Warwick Algebraic Topology Seminar 25/26

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

Term 2

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

Date	Speaker	Affiliation	Title
Jan 13	Oscar Randal-Williams	Cambridge	Stable cohomology of congruence subgroups
Jan 20	Nansen Petrosyan	Southampton	Simple groups with strong fixed-point properties
Jan 27	Raphael Ruimy	Grenoble	Pro-étale motives and solid rigidity
Feb 3	Logan Hyslop	Harvard	Constructible and Homological Spectra
Feb 10	Robert Tang	Xi’an Jiaotong-Liverpool	The metric Rips filtration and universal quasigeodesic cones
Feb 17	Max Blans	Oxford
Feb 24
Mar 3	Nicola Gambino	Manchester
Mar 10
Mar 17	Markus Upmeier	Aberdeen

Oscar Randal-Williams, 13 January

Title: Stable cohomology of congruence subgroups

Abstract: I will explain how to complete and extend an argument proposed by F. Calegari for determining the F_p-cohomology of the congruence subgroiups SL_n(Z, p^m) in a certain range (namely in cohomological degrees * < p-1 and for all large enough n). The result has a uniform description at regular primes, but at irregular primes has interesting correction terms, controlled by torsion in K_*(Z) and by special values of the p-adic L-function. The argument for m>1 turns out to be relatively simple, but for m=1 it involves a delicate analysis of the cohomology of the finite groups SL_n(Z/p) with coefficients in certain modular representations.

Nansen Petrosyan, 20 January

Title: Simple groups with strong fixed-point properties.

Abstract: I will discuss a construction of finitely generated torsion-free simple groups for which any action on any finite-dimensional CW-complex with finite Betti numbers has a global fixed point. In particular, these groups are not in Kropholler's class HF.

Raphael Ruimy, 27 January

Title: Pro-étale motives and solid rigidity

Abstract: Ekedahl's coefficient system given by ℓ-completing étale sheaves allows for a nice description of integral ℓ-adic cohomology, but not of rational ℓ-adic cohomology. Even worse: the ℓ-adic realization functor from étale motives sends Q to the zero object. The problem is actually that étale sheaves cannot handle topological phenomena: we need to use pro-étale sheaves of Bhatt-Scholze, more specifically solid (a kind of topological completeness condition) pro-étale sheaves as defined by Fargues and Scholze in the rigid-analytic setting.

In a joint work with Tubach and Wolf, we introduce solid pro-étale sheaves on schemes and show that they afford a six functors formalism. This contrasts with the original Fargues-Scholze setting where it is not possible. The main ingredient is the introduction of pro-étale motives with coefficients in any topological ring and in which étale motives embed fully faithfully when the ring is discrete. We then use the framework of condensed category theory to define a solidification process and show that solid pro-étale motives and solid pro-étale sheaves coincide with ℓ-adic coefficients (this is a rigidity result) which yields the six operations and a solid realization functor of motives, extending the ℓ-adic realization functor. The solid realization functor is compatible with change of coefficients, which allows one to recover rational ℓ -adic realization functor simply by taking Q-modules. This for instance yields a new definition of integral Nori motives.

Logan Hyslop, 3 February

Title: Constructible and Homological Spectra.

Abstract: In this talk, we will discuss several alternative definitions of homological spectra. Based on forthcoming work with Tobias Barthel and Maxime Ramzi, we will discuss how to realize the underlying set of the homological spectrum as the constructible spectrum in 2-Ring^{rig} when we are rational, together with the interpretation of the image of the comparison map from the constructible spectrum to the Balmer spectrum in general. Time permitting, we will discuss how to realize the homological spectrum as an “E_n-constructible spectrum” in general, and how this can be used to show that any tt-category with an E_n refinement (n>=4) has all points of its homological spectrum being geometric in the category of rigid E_n-1-2-Rings.

Robert Tang, 10 February

Title: The metric Rips filtration and universal quasigeodesic cones.

Abstract: Given a metric space X and a scale parameter σ ≥ 0, the Rips graph Rips_σX has X as its vertex set, with two vertices declared adjacent whenever their distance is at most σ. A classical fact is that X is a quasigeodesic space precisely if it is quasi-isometric to its Rips graph at sufficiently large scale. By considering all possible scales, we obtain a directed system of graphs known as the Rips filtration. How does the large-scale geometry of Rips_σX evolve as σ goes to ∞? Specifically, when does it have a colimit? It turns out that the answers depends on whether we work up to quasi-isometry or coarse equivalence.

The Rips colimit serves as a useful tool for computing universal quasigeodesic cones over a diagram of metric spaces satisfying some uniform control hypotheses. We will illustrate this in the case of the mapping class group equipped with its family of subsurface projections to curve graphs. More generally, our viewpoint recasts hierarchically hyperbolic spaces in terms of a universal property.

Nicola Gambino, 3 March

Title: On the Boardman-Vogt tensor products of operads and their bimodules.

Abstract: The Boardman-Vogt tensor product of operads is a classical tool for constructing operads which is important for many applications. In a 2014 paper, Dwyer and Hess have shown how to extend it to operad bimodules. In the seminar, after reviewing the basic notions, I will explain how the Boardman-Vogt tensor product of operads interacts with composition of operad bimodules, giving rise to an interesting categorical structure. This is based on joint work with Richard Garner and Christina Vasilakopoulou (arXiv:2511.14402), in which we develop a unified theory of commuting tensor products, subsuming also some work of Elmendorf and Mandell.

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	No talk
Nov 18	Jane Turner	University of Southampton	Combinatorial Relative Algebraic K-theory
Nov 25	Inbar Klang	VU Amsterdam	A Thom spectrum approach to real topological Hochschild homology
Dec 2	Markus Land	LMU Munich	Classification of DGAs and applications
Dec 9	Lukas Brantner	University of Oxford	Formal integration of derived foliations

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.

Jane Turner, 18 November

Title: Combinatorial Relative Algebraic K-theory

Abstract: The algebraic K-groups of an exact functor F: M to N between exact categories are classically defined as the homotopy groups of the homotopy fiber of the induced map on the K-theory spaces of M and N. In 2016, Daniel Grayson produced a conjectural presentation of these K-groups using a certain category built using chain complexes. In this talk, I will present work proving that these descriptions agree.

Inbar Klang, 25 November

Title: A Thom spectrum approach to real topological Hochschild homology

Abstract: Real topological Hochschild homology (THR) is an invariant of ring spectra with anti-involution, and can be considered as an equivariant refinement of topological Hochschild homology. In this talk, I will introduce the relevant invariants, and discuss joint work with Asaf Horev and Foling Zou, in which we prove the G-symmetric monoidal properties of equivariant Thom spectra and use them to give a new description of THR of such spectra.

Markus Land, 2 December

Title: Classification of DGAs and applications

Abstract: I will report on joint work with Bayındır on the classification of DGAs whose homology is a polynomial algebra over F_p in a single generator. After putting this into some context, I will briefly explain my original motivation, coming from computations in algebraic K-theory. Then, I will explain the main constructions and proof ingredients for our main results, and end with further applications, for instance to uniqueness questions about dg-enhancements of certain triangulated categories.

Lukas Brantner, 9 December

Title: Formal integration of derived foliations

Abstract: Frobenius’ theorem in differential geometry asserts that, given a smooth manifold M, every involutive subbundle E \subset T_M determines a decomposition of M into smooth leaves tangent to E. I will explain an infinitesimal analogue of this integration phenomenon for suitably nice schemes over coherent base rings, and then discuss an application and an open problem. This talk is based on joint work with Magidson and Nuiten and ties into the work of Jiaqi Fu.