Past Seminars › Seminar Series

Topology Seminar (past)

Mon, 21/05
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
15:45 		Frederic Haglund (Orsay) Topology Seminar Add to calendar L3
Special actions on CAT(0) cube complexes
Mon, 30/04
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
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
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
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
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
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.
Mon, 06/02
13:00 		Ruth Charney (Brandeis) Topology Seminar Add to calendar L3
Length functions of right-angled Artin groups
Morgan and Culler proved in the 1980’s that a minimal action of a free group on a tree is completely determined by its length function. This theorem has been of fundamental importance in the study of automorphisms of free groups. In particular, it gives rise to a compactification of Culler-Vogtmann's Outer Space. We prove a 2-dimensional analogue of this theorem for right-angled Artin groups acting on CAT(0) rectangle complexes. (Joint work with M. Margolis)
Mon, 06/02
03:45 		Ian Leary (Southampton) Topology Seminar Add to calendar L3
Variations on a theme of Eilenberg-Ganea

The Eilenberg-Ganea conjecture is the statement that every group of cohomological dimension two admits a two-dimensional classifying space.  This problem is unsolved after 50 years.  I shall discuss the background to this question and negative answers to some other related questions.  This includes recent joint work with Martin Fluch.
Mon, 30/01
15:45 		Chris Cashen Topology Seminar Add to calendar L3
Line Patterns in Free Groups
I will discuss quasi-isometries of the free group that preserve an equivariant pattern of lines. There is a type of boundary at infinity whose topology determines how flexible such a line pattern is. For sufficiently complicated patterns I use this boundary to define a new metric on the free group with the property that the only pattern preserving quasi-isometries are actually isometries.
Mon, 16/01
15:45 		Richard Hepworth (Aberdeen) Topology Seminar Add to calendar L3
String topology of classifying spaces

Chataur and Menichi showed that the homology of the free loop space of the classifying space of a compact Lie group admits a rich algebraic structure: It is part of a homological field theory, and so admits operations parametrised by the homology of mapping class groups.  I will present a new construction of this field theory that improves on the original in several ways: It enlarges the family of admissible Lie groups.  It extends the field theory to an open-closed one.  And most importantly, it allows for the construction of co-units in the theory.  This is joint work with Anssi Lahtinen.
Mon, 28/11/2011
15:45 		Danny Calegari (Cambridge) Topology Seminar Add to calendar L3
Ziggurats and rotation numbers
I will discuss new rigidity and rationality phenomena (related to the phenomenon of Arnold tongues) in the theory of nonabelian group actions on the circle. I will introduce tools that can translate questions about the existence of actions with prescribed dynamics, into finite combinatorial questions that can be answered effectively. There are connections with the theory of Diophantine approximation, and with the bounded cohomology of free groups. A special case of this theory gives a very short new proof of Naimi’s theorem (i.e. the conjecture of Jankins-Neumann) which was the last step in the classification of taut foliations of Seifert fibered spaces. This is joint work with Alden Walker.
Mon, 21/11/2011
15:45 		Brendan Owens (Glasgow) Topology Seminar Add to calendar L3
Concordance groups of links
The concordance group of classical knots C was introduced over 50 years ago by Fox and Milnor. It is a much-studied and elusive object which among other things has been a valuable testing ground for various new topological (and smooth 4-dimensional) invariants. In this talk I will address the problem of embedding C in a larger group corresponding to the inclusion of knots in links.
Mon, 14/11/2011
15:45 		Henry Wilton Topology Seminar Add to calendar L3
One-ended subgroups of graphs of free groups

A longstanding question in geometric group theory is the following. Suppose G is a hyperbolic group where all finitely generated subgroups of infinite index are free. Is G the fundamental group of a surface? This question is still open for some otherwise well understood classes of groups. In this talk, I will explain why the answer is affirmative for graphs of free groups with cyclic edge groups. I will also discuss the extent to which these techniques help with the harder problem of finding surface subgroups.
Mon, 07/11/2011
15:45 		Ric Wade (Oxford) Topology Seminar Add to calendar L3
Right-angled Artin groups and their automorphisms
Automorphisms of right-angled Artin groups interpolate between $ Out(F_n) $ and $ GL_n(\mathbb{Z}) $. An active area of current research is to extend properties that hold for both the above groups to $ Out(A_\Gamma) $ for a general RAAG. After a short survey on the state of the art, we will describe our recent contribution to this program: a study of how higher-rank lattices can act on RAAGs that builds on the work of Margulis in the free abelian case, and of Bridson and the author in the free group case.
Mon, 31/10/2011
15:45 		Ilya Kazachkov (Oxford) Topology Seminar Add to calendar L3
Group actions on real cubings

We introduce the notion of a real cubing. Roughly speaking, real cubings are to CAT(0) cube complexes what real trees are to simplicial trees. We develop an analogue of the Rips’ machine and establish the structure of groups acting nicely on real cubings.
Mon, 24/10/2011
15:45 		Nick Wright (Southampton) Topology Seminar Add to calendar L3
Asymptotic dimension for CAT(0) cube complexes
In this talk I'll explain how to build CAT(0) cube complexes and construct Lipschitz maps between them. The existence of suitable Lipschitz maps is used to prove that the asymptotic dimension of a CAT(0) cube complex is no more than its dimension.
Mon, 17/10/2011
15:45 		Andrew Baker (Glasgow) Topology Seminar Add to calendar L3
BP: Close encounters of the E-infinity kind
The notion of an E-infinity ring spectrum arose about thirty years ago, and was studied in depth by Peter May et al, then later reinterpreted in the framework of EKMM as equivalent to that of a commutative S-algebra. A great deal of work on the existence of E-infinity structures using various obstruction theories has led to a considerable enlargement of the body of known examples. Despite this, there are some gaps in our knowledge. The question that is a major motivation for this talk is `Does the Brown-Peterson spectrum BP for a prime p admit an E-infinity ring structure?'. This has been an important outstanding problem for almost four decades, despite various attempts to answer it. I will explain what BP is and give a brief history of the above problem. Then I will discuss a construction that gives a new E-infinity ring spectrum which agrees with BP if the latter has an E-infinity structure. However, I do not know how to prove this without assuming such a structure!

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

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