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

Term 1

The talks will take place on Tuesday at 5pm in B3.03.

Date	Speaker	Affiliation	Title	Notes
Oct 1	Julie Bergner	University of Virginia	Compatibility of inputs to Waldhausen's S-constuction
Oct 8	Yusuf Baris Kartal	University of Edinburgh	Recovering generalized cohomology from symplectic cohomology
Oct 15	Robin Stoll	University of Cambridge	The stable cohomology of block diffeomorphisms of connected sums of S^k × S^l
Oct 22	Ming Ng	Queen Mary University of London	K1(Var) is generated by Quasi-Automorphisms
Oct 29	Matt Booth	Imperial College London
Nov 5	Marco La Vecchia	University of Warwick
Nov 19
Nov 26	Markus Hausmann	University of Bonn
Dec 3	Irakli Patchkoria	University of Aberdeen

Abstracts

Julie Bergner: Compatibility of inputs to Waldhausen's S-constuction

Campbell and Zakharevich developed the theory of CGW categories to provide a very general framework for doing algebraic K-theory that encompasses examples such as the K-theory of varieties. On the other hand, in joint work with Osorno, Ozornova, Rovelli, and Scheimbauer, we proved that augmented stable double Segal spaces provide a universal input for Waldhausen's S-construction, in such a way that the output has the structure of a 2-Segal space. In joint work with Shapiro and Zakharevich, we show how CGW categories can be characterized in this universal framework, enabling interplay of the features and examples of both approaches.

Yusuf Baris Kartal: Recovering generalized cohomology from symplectic cohomology

Symplectic cohomology is a powerful invariant associated to open symplectic manifolds. It is essential in modern symplectic dynamics and plays an important role in mirror symmetry. However, it is not very sensitive to the homotopy type of the underlying manifold: it can even vanish for manifolds with arbitrarily complicated topology. When a natural filtration and circle action on it is remembered, the rational homology can be recovered as a variant of Tate cohomology, but the torsion information is completely lost. In this talk, I will explain how to recover further information, including torsion part of the homology, complex K-theory and Morava K-theory from an enhanced version of symplectic cohomology. This is joint work with Laurent Cote.

Robin Stoll: The stable cohomology of block diffeomorphisms of connected sums of S^k × S^l

I will explain an identification of the stable rational cohomology of the classifying spaces of self-equivalences as well as block diffeomorphisms of connected sums of S^k × S^l (relative to an embedded disk), where 2 < k < l < 2k–1. The result is expressed in terms of versions of Lie graph complex homology, the constructions of which I will recall. This also leads to a computation, in a range of degrees, of the stable rational cohomology of the classifying spaces of diffeomorphisms of these manifolds. In the case l = k+1, this recovers and extends results of Ebert-Reinhold. If time permits, I will explain parts of the proof; this includes in particular work joint with Berglund on a certain type of algebraic models for relative self-equivalences of bundles, inspired by results of Berglund-Zeman.

Ming Ng: K1(Var) is generated by Quasi-Automorphisms

Our understanding of K-theory is changing. In recent years, much work has been done to extend various tools from algebraic K-theory to various non-additive settings. One particular highlight: in the same way one can define the K-theory spectrum of an exact category, one can construct a K-theory spectrum K(Var) recovering the Grothendieck ring of varieties as ¥pi_0 [Zakharevich, Campbell]. Up until recently, no complete characterisation of K_n(Var) was known except for n=0. This talk will discuss a new result that shows K_1(Var) is generated by an interesting generalisation of automorphisms of varieties, and present its full relations. In our language: given any pCGW category C (a generalisation of exact categories that also includes finite sets, varieties, definable sets, etc.), the group K1(C) is generated by double exact squares (which we also call quasi-automorphisms). Time permitting, we discuss future applications, as well as a technical subtlety regarding how composition of 1-simplices split in K1(Var), and compare this with Zakharevich’s original presentation of K1(Var).