Warwick algebraic topology seminar 2223
A list of the seminar talks of the previous years can be found here.
Term 3
Date  Speaker  Affiliation  Title  Notes 

Apr 25  Martin Gallauer  University of Warwick  Permutation generation  
May 2  John Jones  University of Warwick  On the topology of the Rosenfeld Projective Planes  
May 9  Fabian Hebestreit  University of Aberdeen  KaroubiGrothendieckWitt spectra  
May 16  No Seminar  
May 23  Julius Frank  University of Aberdeen 
DGAs with polynomial homology and E₁quotients 

May 30  Niall Taggart  Utrecht University  Convergence and analyticity in orthogonal calculus  
Jun 6  No Seminar  
Jun 13  No Seminar  
Jun 20  Sil Linskens  University of Bonn  Globally equivariant topological modular forms  
Jun 27  Maxime Ramzi  University of Copenhagen  Separability in homotopical algebra 
Abstracts
Martin Gallauer: Permutation generation
Let R be a ring (spectrum) and G a (finite) group. We are concerned with Rlocal systems on BG, and more specifically with the following question: When can they be built out of permutational ones? (Based on work with Paul Balmer and Tobias Barthel.)
John Jones: On the topology of the Rosenfeld Projective Planes
The aim of this project is to study the topological invariants of the manifolds R_4, R_5, R_6, and R_7 of dimension 16, 32, 64, 128 which are known as the four exceptional Rosenfeld projective planes. These manifolds are homogeneous spaces for the exceptional Lie groups F_4, E_6, E_7, E_8 respectively.
The main idea is to exploit the fact that there are rather straightforward conceptual descriptions of the Ktheory of Lie groups and homogeneous spaces in terms of representation theory. Since so much is known about the representation rings of the simply connected compact Lie groups, this gives a very efficient method of systematically calculating topological invariants of Lie groups and homogeneous spaces.
The talk will start with an introduction to the Rosenfeld projective planes and go on to explain how using ideas from Ktheory really does help to understand them.
This is joint work with Dmitriy Rumynin and Adam Thomas.
Fabian Hebestreit: KaroubiGrothendieckWitt spectra
(jt with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus, W.Steimle)
In previous work we related the GrothendieckWittspectra of a ring to its (connective) algebraic K and (somewhat periodic) Lspectra. For many applications both in geometric topology and algebraic geometry these spectra do, however, have the defect that they are not invariant under idempotent completion (when applied to sufficiently categorified input). Invariance under idempotent completion can, however, be universally enforced and this classically leads one from connective to nonconnective algebraic Kspectra, a seemingly minor modification. The analogue for GrothendieckWitt spectra is, however, more complicated, not least notationally, in that one already starts out with nonconnective spectra; the resulting objects are the KaroubiGrothendieckWitt spectra from the title.
Julius Frank: DGAs with polynomial homology and E₁quotients
Differential graded algebras whose homology is a graded polynomial algebra in one variable are surprisingly sparse. One example can be obtained by taking the E₁quotient of a commutative ring by some prime p. This construction also shows up in some algebraic Ktheory calculations, which motivates investigating this algebra with trace methods.
Examples are necessarily noncommutative: I will explain how all E₂DGAs with polynomial homology over a fixed perfect ring are equivalent as ring spectra.
Niall Taggart: Convergence and analyticity in orthogonal calculus
Orthogonal calculus is a brand of functor calculus that studies functors from Euclidean spaces to spaces or spectra. It was originally developed by Weiss to study spaces coming from geometry such as the space of homotopy automorphisms of spheres and classifying spaces of certain diffeomorphism groups. The calculus produces a tower of approximations to the input functor but unlike Goodwillie calculus or manifold calculus, not much (if anything at all) is known about the convergence of the tower of approximations in orthogonal calculus. The predominant reason for this is that both Goodwillie calculus and manifold calculus have the notion of analytic functors  functors that satisfy and easily checkable condition which in turn implies that their tower of approximations converges to the input functor.
In this talk I will discuss joint work with Arone in which we develop the notion of an analytic functor in the setting of orthogonal calculus, and if time permits discuss how this relates to analytic functors in the other calculi.
Sil Linskens: Globally equivariant topological modular forms
An extension of elliptic cohomology theories to a cohomology theory for equivariant spaces has been a dream for many years, ever since the axiomatics for such a theory were laid down by GinzburgKapranovVasserot in 1995. Recently, a rigorous construction of such a theory was suggested by Lurie using the theory of spectral algebraic geometry, and carried out by GepnerMeier. Given an oriented elliptic curve E> M and an abelian compact Lie group G, the result of the construction is an equivariant cohomology theory Ell_G(): Top_G > Sp, which they show is represented by a (genuine) Gspectrum E_G when G is a torus. However there is a more basic object at play. The cohomology theory Ell_G() is in fact given by restricting a functor Ell():Stk —> Sp defined on stacks along the quotient stack construction ()//G:Top_G—>Stk. One may hope that the functor Ell() should itself be a “(genuine) cohomology theory for stacks”. Such theories are represented by global spectra in the sense of Schwede. In this talk I will explain how one can show that the functor Ell() is in fact represented by a global spectrum E_gl, i.e. defines a genuine cohomology theory on stacks. This is joint work with David Gepner and Luca Pol. The construction crucially makes use of an alternative model of global spectra, recently introduced in joint work with Denis Nardin and Luca Pol. I will also introduce this model in the talk.
Maxime Ramzi: Separability in homotopical algebra
Separable algebras are a generalization of étale algebras that can be defined in more general homotopical contexts, and have been studied in tensortriangular geometry, partly due to their nice properties in this context. In this talk, I will attempt to explain some of these nice properties in tensortriangulated categories, by showing that they come from surprising features of separable algebras in stable oocategories, in particular showing that all separable uptohomotopy algebras lift (almost) uniquely to homotopy coherent algebras. If time permits, I will also mention how the basics of the classical theory of separable algebras extend to homotopical algebra.
Term 2
Date  Speaker  Affiliation  Title  Notes 

Jan 10  Ismael Sierra  University of Cambridge  Homological stability of diffeomorphism groups using Ek algebras  B3.02 
Jan 17  Foling Zou  University of Michigan  Nonabelian Poincare duality theorem in equivariant factorization homology  TeamsLink opens in a new window/B3.02 
Jan 24  No Seminar  
Jan 31  Neil Strickland  University of Sheffield  Questions around chromatic splitting  B3.02 
Feb 7  Luca Pol  University of Regensburg  Quillen stratification in equivariant homotopy theory  Teams/B3.02 
Feb 14  No Seminar  
Feb 21  Constanze Roitzheim  University of Kent  How algebraic is a stable model category?  B3.02 
Feb 28  Gonçalo Tabuada  University of Warwick  Grothendieck classes of quadric hypersurfaces and involution varieties  B3.02 
Mar 7  Ben Briggs  University of Copenhagen  Syzygies of the cotangent complex  B3.02 
Mar 14  Jeffrey Carlson  Imperial College  Products on Tor and strong homotopy commutativity  B3.02 
Abstracts
Ismael Sierra: Homological stability of diffeomorphism groups using Ek algebras
I will state some recent results about homological stability of diffeomorphism groups of manifolds and give an outline of their proof. In particular, I will talk about the connection to Ek algebras, and about certain complexes, called "splitting complexes", whose highconnectivities are essential to the proof. Finally I will sketch the proof of the highconnectivity of the splitting complexes, which is the most substantial part of the whole argument.
Foling Zou: Nonabelian Poincare duality theorem in equivariant factorization homology
The factorization homology are invariants of $n$dimensional manifolds with some fixed tangential structures that take coefficients in suitable $E_n$algebras. In this talk, I will give a definition for the equivariant factorization homology of a framed manifold for a finite group $G$ by monadic bar construction following KupersMiller. Then I will prove the equivariant nonabelian Poincare duality theorem in this case. As an application, in joint work with Asaf Horev and Inbar Klang, we compute the equivariant factorization homology on equivariant spheres for certain Thom spectra. In particular, we recover a computation of the Real topological Hoschild homology by DottoMoiPatchkoriaReeh.
Neil Strickland: Questions around chromatic splitting
The Chromatic Splitting Conjecture of Hopkins states that if we take the sphere spectrum, localise with respect to the Morava Ktheory K(n), then localise again with respect to E(n1), then the result splits as a coproduct of 2^n pieces, each of which is a sphere localised with respect to E(m) for some m < n. This is known to be false (by work of Beaudry) when n=p=2, but remains open (and mysterious) in general. We will explain how the conjecture, when combined with known phenomena such as chromatic fracture squares, predicts some calculations with a very intricate combinatorial/algebraic structure. These calculations appear to be selfconsistent, which could easily have failed to be the case; this suggests that the conjecture may be true, or closely related to the truth.
Luca Pol: Quillen stratification in equivariant homotopy theory
Quillen’s celebrated stratification theorem provides a geometric description of the Zariski spectrum of the cohomology ring of any finite group with coefficients in a field in terms of information coming from its elementary abelian psubgroups. The goal of this talk is to discuss an extension of Quillen's result to the world of equivariant tensortriangular geometry. For the category of equivariant modules over a commutative equivariant ring spectrum we obtain a stratification result in the terms of the geometric fixed points equipped with their Weylgroup actions for all subgroups, and hence a classification of localizing tensor ideals. Finally, I will apply these methods to several examples of interests such as Borelequivariant Morava Etheory and equivariant topological Ktheory. This is joint work with Tobias Barthel, Natalia Castellana, Drew Heard and Niko Naumann.
Constanze Roitzheim: How algebraic is a stable model category?
There are many different notions of "being algebraic" used in stable homotopy theory. The relationships between those turn out to be unexpectedly subtle. We will explain the different ways in which a model category of interest can be algebraic, explore the different implications between them and illustrate those with plenty of examples.
(This is joint work with Jocelyne Ishak and Jordan Williamson.)
Gonçalo Tabuada: Grothendieck classes of quadric hypersurfaces and involution varieties
The Grothendieck ring of varieties, introduced in a letter from Alexander Grothendieck to JeanPierre Serre (August 16th 1964), plays an important role in algebraic geometry. However, despite the efforts of several mathematicians, the structure of this ring still remains poorly understood. In this talk, in order to better understand the Grothendieck ring of varieties, I will describe some new structural properties of the Grothendieck classes of quadric hypersurfaces and involution varieties. More specifically, by combining the recent theory of noncommutative motives with the classical theory of motives, I will show that if two quadric hypersurfaces (or, more generally, two involution varieties) have the same Grothendieck class, then they have the same even Clifford algebra and the same signature. As an application, this implies in numerous cases (e.g., when the base field is a local or global field) that two quadric hypersurfaces (or, more generally, two involution varieties) have the same Grothendieck class if and only if they are isomorphic.
Ben Briggs: Syzygies of the cotangent complex
The cotangent complex is an important object from commutative algebra. It was defined by Quillen using homotopical methods, and is usually extremely difficult to compute. It is connected with some more tractable invariants: the module of differential forms, the conormal module, and Koszul homology can all be seen as syzygies of the cotangent complex. One can try to establish higher analogues of the Jacobian criterion by characterising geometric conditions in terms of homological properties of these syzygies. I will explain how thinking along these lines leads to a new proof of Quillen's conjecture on the cotangent complex and Vasconcelos' conjecture on the conormal module. I'll also try to explain some of the parallels in rational homotopy theory. This is joint work with Srikanth Iyengar.
Jeffrey Carlson: Products on Tor and strong homotopy commutativity
The Eilenberg–Moore spectral sequence converges from the classical Tor of a span of cohomology rings to the differential Tor of a span of cochain algebras (which is the cohomology of the homotopy pullback). These are both rings, for very different reasons: the first structure comes about because cohomology rings are commutative, and the second arises as a corollary of the Eilenberg—Zilber theorem.
One might well ask when a more general differential Tor of DGAs admits a ring structure, though apparently no one did. We will show that when the DGAs in question admit a certain sort of $E_3$algebra structure generalizing the previous examples, Tor is a commutative graded algebra.
We have not done this out of an innocent interest in homotopycommutative algebras. In the 1960s and '70s there was a flurry of activity developing Ainfinityalgebraic techniques with an aim toward computing the Eilenberg–Moore spectral sequence (for example, of a loop space or homogeneous space). Arguably the most powerful result this program produced was the 1974 theorem of Munkholm that the sequence collapses when the three input spaces have polynomial cohomology over a given principal ideal domain, which however only gives the story on cohomology groups. Our result shows that Munkholm's map is in fact an isomorphism of rings.
This work is joint with several large commutative diagrams.
Term 1
Date  Speaker  Affiliation  Title  Notes 

Oct 11  Sebastian Chenery  Southampton  On PushoutPullback Fibrations  B3.03 
Oct 18  Thomas Read  Warwick  Gtypical Witt vectors with coefficients and the norm  B3.03 
Oct 25  Severin Bunk  Oxford  Functorial field theories from differential cocycles  B3.03 
Nov 1  No Seminar  
Nov 8  Thibault Décoppet  Oxford  Fusion 2Categories associated to 2groups  B3.03 
Nov 15  Foling Zou  University of Michigan  Nonabelian Poincare duality theorem in equivariant factorization homology  TeamsLink opens in a new window/B3.03 
Nov 22  Irakli Patchkoria  Aberdeen  Morava Ktheory of infinite groups and Euler characteristic  B3.03 
Nov 29  Florian Naef  Trinity College Dublin  Relative intersection product, Whiteheadtorsion and string topology  B3.03 
Dec 6  Lucy Yang  Harvard  A real HochschildKostantRosenberg theorem  TeamsLink opens in a new window/B3.03 
Abstracts
Sebastian Chenery: On PushoutPullback Fibrations
We will discuss recent work inspired by a paper of Jeffrey and Selick, where they ask whether the pullback bundle over a connected sum can itself be homeomorphic to a connected sum. We provide a framework to tackle this question through classical homotopy theory, before pivoting to rational homotopy theory to give an answer after taking based loop spaces.
Thomas Read: Gtypical Witt vectors with coefficients and the norm
The norm is an important construction on equivariant spectra, most famously playing a key role in the work of Hill, Hopkins and Ravenel on the Kervaire invariant one problem. Witt vectors are an algebraic construction first used in Galois theory in the 1930s, but later finding applications in stable equivariant homotopy theory. I will describe a new generalisation of Witt vectors that can be used to compute the zeroth equivariant stable homotopy groups of the norm $N_e^G Z$, for $G$ a finite group and $Z$ a connective spectrum.
Severin Bunk: Functorial field theories from differential cocycles
In this talk I will demonstrate how differential cocycles give rise to (bordismtype) functorial field theories (FFTs). I will discuss some background on smooth FFTs, differential cohomology and higher gerbes with connection as a geometric model for differential cocycles before explaining the general principle for how to obtain smooth FFTs from higher gerbes. In the second part, I will focus on the twodimensional case. Here I will present a concrete, geometric construction of twodimensional smooth FFTs on background manifolds, starting from gerbes with connection. This is related to WZW theories. If time permits, I will comment on an extension of this construction which produces openclosed field theories.
Thibault Décoppet: Fusion 2Categories associated to 2groups
Motivated by the cobordism hypothesis, which provides a correspondence between fully dualizable objects and fully extended framed TQFTs, it is natural to seek out interesting examples of fully dualizable objects. In dimension four, the fusion 2categories associated to 2groups are examples of fully dualizable objects. In my talk, I will begin by reviewing the 2categorical notion of Cauchy completion, and recall the definition of a fusion 2category in detail. Then, I will explain how one can construct a fusion 2category of 2vector spaces graded by 2group, and how this construction can be twisted using a 4cocycle. Finally, it is important to understand when two such fusion 2categories yield equivalent TQFTs. The answer is provided by the notion of Morita equivalence between fusion 2categories, which will be illustrated using some examples.
Foling Zou: Nonabelian Poincare duality theorem in equivariant factorization homology
The factorization homology are invariants of $n$dimensional manifolds with some fixed tangential structures that take coefficients in suitable $E_n$algebras. In this talk, I will give a definition for the equivariant factorization homology of a framed manifold for a finite group $G$ by monadic bar construction following KupersMiller. Then I will prove the equivariant nonabelian Poincare duality theorem in this case. As an application, in joint work with Asaf Horev and Inbar Klang, we compute the equivariant factorization homology on equivariant spheres for certain Thom spectra. In particular, we recover a computation of the Real topological Hoschild homology by DottoMoiPatchkoriaReeh.
Irakli Patchkoria: Morava Ktheory of infinite groups and Euler characteristic
Given an infinite discrete group G with a finite model for the classifying space for proper actions, one can define the Euler characteristic of G and the orbifold Euler characteristic of G. In this talk we will discuss higher chromatic analogues of these invariants in the sense of stable homotopy theory. We will study the Morava Ktheory of G and associated Euler characteristic, and give a character formula for the LubinTate theory of G. This will generalise the results of HopkinsKuhnRavenel from finite to infinite groups and the Ktheoretic results of Adem, Lück and Oliver from chromatic level one to higher chromatic levels. Along the way we will give explicit computations for amalgamated products of finite groups, right angled Coxeter groups and certain special linear groups. This is all joint with Wolfgang Lück and Stefan Schwede.
Florian Naef: Relative intersection product, Whiteheadtorsion and string topology
Given a closed oriented manifold one can define an intersection product on the homology. This can be extended to local coefficient, and further made relative to the diagonal. I will explain how such a relative selfintersection product is not homotopy invariant (in contrast to the ordinary intersection product) and how this is picked up by string topology. Eventually, we will identify the error term with the trace of Whitehead torsion. More precisely, we will extract an invariant from a Poincare embedding of the diagonal (in the sense of J. Klein) that is the trace of (a version of) Reidemeister torsion. This is based on joint work with P. Safronov.
Lucy Yang: A real HochschildKostantRosenberg theorem
GrothendieckWitt and real Ktheory are enhancements of Ktheory in the presence of duality data. Similarly to ordinary Ktheory, real Ktheory admits homological approximations, known as real trace theories. In this talk, I will identify a filtration on real Hochschild homology and compute the associated graded in terms of an analogue of de Rham forms. We will see how C₂ genuine equivariant algebra is the natural setting for these theories, provide equivariant enhancements of the cotangent and de Rham complexes, and sketch the proof of the main theorem. This work is both inspired by and builds on that of Raksit.