# Warwick Algebraic Topology Seminar 24/25

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 | Calabi-Yau structures and Koszul duality | |

Nov 5 | Marco La Vecchia | University of Warwick | Twisted equivariant Chern Classes and Equivariant Formal Group Laws | |

Nov 12 | John Greenlees | University of Warwick | An algebraic model for rational SU(3)-spectra in 18 blocks | |

Nov 19 | Martin Gallauer | University of Warwick | ||

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).

###### Matt Booth: Calabi-Yau structures and Koszul duality

I'll give a reminder of Koszul duality, before talking about a generalised notion of Calabi-Yau structure for dg (co)algebras and indicating why it is Koszul dual to a symmetric Frobenius condition. There is also an analogous one-sided version: Gorenstein (co)algebras are Koszul dual to Frobenius (co)algebras. This leads to a surprising example: the ring k[[x]] of formal power series, equipped with its natural topology, is a pseudocompact Frobenius algebra. As an application of the above theory, we obtain a new characterisation of Poincaré duality spaces, which for simply connected spaces recovers Félix-Halperin-Thomas's notion of Gorenstein space. This is joint work with Joe Chuang and Andrey Lazarev, to appear on the ArXiv soon.

###### Marco La Vecchia: Twisted equivariant Chern Classes and Equivariant Formal Group Laws

Chern classes are classical invariants that play a fundamental role in the study of vector bundles. Recently, Schwede introduced $U(n)$-equivariant Chern classes within the context of equivariant bordism. In this talk, I will extend this framework by defining $G$-twisted equivariant Chern classes in the setting of $G \times U(n)$-equivariant bordism. I will then explore the connection between these twisted equivariant Chern classes and the theory of $G$-equivariant formal group laws, where $G$ is any compact Lie group.

###### John Greenlees: An algebraic model for rational SU(3)-spectra in 18 blocks

For each compact Lie group G, one may hope to construct an algebraic category A(G) which is Quillen equivalent to the category of rational G-equivariant cohomology theories. A(G) takes the form of a category of sheaves over a space X_G of conjugacy classes of subgroups of G. When G is SU(3) there is a partition of X_G into 18 blocks, over each of which one may make A(G) explicit. This example is small enough to be explicit and large enough to illustrate some general techniques.