Warwick Algebraic Topology Seminar 25/26

A list of the seminar talks of the previous years can be found here.

Term 1

The talks will take place on Tuesday at 4pm in B3.03.

Date	Speaker	Affiliation	Title
Oct 7	Matt Booth	Imperial College London	How to invert well-pointed endofunctors
Oct 14	Baylee Schutte	University of Aberdeen	Complex line fields on almost-complex manifolds
Oct 21	Tom Peirce	University of Warwick	Proxy-Small Geometric Functors
Oct 28	Jesse Pajwani	University of Bristol	Arithmetic information in a higher Euler characteristic
Nov 4	Ric Wade	University of Warwick	Mapping class groups of handlebodies are virtual duality groups
Nov 11
Nov 18
Nov 25	Inbar Klang	VU Amsterdam
Dec 2	Markus Land	LMU Munich
Dec 9	Lukas Brantner	University of Oxford

Matt Booth, 7 October

Title: How to invert well-pointed endofunctors.

Abstract: In 1980, Max Kelly showed that many transfinite constructions of free objects can be reduced to the special case of free algebras for well-pointed endofunctors. This yields - in a fairly simple way - a recipe to construct certain kinds of (enriched) localisations. I'll talk about how this works, before observing that this generalises constructions of Keller, Seidel, and Chen--Wang in representation theory and symplectic geometry. I'll relate this (in a rather naive way!) to a very general construction of spectra, which are, similarly, a way to invert endofunctors. If there's time or appetite, I can also say a few things about cospectra.

Baylee Schutte, 14 October

Title: Complex line fields on almost-complex manifolds.

Abstract: The content of this talk is joint work with Nikola Sadovek [arxiv:2411.14161Link opens in a new window]. We study linearly independent complex line fields on almost-complex manifolds, which is a topic of long-standing interest in differential topology and complex geometry. A necessary condition for the existence of such fields is the vanishing of appropriate virtual Chern classes. This condition is also sufficient for the existence of one, two, or three linearly independent complex line fields over certain manifolds. Finally, we apply our results to obtain a refinement of the Schwarzenberger condition that dictates which cohomology classes can be the Chern classes of a complex vector bundle (with prescribed line bundle splitting properties) over complex projective space

Tom Peirce, 21 October

Title: Proxy-Small Geometric Functors

Abstract: Proxy-Smallness is a finiteness property satisfied by geometric functors between tensor-triangulated categories, and in this talk I explain how we use this to generalise the homotopical commutative algebra of Dwyer--Greenlees--Iyengar in particular how it provides a unifying approach to generalising Gorenstein ring spectra. Along the way, I will explain how with proxy-smallness properties such as rigidity and invertibility are reflected in the corresponding torsion category, and on a certain subcategory one recovers a generalised form of Grothendieck duality in the sense of Balmer--Dell'Ambrogio--Sanders. This is joint work with Jordan Williamson.

Jesse Pajwani, 28 October

Title: Arithmetic information in a higher Euler characteristic

Abstract: For k a field, the A1 Euler characteristic, constructed using motivic homotopy theory, furnishes a ring homomorphism K_0(Var_k) -> GW(k), which both refines the classical Euler characteristic of a CW complex and contains arithmetic information. Recent work by Nanavaty and Röndigs shows that this ring homomorphism lifts to a morphism of spectra from the K theory spectrum of varieties, K(Var_k), to the endomorphisms of the motivic sphere spectrum over k. This, in turn, induces maps between higher homotopy groups of these spectra. In this talk, we study the induced morphism on the level of \pi_1. We obtain an explicit homotopical description for this morphism, relate it to an invariant coming from Hermitian K theory, and give a few examples. This is joint work in progress with Ran Azouri, Stephen McKean, and Anubhav Nanavaty.

Ric Wade, 4 November

Title: Mapping class groups of handlebodies are virtual duality groups.

Abstract: Harer showed that mapping class groups of surfaces are virtual duality groups. This implies that there is a twisted isomorphism between the rational homology and cohomology of the group, controlled by a module called the dualising module. In the case of the mapping class group, Harer showed that the dualising module is given by the action on the homology of the curve complex.

In joint work with Petersen we show that mapping class groups of handlebodies are also virtual duality groups, and their dualising modules correspond to the action on the homology of the complex of 'non-simple disc systems'. The result centres around an appropriate contractible submanifold of Teichmüller space on which the group acts cocompactly. The proof proceeds by finding an appropriate stratification of this manifold and applying Quillen’s fibre lemma to analyse the topology of the poset associated to this stratification.