Topology Seminars Spring 2014

Monday 03 February
2014 The topology of Stein fillable contact manifolds in higher dimensions
Diarmuid Crowley (Max Planck Institute, Bonn)
4pm FRANK ADAMS 2Abstract (click to view)An almost contact manifold M is a closed oriented (2q+1)manifold with a reduction of its structure group to U(q). It is an open question in dimensions 7 and higher whether every almost contact manifold admits an actual contact structure. A special class of contact structures arise when M is the boundary of a Stein domain and Eliashberg's hprinciple, a deep result in the subject, characterises Stein domains. In this talk I will report on a joint project with Jonathan Bowden and Andras Stipsicz where we organise Eliashberg's hprinciple in the setting of Kreck's modified surgery. As a consequence, we obtain a bordismtheoretic characterisation of which almost contact manifolds admit Stein fillings. As an application, we show that every simply connected almost contact 7manifold with torsion free second homotopy admits a Stein filling.

Monday 10 February
2014 Towards the GrothendieckTeichmuller Group
Goran Malic (University of Manchester)
4pm FRANK ADAMS 2Abstract (click to view)One of the most mysterious objects in mathematics is the absolute Galois group G:=Gal(Q/Q), the group of automorphisms of Q which fix Q pointwise. This group encodes Galois theory over Q. From 1972 to about 1984 Grothendieck sketched out an approach to studying this group via its action on Dehn twists and connected bipartite graphs cellularly embedded onto compact and closed Riemann surfaces. In 1990 Drinfeld constructed a certain group called the GrothendieckTeichmuller group, denoted by GT, which acts on a certain braided tensor category. It is conjectured that G and GT are isomorphic. In this talk I will describe some elementary aspects of the action of G on bipartite graphs cellularly embededd onto Riemann surfaces, and demonstrate an elementary construction of GT.

Monday 17 February
2014 No Seminar
Room in use for President's visit.
4pm FRANK ADAMS 2

Monday 24 February
2014 An introduction to stunted weighted projective space
Beverley O'Neill (University of Manchester)
4pm FRANK ADAMS 2Abstract (click to view)Weighted projective spaces provide the most basic examples of toric orbifolds. Although they belong to such a restricted class of algebraic varieties, their simplicity has made them attractive objects to study in algebraic and differential geometry as well as theoretical physics. Their integral cohomology ring was computed in the pioneering work of Kawasaki in 1973 but has a chaotic feel. For many years since then algebraic topologists have paid little attention to weighted projective spaces. By bringing some order to Kawasaki's work, I will introduce stunted weighted projective space, a generalisation of stunted complex projective space, and extend his results to compute their integral cohomology rings. It is possible to impose a CWstructure and identify the corresponding homology generators in terms of cellular cycles, but this is usually rather complicated. If time permits, I shall give a brief overview of how it may be achieved.

Monday
03
March
2014 Seminar Cancelled, owing to unforseen circumstances
4pm in FRANK ADAMS 2Abstract (click to view) 
Monday
10
March
2014 Thom spaces and Thom isomorphisms; Spin, Spin^C and all that
Nigel Ray (University of Manchester)
4pm in FRANK ADAMS 2Abstract (click to view)A remarkable number of spaces that arise in complex and quaternionic geometry are related in some way or other to Thom spaces of vector bundles. Such relationships often give excellent insight into the deeper topological structure of the spaces in question; for example, they may facilitate computations of their E*() cohomology rings. Here, E ranges from the basic examples of singular cohomology with Z/2 or Z coefficients, via real and complex K theory, through to complex and quaternionic cobordism. The key ingredient in all these cases is the Thom isomorphism  whose existence depends on the orientability of the bundle with respect to E*(). Conditions for orientability vary from theory to theory, and lead us naturally to the Lie groups Spin and SpinC. Rather than focus on the details of these ideas, I shall try to interest the audience in more general aspects of key examples (which can be difficult to extract from the literature).

Monday
17
March
2014 The toric structure of (2n,k)manifolds
Victor M Buchstaber (Moscow State University and The Steklov Institute)
4pm in FRANK ADAMS 2Abstract (click to view)This talk is based on recent results obtained with Svjetlana Terzic. It aims to axiomatise a notion of generalised quasitoric manifold, so as to include rich and combinatorially beautiful examples such as complex Grassmannians. We apply and develop methods and results from the algebraic geometry of homogeneous spaces, and from toric topology

Monday
24
March
2014 Moduli spaces of labelled graphs
James Griffin (University of Glasgow)
4pm in FRANK ADAMS 2Abstract (click to view)Although Aut(Fr) has a finite simplicial set as a classifying space, its homology is extremely difficult to calculate and the problem just gets worse as the rank r increases. However by a result of Hatcher and Vogtmann the homology is known to be stable, and Galatius computed this stable homology to be that of the sphere spectrum. More generally Hatcher and Wahl conjectured that automorphism groups Aut(H*G*...*G) of free products of groups are homologically stable. I'll prove this via a moduli space of labelled graphs and a little category theory.

Monday
31
March
2014 Topology of spaces of symmetric loops
James Montaldi (University of Manchester)
4pm in FRANK ADAMS 2Abstract (click to view)There is a classical result that the set of connected components of the loop space of a manifold is in 11 correspondence with the conjugacy classes of the fundamental group of the manifold. Motivated by the study of symmetries of planar choreographies, I will describe the extension to symmetric loops (periodic orbits) of this result, when the manifold has a group action. This is joint work with my PhD student Katie Steckles.

Monday
28
April
2014 An introduction to homotopy type theory
James Cranch (University of Sheffield)
4pm in FRANK ADAMS 2Abstract (click to view)Homotopy type theory is an recent idea under vigorous development, connecting homotopy theory to aspects of logic. While the basic theory is not yet complete, it is already providing applications to programming language design. From the point of view of pure mathematics, it offers an novel foundation for homotopy theory, with the intriguing feature that spaces are more fundamental objects than sets. I'll give a purely introductory talk geared up for topologists: I'll use some of the language of homotopy theory, but assume nothing whatsoever about type theory.

Monday
5
May
2014 No Seminar  Bank Holiday
4pm in FRANK ADAMS 2Abstract (click to view) 
Monday
12
May
2014 Adjunctions as homotopy
Amit Kuber and David Wilding (Double act  University of Manchester)
4pm in FRANK ADAMS 2Abstract (click to view)When treating (small) categories as topological spaces (obtained by realizing the nerve of the category), adjunctions yield homotopy equivalence. Since homotopy is an equivalence relation, one forgets about which adjoint is the left one. In this talk, we will explain some ideas from our joint work over the past year about an ordered version of homotopy theory. This definition of homotopy uses adjoints in a way that distinguishes between left and right adjoints. We will describe a way to assign a sequence of posets to a (small) category, and we will explain their significance using several examples from the (2) category of posets. We will also discuss the classes of maps on which this assignment is functorial. These constructions do not generalise the topological homotopy theory and are not based on any model structure; they are supposed to capture a different notion of deformation of a map into another.

Monday
19
May
2014 Global relations for toric singularities
Oleg Karpenkov (University of Liverpool)
4pm in FRANK ADAMS 2Abstract (click to view)In this talk we will discuss a link between geometry of continued fractions and global relations for singularities of complex projective toric surfaces. The results are based on recent development of lattice trigonometric functions that are invariant with respect to Aff(2,Z)group action.
Seminars will begin this semester on Monday 3 February. They will be held at 4.00pm in Frank Adams 2, preceded by highclass tea, coffee and biscuits from 3.30pm on the Atrium Bridge. We will visit Sandbar after each meeting, for refreshments and further discussion...
Further information
For further information please contact the seminar organiser.