to calendar L3
Unital associahedra and homotopy unital homotopy associative algebras
The classical associahedra are cell complexes, in fact polytopes, introduced by Stasheff to parametrize the multivariate operations naturally occurring on loop spaces of connected spaces. They form a topological operad $ Ass_\infty $ (which provides a resolution of the operad $ Ass $ governing spaces-with-associative-multiplication) and the complexes of cellular chains on the associahedra form a dg operad governing $ A_\infty $-algebras (that is, a resolution of the operad governing associative algebras). In classical applications it was not necessary to consider units for multiplication, or it was assumed units were strict. The introduction of non-strict units into the picture was considerably harder: Fukaya-Ono-Oh-Ohta introduced homotopy units for $ A_\infty $-algebras in their work on Lagrangian intersection Floer theory, and equivalent descriptions of the dg operad for homotopy unital $ A_\infty $-algebras have now been given, for example, by Lyubashenko and by Milles-Hirsch. In this talk we present the "missing link": a cellular topological operad $ uAss_\infty $ of "unital associahedra", providing a resolution for the operad governing topological monoids, such that the cellular chains on $ uAss_\infty $ is precisely the dg operad of Fukaya-Ono-Oh-Ohta. (joint work with Fernando Muro, arxiv:1110.1959, to appear Forum Math)
Mon, 27/02/2012
15:45 		Ieke Moerdijk (Utrecht and Sheffield) Topology Seminar Add to calendar L3
Infinity categories and infinity operads
I will discuss some aspects of the simplicial theory of infinity-categories which originates with Boardman and Vogt, and has recently been developed by Joyal, Lurie and others. The main purpose of the talk will be to present an extension of this theory which covers infinity-operads. It is based on a modification of the notion of simplicial set, called 'dendroidal set'. One of the main results is that the category of dendroidal sets carries a monoidal Quillen model structure, in which the fibrant objects are precisely the infinity operads,and which contains the Joyal model structure for infinity-categories as a full subcategory. (The lecture will be mainly based on joint work with Denis-Charles Cisinski.)
Mon, 20/02/2012
15:45 		Dawid Kielak (Oxford) Topology Seminar Add to calendar L3
Free and linear representations of Out(F_n)
For a fixed n we will investigate homomorphisms Out(F_n) to Out(F_m) (i.e. free representations) and Out(F_n) to GL_m(K) (i.e. K-linear representations). We will completely classify both kinds of representations (at least for suitable fields K) for a range of values $ m $.
Mon, 13/02/2012
15:45 		Karen Vogtmann (Cornell) Topology Seminar Add to calendar L3
The topology and geometry of automorphism groups of free groups II
Free groups, free abelian groups and fundamental groups of closed orientable surfaces are the most basic and well-understood examples of infinite discrete groups. The automorphism groups of these groups, in contrast, are some of the most complex and intriguing groups in all of mathematics. In these lectures I will concentrate on groups of automorphisms of free groups, while drawing analogies with the general linear group over the integers and surface mapping class groups. I will explain modern techniques for studying automorphism groups of free groups, which include a mixture of topological, algebraic and geometric methods.

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

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