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Warwick Algebraic Topology Seminar 24/25

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

Term 2

The talks will take place on Tuesdays at 4pm in B3.02.

Date Speaker Affiliation
Title
Notes
Jan 7        
Jan 14        
Jan 21 Sofia Marlasca Aparicio Oxford Ultrasolid Geometry and Deformation Theory  
Jan 28 Paul Balmer UCLA Completion in tensor-triangular geometry  
Feb 4 Jérôme Scherer EPFL Module spaces over homology theories and completion towers  
Feb 11 Ran Levi Aberdeen Foundations of Differential Calculus for modules over Posets  
Feb 18 Benjamin Dünzinger Regensburg Categorified Assembly  
Feb 25 Niall Taggart Nijmegen    
Mar 4 Oliver Röndigs Osnabrück    
Mar 11 Teena Gerhardt Michigan State    
Abstracts
Sofia Marlasca Aparicio: Ultrasolid Geometry and Deformation Theory

We will introduce ultrasolid modules, a variant of the solid modules of Clausen and Scholze, which generalise complete modules over a field k. In this setting, we show some commutative algebra results like an ultrasolid variant of Nakayama's lemma. We also explore higher algebra in the form of animated and E∞ ultrasolid k-algebras. Finally, we will apply this to generalise the Lurie-Schlessinger criterion in deformation theory (no previous knowledge of deformation theory is assumed).

Paul Balmer: Completion in tensor-triangular geometry

We'll review the notion of completion, à la Bousfield, and discuss its significance in tensor-triangular geometry. The problem appears in the very fundamental situation where one tries to patch the spectrum together from a partition consisting of an open and its closed complement. We shall also discuss examples, in commutative algebra and in representation theory of finite groups.

Jérôme Scherer: Module spaces over homology theories and completion towers

This is joint work with W. Chacholski and W. Pitsch. To any coaugmented functor X=>EX one can associate its "modules", i.e. spaces X which are retracts of EX. When E is ordinary homology, this functor is the infinite symmetric product and modules are generalized Eilenberg-Mac Lane spaces (GEMs). Given two such functors E and F we introduce a new one, we denote by a bracket [E, F], as a pullback of two natural transformations E=>EF and F=>EF. We study modules over such brackets and construct two towers by bracketing systematically either on the left or the right. For ordinary homology we obtain two well-known towers: the Bousfield-Kan completion tower and the modified one as introduced by Dror Farjoun.

Ran Levi: Foundations of Differential Calculus for Modules over Posets

Let 𝑘 be a field and let C be a small category. A 𝑘-linear representation of C, or a 𝑘C-module, is a functor from C to the category of finite dimensional vector spaces over 𝑘. Unsurprisingly, it turns out that when the category C is more general than a linear order, then its representation type is generally infinite and in most cases wild. Hence the task of understanding such representations in terms of their indecomposable factors becomes difficult at best, and impossible in general. In this talk I will report on a project, joint with Jacek Brodzki, Henri Rihiimaki and Markus Kelmetti, which offers a new set of ideas designed to enable studying modules locally. Specifically, inspired by work in discrete calculus on graphs, we set the foundations for a calculus type analysis of 𝑘C-modules, under some restrictions on the category C.

Benjamin Dünzinger: Categorified Assembly

In the context of algebraic K-theory, a classical question is whether for a group G the assembly map H(BG,K(S)) -> K(S[G]) is split injective. This talk aims to show that, in some cases, this result can be proven by establishing a corresponding decomposition of localizing motives for categories with G-action. I will explain that this method works under the assumption of Carlsson-Pedersen that an appropriate equivariant compactification of a model of EG exists. One can interpret this method as an analogue of the Gamma element method, which is used to prove split injectivity of the Baum-Connes assembly map. The talk is based on joint work with Georg Lehner.

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 Derived commutative algebraic geometry  
Nov 26 Markus Hausmann University of Bonn The universal property of bordism rings of manifolds with commuting involutions  
Dec 3 Irakli Patchkoria University of Aberdeen On the Farrell-Tate K-theory of Out(F_n)  
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.

Martin Gallauer: Derived commutative algebraic geometry

Venerable algebraic geometry (AG) has many descendants, including relatively recently "derived noncommutative AG" (Kapranov, Bondal, Orlov, Kontsevich,...). In this expository talk I will discuss an even younger one that one might call "derived commutative AG" and which Balmer introduced as "tensor-triangular geometry". Comparing the main features I'll try to pitch it as an attractive addition to the family.

Markus Hausmann: The universal property of bordism rings of manifolds with commuting involutions

My talk concerns bordism rings of compact smooth manifolds equipped with a smooth action by a finite group. I will start by recalling classical results on the subject from the 60's and 70's, mostly due to Conner-Floyd, Boardman, Stong and Alexander. Afterwards I will discuss joint work with Stefan Schwede in which we prove an algebraic universal property for the collection of bordism rings of manifolds with commuting involutions, in the representation-graded sense. This universal property can be viewed as a delocalization of the corresponding one for homotopical equivariant bordism rings.

Irakli Patchkoria: On the Farrell-Tate K-theory of Out(F_n)

This is joint work with Naomi Andrew. The classical Farrell-Tate cohomology measures the failure of duality in group (co)homology. Brown in 70s gave a general method for computing the p-local part of the Farrell-Tate cohomology. Using Brown’s methods Farrell-Tate cohomology has been computed for various arithmetic groups, mapping class groups and Out(F_n)-s, outer automorphism groups of Free groups. Later Klein introduced generalised Farrell-Tate cohomology with coefficients in an arbitrary spectrum. In this project we investigate the Farrell-Tate K-theory of Out(F_n). We will show that for any discrete group with finite classifying space for proper actions, the p-adic Farrell-Tate K-theory is rational. Then using Lück’s Chern character, we will give a general formula for the p-adic Farrell-Tate K-theory in terms of centralisers. In particular, we apply this formula to Out(F_{p+1}) which has curious p-torsion behaviour: It has exactly one conjugacy class of a p-torsion element which does not come from Aut(F_{p+1}). Computing the rational cohomology of the centraliser of this element allows us to fully compute the p-adic Farrell-Tate K-theory of Out(F_{p+1}). As a consequence we show for example that the 11-adic Farrell-Tate K-theory of Out(F_{12}) is non-trivial, thus detecting a non-trivial class in odd K-theory of Out(F_{12}) without using any computer calculations.