Warwick Algebraic Topology Seminar 24/25

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

Term 3

Due to staff being away this term, we will run a reduced version of the seminar, with fewer talks. They happen in B3.03.

Date	Speaker	Affiliation	Title	Notes
Apr 29	Luca Pol	Regensburg	New phenomena in the tt-geometry of global representations

Abstracts

Luca Pol: New phenomena in the tt-geometry of global representations

A global representation over a field is a compatible collection of representations of the outer automorphism groups of the groups belonging to some collection of finite groups U. Global representations assemble into an abelian category AU which simultaneously generalizes classical representation theory and the category of VI-modules appearing in the representation theory of the general linear groups. The derived category of global representations D(U) is an example of a non-rigid compactly generated tensor-triangulated category, which is known to be equivalent to the category of rational global spectra for the family U. In this talk I would like to present some new phenomena that we encounter and explain some calculations of the Balmer spectrum in this setting. This is based on joint work with Miguel Barrero, Tobias Barthel, Neil Strickland and Jordan Williamson.

Term 2

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

Date	Speaker	Affiliation	Title
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	The Weiss calculus of vector bundles
Mar 4	Oliver Röndigs	Osnabrück	Homotopy of the special linear group
Mar 11	Teena Gerhardt	Michigan State	Topological Hochschild homology, Witt vectors, and equivariance

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.

Niall Taggart: The Weiss calculus of vector bundles

Abstract: It is a classical problem in topology to count the number rank d vector bundles over a space X having fixed characteristic class data. In this talk I will explain how the implementation of Weiss calculus make this a tractable problem in certain cases, i.e., for stably trivial bundles and under localisation, and discuss some of the curious relationships between the real and complex case. This is joint work with Guy Boyde.

Oliver Roendigs: Homotopy of the special linear group

The special linear group SL_n, considered as an algebraic variety, defines an algebraic homotopy type in the Morel-Voevodsky unstable homotopy category over a field F. Hence it has homotopy groups (or rather sheaves of such) indexed by natural numbers. If the index m is less than n-1 and n at least 3, then the m-th homotopy group of SL_n coincides with the (m+1)-th algebraic K-group of F, by results of Morel and Voevodsky. In particular, it does not depend on n anymore, the so-called stable range. The talk will discuss the first non-stable case for small n.

Teena Gerhardt: Topological Hochschild homology, Witt vectors, and equivariance

Hochschild homology of a ring has a topological analogue for ring spectra, topological Hochschild homology (THH), which plays an essential role in the trace method approach to algebraic K-theory. Topological Hochschild homology is closely related to Witt vectors, and this relationship has facilitated algebraic K-theory calculations. For equivariant rings (or ring spectra) there is a theory of twisted topological Hochschild homology that builds upon Hill, Hopkins, and Ravenel's work on equivariant norms. This twisted THH is closely related to an equivariant version of Witt vectors. Indeed, in this talk I will discuss recent work showing that the equivariant homotopy of twisted THH forms an equivariant Witt complex. This is joint work with Bohmann, Krulewski, Petersen, and Yang.

Term 1

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

Date	Speaker	Affiliation	Title
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.