# Warwick algebraic topology seminar 21-22

The current seminar is here.

###### Term 3

The talks will take place on Tuesday at 4pm. Some of the talks will be in person, some will be given by a remote speaker on MS Teams and streamed in B3.03.

Date Speaker Affiliation Title Notes
Apr 26 Geoffroy Horel Paris 13 Galois symmetries of knot spaces VideoLink opens in a new window
May 3 Luca Pol University of Regensburg Global homotopy theory via partially lax limits VideoLink opens in a new window
May 10 Markus Szymik Sheffield Quandles, Real Algebra, and Symmetries B3.03
May 17 Özgür Bayındır City University of London Adjoining roots to ring spectra and algebraic K-theory B3.03
May 24 Rachael Boyd Cambridge Embedding spaces of split links B3.03
May 31 Shane Kelly University of Tokyo
Nilpotent sensitive cohomology theories with compact support
B3.03
Jun 7 Ferdinando Zanchetta University of Bologna Exterior power operations on algebraic K-theory: a comparison B3.03
Jun 14 Achim Krause Münster On the K-theory of Z/p^n VideoLink opens in a new window
Jun 21 No seminar
Jun 28 Matteo Barucco Warwick Computing S^1-equivariant elliptic cohomology of complex projective spaces B3.03
##### Abstracts
###### Geoffroy Horel: Galois symmetries of knot spaces

I will explain how the action of the absolute Galois group of Q on the profinite completion of the braid groups, originally studied by arithmetic geometers, induces an interesting action on the manifold calculus tower for long knots. Using this action together with work of Volić and Kosanović, this can be used to deduce some integral results about the universal finite type invariant for knots. This is joint work with Pedro Boavida de Brito and Danica Kosanović.

###### Luca Pol: Global homotopy theory via partially lax limits

Global homotopy theory is the study of equivariant objects which exist uniformly and compatibly for all compact Lie groups in a certain family, and which exhibit extra functoriality. In this talk I will present new infty-categorical models for unstable and stable global homotopy theory. We use the notion of partially lax limit to formalize the idea that a global object is a collection of G-objects, one for each compact Lie group G, which are compatible with the restriction-inflation functors. This is joint work with Sil Linskens and Denis Nardin.

###### Markus Szymik: Quandles, Real Algebra, and Symmetries

It is a truism that groups are generally the mathematical structure that best describes symmetries, and I will not argue against that. I will, however, present arguments that quandles, a closely related algebraic structure, sometimes beat them at that, as I will illustrate with real applications coming from knot spaces and Galois symmetries. Along the way, I will give a leisurely introduction to the basic definitions and discuss some open questions on the precise relationship between quandles and groups.

###### Özgür Bayındır: Adjoining roots to ring spectra and algebraic K-theory

In this work, we start with a new method to adjoin roots to E_2 ring spectra. Upon making a new definition of logarithmic THH, we prove that root adjunction is logarithmic THH \'etale and that it results in interesting splittings in algebraic K-theory. For instance, we obtain that T(n+1)-locally, the algebraic K-theory of the nth Morava E-theory contains the algebraic K-theory of the Johnson-Wilson spectrum with coefficients as a summand.

This is a joint work with Tasos Moulinos and Christian Ausoni.

###### Rachael Boyd: Embedding spaces of split links

This talk reports on work in progress with Corey Bregman. We study the homotopy type of embedding spaces of unparameterised links, inspired by work of Brendle and Hatcher. Our main innovation is a semi-simplicial space of separating spheres, which is a combinatorial object that provides a gateway to studying the homotopy type of configurations of split links via the homotopy type of their individual pieces. We apply this tool to find a simple description of the fundamental group.

###### Shane Kelly: Nilpotent sensitive cohomology theories with compact support.

This is joint work in progress with Shuji Saito. In algebraic geometry, cohomology with compact support is often defined using compactifications. This approach needs a kind of "invariance of the compactification" property of the cohomology theory. One way of formalising this is using the cdh-topology or h-topology. These topologies appeared in Voevodsky's proof of the Bloch-Kato conjecture, and more recently in Beilinson's simple proof of Fontaine's CdR conjecture, and in Bhatt and Scholze's work on projectivity of the affine Grassmanian.

A feature of these topologies which often turns out to be a bug is that the associated sheaves cannot see nilpotents. In this talk I will discuss a modification which can see nilpotents, and which can still be used to define cohomology with compact support.

###### Ferdinando Zanchetta: Exterior power operations on algebraic K-theory: a comparison

Exterior power operations provide additional structure on K-theory that has been very useful for many purposes (e.g. in Riemann-Roch theory). In the context of higher algebraic K-theory of schemes, many a priori different definitions of them have been given over the last decades. After a short introduction, I will explain how it is possible to show that, for reasonably general schemes, they agree. Time permitting, I’ll end up discussing applications of such comparison. This is joint with B. Koeck.

###### Achim Krause: On the K-theory of Z/p^n

In recent work with Ben Antieau and Thomas Nikolaus, we develop new methods to compute K-theory of Z/p^n and related rings, based on prismatic cohomology. This approach can be turned into an algorithm, which we implement. The same methods also allow us to prove that K-theory of Z/p^n vanishes in large enough even degrees, and to give an explicit formula for the orders in large odd degrees. In this talk, I want to give an overview over the ingredients of these computations.

###### Matteo Barucco: Computing S^1-equivariant elliptic cohomology of complex projective spaces

Equivariant elliptic cohomology is an interesting cohomology theory that has found numerous applications in physics and representation theory, but that often lacks geometric interpretation. In this talk I will present a construction for rational T^2-equivariant elliptic cohomology for the 2-torus T^2, using algebraic models, that has the advantage of a clear interpretation in terms of the geometry of the elliptic curve we start with. We will apply this construction to compute rational S^1-equivariant elliptic cohomology of CP(V) for a complex S^1-representation V.

###### Term 2

The talks will take place on Tuesday at 4pm. Some of the talks will be in person, some will be given by a remote speaker on MS Teams and streamed in B3.02.

Date Speaker Affiliation Title Notes
Jan 11 Dave Benson University of Aberdeen The singularity category of $C^*BG$, $G$ a finite group B3.02
Jan 18 Andreas Stavrou University of Cambridge Cohomology of configuration spaces of surfaces as mapping class group representations B3.02
Jan 25 Tyler Lawson University of Minnesota Actions and algebras VideoLink opens in a new window
Feb 1 Andrey Lazarev   Talk postponed to March 1st
Feb 8 Guy Boyde University of Southampton Growth of homotopy groups of spheres with Goodwillie Calculus B3.02
Feb 15 No Seminar
Feb 22 Inbar Klang Columbia University Equivariant Hochschild theories from a shadow perspective VideoLink opens in a new window
Mar 1 Andrey Lazarev Lancaster University On differential graded Koszul duality B3.02
Mar 8 Fabian Hebestreit University of Muenster Symplectic groups and cobordism categories VideoLink opens in a new window
Mar 15 Jelena Grbic University of Southampton Higher Whitehead maps and relations amongst them B3.02
##### Abstracts
###### Dave Benson: The singularity category of $C^*BG$, $G$ a finite group

I shall talk about the $A_\infty$ structure of $C^*BG$, the cochains on the classifying space of a finite group, and its Koszul dual $C_*\Omega BG^{^\wedge}_p$, the chains on the loop space of the $p$-completion of $BG$. This will lead to a discussion of the singularity category of $C^*BG$, which is equivalent to the cosingularity category of $C_*\Omega BG^{^\wedge}_p$. The case of cyclic Sylow $p$-subgroups will be used as an example. This is joint work with John Greenlees.

###### Andreas Stavrou: Cohomology of configuration spaces of surfaces as mapping class group representations

Recently there has been intense study of the homology of configuration spaces of manifolds and many extra structures have been imposed on it to extract more information. In this talk I will focus on the action of the mapping class group and will present a result for general manifolds. In the case of compact oriented surfaces with one boundary component, I will present the complete answer for the cohomology of its configuration spaces as mapping class group representations. I will motivate the answer pictorially and give an overview of the steps involved in the proof.

###### Tyler Lawson: Actions and algebras

At the prime 2, the dual Steenrod algebra is a graded ring that is polynomial on infinitely many generators over Z/2, and it appears as the homology of a spectrum H Z/2. In this talk I'll describe a minor mystery about the structure of modules and algebras over it that arose in joint work with Beaudry-Hill-Shi-Zeng, and how this can be resolved by a general result relating pushouts of E_k-algebras with relative tensors over E_{k+1}-algebras. (Joint work with Michael Hill.)

###### Guy Boyde: Growth of homotopy groups of spheres with Goodwillie Calculus

How do the homotopy groups of spheres grow? Can we bound their size? Various people have produced exponential upper bounds, using the EHP sequence, but is this the best we can do? Goodwillie Calculus gives a notion of a "Taylor Series" decomposition of a space, beginning from its stable homotopy type and converging to its "true" unstable homotopy type. Mark Behrens combined the Goodwillie Tower and the EHP sequence to give a modern take on classical work of Toda. I'll explain how one can use his ideas to exhibit an exponential upper bound for the unstable homotopy groups as a "Taylor Series", in which (loosely) the size of the stable homotopy groups places an upper bound on the size of the unstable ones.

###### Inbar Klang: Equivariant Hochschild theories from a shadow perspective

Bicategorical shadows, de fined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. I'll begin the talk by reviewing this framework and introducing the equivariant Hochschild invariants we will discuss, twisted THH and Hochschild homology for Green functors. I will then talk about joint work with Adamyk, Gerhardt, Hess, and Kong, in which we prove that these equivariant Hochschild invariants are bicategorical shadows, from which we deduce that they satisfy Morita invariance. We also show that they receive trace maps from an equivariant version of algebraic K-theory.

###### Andrey Lazarev: On differential graded Koszul duality

Differential graded (dg) Koszul duality is a Quillen equivalence between the category of dg algebras and conilpotent dg coalgebras. There is also a linear version of it involving modules and comodules. In this talk I will explain what happens when one drops the condition of conilpotency on the coalgebra side. This represents partly joint work with A. Guan and partly work in progress.

###### Fabian Hebestreit: Symplectic groups and cobordism categories

The goal of the talk will be to explain joint work with M.Land and T.Nikolaus, in which we compute the so-called stable part of the cohomology of symplectic and orthogonal groups over the integers to a large extent, in particular at the previously mysterious prime 2. Our approach is via the group completion theorem, which relates the stable cohomology of these arithmetic groups (over general rings) to that of certain Grothendieck-Witt spaces. Recent work with W.Steimle identifies these in terms of algebraic cobordism categories, whence they can be analysed using the tools set-up in our previous joint work with B.Calmès, E.Dotto, Y.Harpaz, K.Moi and D.Nardin.

###### Jelena Grbic: Higher Whitehead maps and relations amongst them

Understanding the groups of homotopy classes of maps is the Holy Grail of homotopy theory. In our quest, we exploit the combinatorial nature of polyhedral products to unlock the mystery of relations between higher Whitehead maps.This is joint work with George Simmons and Matthew Staniforth.

###### Term 1

The talks will take place on Tuesday at 4pm in B3.03. Some of the talks will be in person, some will be given by a remote speaker on MS Teams and streamed in B3.03. You can join the teams talks with the code hud33su, or email me at emanuele.dotto@warwick.ac.uk.

Date Speaker Affiliation Title Notes
Oct 5 Nima Rasekh EPFL Shadows are bicategorical Traces VideoLink opens in a new window
Oct 12 Andrew Macpherson University of Warwick Designer ∞-categories via homotopy theory of diagrams B3.03
Oct 19 Rajan Mehta Smith College Frobenius objects in categories of relations and spans VideoLink opens in a new window
Oct 26 Anna Marie Bohmann Vanderbilt University Algebraic K-theory for Lawvere theories: assembly and Morita invariance B3.03/Teams
Nov 2 Irakli Patchkoria University of Aberdeen Franke’s conjecture and derived infinity categories B3.03
Nov 9 Oscar Randal-Williams University of Cambridge The second Weiss derivative of BTop(-) B3.03
Nov 16 Mingcong Zeng MPI Bonn Real bordism and its friends VideoLink opens in a new window
Nov 23 Angélica Osorno Reed college Counting model structures on [n] VideoLink opens in a new window
Nov 30 Marco La Vecchia Warwick The completion and local homology theorems for equivariant complex bordism for all compact Lie groups B3.03
Dec 7 No Seminar
##### Abstracts
###### Nima Rasekh: Shadows are bicategorical Traces

Topological Hochschild homology (THH), first defined for ring spectra and then later dg-categories and spectrally enriched categories, is an important invariant with connections to algebraic K-theory and fixed point methods. The existence of THH in such diverse contexts motivated Ponto to introduce a notion that can encompass the various perspectives: a shadow of bicategories. On the other side, many versions of THH have been generalized to the homotopy coherent setting providing us with motivation to develop an analogous (∞,2)-categorical generalization of shadows.

The goal of this talk is to use an appropriate bicategorical notion of THH to prove that a shadow on a bicategory is equivalent to a functor out of THH of that bicategory. We then use this result to give an alternative conceptual understanding of shadows and, in particular, an alternative proof for the Morita invariance. Moreover, we use this alternative characterization to give an appropriate definition of a homotopy coherent shadow of (∞,2)-categories.

This is joint work with Kathryn Hess

###### Andrew Macpherson: Designer ∞-categories via homotopy theory of diagrams

In this talk, I will discuss two ways - Ehresmann's colimit sketches and Quillen's homotopy theory of categories - that *diagrams* are used in the construction of new categories with prescribed universal properties, and how in the world of higher categories, they merge and become essentially the same subject. Studying this as a general framework lets us give names to questions or phenomena - compositionality, filteredness, plus constructions - that pervade cocompletion constructions in category theory, and study the relations between them. Since all this is rather abstract, I'll try to couch it in terms of the examples that got me thinking about this.

###### Rajan Mehta: Frobenius objects in categories of relations and spans

Frobenius algebras can be given a category-theoretic definition in terms of the monoidal category of vector spaces, leading to a more general definition of Frobenius object in any monoidal category. In this talk, I will describe Frobenius objects in categories where the objects are sets and the morphisms are relations or spans. These categories can be viewed as toy models for the symplectic category. The main result is that, in both cases, it is possible to construct a simplicial set that encodes the data of the Frobenius structure. The simplicial sets that arise in this way satisfy conditions that are closely connected to the 2-Segal conditions of Dyckerhoff-Kapranov and Galvez-Carrillo-Kock-Tonks.

Commutative Frobenius objects in these categories give surface invariants that can be Boolean or natural number valued. I will give some explicit examples where the invariants can be computed. This work is a very small first step in a bigger program aimed at better understanding the relationship between Poisson/symplectic geometry and topological field theory. Part of the talk will be devoted to giving an overview of this program.

This is based on work with Ruoqi Zhang (arXiv:1907.00702), with Ivan Contreras and Molly Keller (arXiv:2106.14743), and work in progress with Ivan Contreras, Adele Long, and Sophia Marx

###### Anna Marie Bohmann: Algebraic K-theory for Lawvere theories: assembly and Morita invariance

Much like operads and monads, Lawvere theories are a way of encoding algebraic structures, such as those of modules over a ring or sets with a group action. In this talk, we discuss the algebraic K-theory of Lawvere theories, which contains information about automorphism groups of these structures. We'll discuss both particular examples and general constructions in the K-theory of Lawvere theories, including examples showing the limits of Morita invariance and the construction of assembly-style maps. This is joint work with Markus Szymik.

###### Irakli Patchkoria: Franke’s conjecture and derived infinity categories

For a “nice enough” homology theory on a stable infinity category, we introduce a derived infinity category which encodes the Adams spectral sequence associated to the homology theory. By running the Goerss-Hopkins obstruction theory in the later, we prove Franke’s algebraicity conjecture from 1996 which asserts the following: Suppose we are given a “nice enough” homology theory on a stable infinity category C with values in an abelian category A and assume that A has finite cohomological dimension and is sufficiently sparse. Then the homotopy category of C is equivalent to the homotopy category of differential complexes in A. In fact it turns out that up to some fixed level, higher homotopy categories are equivalent too. This gives algebraic models in the following special cases: Modules over sufficiently sparse ring spectra, chromatic stable homotopy category for large primes, diagram categories such as filtered objects or towers, and chromatic spectral Mackey functors in the coprime case for large primes. This is all joint with Piotr Pstrągowski.

###### Oscar Randal-Williams: The second Weiss derivative of BTop(-)

One of the motivating examples for Weiss' orthogonal calculus is the functor sending an inner product space V to BTop(V). The zeroth derivative of this functor is the space BTop, which is rationally equivalent to BO which has well-known rational homotopy groups. The first derivative is Waldhausen's A(*), which is rationally equivalent to K(Z) and also has well-known, though much more difficult to calculate, rational homotopy groups. I will explain joint work with Manuel Krannich in which we determine the second derivative, rationally.

###### Mingcong Zeng: Real bordism and its friends

The Real bordism spectrum and its norms are fundamental objects in equivariant stable homotopy theory. Real bordism is defined and investigated in the 60s and 70s by Fuji and Landweber, then investigated in the 2000s by Hu and Kriz. Real bordism and its norms are essential components in the Hill--Hopkins--Ravenel solution of the Kervaire invariant one problem.

In this talk, I will discuss different aspects of Real bordism and its norms. I will show how they can help in understanding computations in chromatic homotopy theory and the Segal conjecture, and how the latter can feed back into giving new computations about Real bordism. The essential computational device in this approach is the localized slice spectral sequence, which fits into a picture of the "Tate diagram of spectral sequences".

Material in this talk are taken from various joint work with Agnes Beaudry, Mike Hill, Tyler Lawson, Lennart Meier and XiaoLin Danny Shi.

###### Angélica Osorno: Counting model structures on [n]

We enumerate the collection of Quillen model structures on the category [n] associated with the finite linear order 0 < 1 < ··· < n, exhibiting surprising connections with Catalan-type combinatorics and the Tamari lattice (i.e., associahedron) along the way. This uses previous work of the authors on the theory of N∞-operads for cyclic groups of prime power order. This is joint work with Scott Balchin, Kyle Ormsby and Constanze Roitzheim.

###### Marco La Vecchia: The completion and local homology theorems for equivariant complex bordism for all compact Lie groups

We generalize the local homology theorem for the equivariant spectrum MU from finite extensions of a torus to compact Lie groups using the splitting of global functors proved by Schwede. This proves a conjecture of Greenlees and May.