Warwick Algebraic Topology Seminar 21/22
A list of the seminar talks of the previous years can be found 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 Ktheory  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  
Jun 14  Achim Krause  Münster  MS Teams  
Jun 21  
Jun 28 
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 inftycategorical 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 Gobjects, one for each compact Lie group G, which are compatible with the restrictioninflation 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 Ktheory
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 Ktheory. For instance, we obtain that T(n+1)locally, the algebraic Ktheory of the nth Morava Etheory contains the algebraic Ktheory of the JohnsonWilson 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 semisimplicial 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 cdhtopology or htopology. These topologies appeared in Voevodsky's proof of the BlochKato 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.
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  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 BeaudryHillShiZeng, and how this can be resolved by a general result relating pushouts of E_kalgebras 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 Ktheory.
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 socalled 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 GrothendieckWitt spaces. Recent work with W.Steimle identifies these in terms of algebraic cobordism categories, whence they can be analysed using the tools setup 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 Ktheory 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 RandalWilliams  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 dgcategories and spectrally enriched categories, is an important invariant with connections to algebraic Ktheory 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 categorytheoretic 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 2Segal conditions of DyckerhoffKapranov and GalvezCarrilloKockTonks.
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 Ktheory 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 Ktheory of Lawvere theories, which contains information about automorphism groups of these structures. We'll discuss both particular examples and general constructions in the Ktheory of Lawvere theories, including examples showing the limits of Morita invariance and the construction of assemblystyle 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 GoerssHopkins 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 RandalWilliams: 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 wellknown rational homotopy groups. The first derivative is Waldhausen's A(*), which is rationally equivalent to K(Z) and also has wellknown, 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 HillHopkinsRavenel 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 Catalantype 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.