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	Karoubi-Grothendieck-Witt spectra
May 16				No Seminar
May 23	Julius Frank	University of Aberdeen	DGAs with polynomial homology and E₁-quotients
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 R-local 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 K-theory 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 K-theory really does help to understand them.

This is joint work with Dmitriy Rumynin and Adam Thomas.

Fabian Hebestreit: Karoubi-Grothendieck-Witt 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 Grothendieck-Witt-spectra of a ring to its (connective) algebraic K- and (somewhat periodic) L-spectra. 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 non-connective algebraic K-spectra, a seemingly minor modification. The analogue for Grothendieck-Witt spectra is, however, more complicated, not least notationally, in that one already starts out with non-connective spectra; the resulting objects are the Karoubi-Grothendieck-Witt spectra from the title.

Julius Frank: DGAs with polynomial homology and E₁-quotients

Differential graded algebras whose homology is a graded polynomial algebra in one variable are surprisingly sparse. One example can be obtained by taking the E₁-quotient of a commutative ring by some prime p. This construction also shows up in some algebraic K-theory calculations, which motivates investigating this algebra with trace methods.

Examples are necessarily non-commutative: I will explain how all E₂-DGAs 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 Ginzburg-Kapranov-Vasserot 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 Gepner-Meier. 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) G-spectrum 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 tensor-triangular geometry, partly due to their nice properties in this context. In this talk, I will attempt to explain some of these nice properties in tensor-triangulated categories, by showing that they come from surprising features of separable algebras in stable oo-categories, in particular showing that all separable up-to-homotopy 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 high-connectivities are essential to the proof. Finally I will sketch the proof of the high-connectivity 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 Kupers--Miller. 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 Dotto--Moi--Patchkoria--Reeh.

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 K-theory K(n), then localise again with respect to E(n-1), 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 self-consistent, 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 p-subgroups. The goal of this talk is to discuss an extension of Quillen's result to the world of equivariant tensor-triangular 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 Weyl-group 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 Borel-equivariant Morava E-theory and equivariant topological K-theory. 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 Jean-Pierre 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 homotopy-commutative algebras. In the 1960s and '70s there was a flurry of activity developing A-infinity-algebraic 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 Pushout-Pullback Fibrations	B3.03
Oct 18	Thomas Read	Warwick	G-typical 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 2-Categories associated to 2-groups	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 K-theory of infinite groups and Euler characteristic	B3.03
Nov 29	Florian Naef	Trinity College Dublin	Relative intersection product, Whitehead-torsion and string topology	B3.03
Dec 6	Lucy Yang	Harvard	A real Hochschild--Kostant--Rosenberg theorem	TeamsLink opens in a new window/B3.03

Abstracts

Sebastian Chenery: On Pushout-Pullback 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: G-typical 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 (bordism-type) 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 two-dimensional case. Here I will present a concrete, geometric construction of two-dimensional 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 open-closed field theories.

Thibault Décoppet: Fusion 2-Categories associated to 2-groups

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 2-categories associated to 2-groups are examples of fully dualizable objects. In my talk, I will begin by reviewing the 2-categorical notion of Cauchy completion, and recall the definition of a fusion 2-category in detail. Then, I will explain how one can construct a fusion 2-category of 2-vector spaces graded by 2-group, and how this construction can be twisted using a 4-cocycle. Finally, it is important to understand when two such fusion 2-categories yield equivalent TQFTs. The answer is provided by the notion of Morita equivalence between fusion 2-categories, 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 Kupers--Miller. 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 Dotto--Moi--Patchkoria--Reeh.

Irakli Patchkoria: Morava K-theory 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 K-theory of G and associated Euler characteristic, and give a character formula for the Lubin-Tate theory of G. This will generalise the results of Hopkins-Kuhn-Ravenel from finite to infinite groups and the K-theoretic 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, Whitehead-torsion 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 self-intersection 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 Hochschild--Kostant--Rosenberg theorem

Grothendieck--Witt and real K-theory are enhancements of K-theory in the presence of duality data. Similarly to ordinary K-theory, real K-theory 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.