Term 3

The talks will take place on Tuesday at 4 or 5pm (note the change of tim from the other terms), in B3.02.

Date	Speaker	Affiliation	Title	Notes
Apr 23	Thomas Wasserman	University of Oxford	Cohen-Macaulay simplicial complexes and duality groups	5pm
Apr 30	Stephen Theriault	University of Southampton	Homotopy theoretic properties of Poincaré Duality complexes	4pm
May 7	Elizabeth Tatum	University of Bonn	Applications of Equivariant Brown-Gitler Spectra	5pm
May 14	Christian Dahlhausen	Heidelberg University	Towards A^1-homotopy theory of rigid analytic spaces	5pm
May 21	Cheuk Yu Mak	University of Southampton	Loop group action on symplectic cohomology	4pm
May 28	Rhiannon Savage	University of Oxford	A Representability Theorem for Stacks in Derived Geometry Contexts	5pm
Jun 4	Daniel Graves	University of Leeds	Reflexive homology and beyond	5pm
Jun 11	Jack Davies	University of Bonn	Geometric norms on equivariant elliptic cohomology	5pm
Jun 18	Ramón Flores	Universidad de Sevilla	Baum-Connes isomorphism for certain groups of matrices	4pm
Jun 25				No Seminar

Abstracts

Thomas Wasserman: Cohen-Macaulay simplicial complexes and duality groups

Duality groups are groups that admit a Poincaré-duality-like relationship between their cohomology and their homology twisted by a module known as the dualising module. Many interesting groups are (virtually) duality groups, like free groups and their (outer) automorphisms, mapping class groups, and the general linear groups over the integers. Knowing duality is useful for homology computations, and the dualising module often has a nice interpretation. In this talk I will discuss work with Ric Wade where we explore the relationship between groups acting on simplicial complexes that are locally Cohen-Macaulay (meaning that their local homology is concentrated in a single degree and free) and groups having duality, and the implications this has for the dualising module.

Stephen Theriault: Homotopy theoretic properties of Poincaré Duality complexes

Let M be a simply-connected closed Poincaré Duality complex of dimension n. The goal is to gain insight into the homotopy theory of M by determining properties of its based loop space. An approach will be outlined that has had some success for particular families of Poincaré Duality complexes.

We will go on to discuss a new development. Rationally, it is known that if X is the (n-1)-skeleton of M then the based loops on M retracts off the based loops on X, provided the rational cohomology ring of M is not generated by a single element. We will show that under certain conditions this is true integrally. Families for which the integral statement holds include moment-angle manifolds and quasi-toric manifolds.

Elizabeth Tatum: Applications of Equivariant Brown-Gitler Spectra

In the 1980s, Mahowald used integral Brown-Gitler spectra to construct a splitting of the cooperations algebra for ko, connective real k-theory. Mahowald's splitting helped make it feasible to do computations using the ko-based Adams spectral sequence. This led to many interesting results on v_1-periodicity, including a proof that the Telescope Conjecture holds at height 1.

In recent work, Guchuan Li, Sarah Petersen, and I have constructed models for C_2-equivariant analogues of the integral Brown-Gitler spectra. In this talk, I will report on our progress towards using these spectra to construct C_2-equivariant analogues of Mahowald's splitting, and related results.

Christian Dahlhausen: Towards A^1-homotopy theory of rigid analytic spaces

In this talk, I will report about work in progress with Can Yaylali (Darmstadt/Orsay) towards A^1-homotopy theory of rigid analytic spaces. In the beginning, I will recall the motivating algebraic theory such as Morel-Voevodsky's seminal work on unstable A^1-homotopy of schemes, Voevodsky's stable version, and Ayoub's proof of a six-functor formalism for that. We seek to study a rigid analytic analogue using the rigid affine line A^1 as an interval. For this purpose, I will give some background on rigid analytic spaces where there are (at least) two canonical interval objects for doing homotopy theory, the closed unit disc B^1 and the rigid affine line A^1. The B^1-homotopy category has already been defined and studied by Ayoub and a full six-functor formalism was established by Ayoub-Gallauer-Vezzani. One drawback of the B^1-invariant theory is that analytic K-theory for rigid analytic spaces (as defined and studied by Kerz-Saito-Tamme) is not representable since it is not B^1-invariant. Thus we aim for an A^1-invariant version with coefficients in any presentable category. For the stable theory, we can prove the existence of a partial six-functor formalism for analytifications of schemes and algebraic morphisms between them by using the results of Ayoub's thesis. Furthermore, using coefficients in condensed categories, we render analytic K-theory representable in the unstable category and identify it with Z x BGL, in analogy to the case of schemes.

Cheuk Yu Mak: Loop group action on symplectic cohomology

For a compact Lie group G, its massless Coulomb branch algebra is the G-equivariant Borel-Moore homology of its based loop space. This algebra is the same as the algebra of regular functions on the BFM space. In this talk, we will explain how this algebra acts on the equivariant symplectic cohomology of Hamiltonian G-manifolds when the symplectic manifolds are open and convex. This is a generalization of the closed case where symplectic cohomology is replaced with quantum cohomology. Following Teleman, we also explain how it relates to the Coulomb branch algebra of cotangent-type representations. This is joint work with Eduardo González and Dan Pomerleano.

Rhiannon Savage: A Representability Theorem for Stacks in Derived Geometry Contexts

The representability theorem for stacks, due to Artin in the underived setting and Lurie in the derived setting, gives conditions under which a stack is representable by an n-geometric stack. In recent work of Ben-Bassat, Kelly, and Kremnizer, a new theory of derived analytic geometry has been proposed as geometry relative to the ∞-category of simplicial commutative Ind-Banach R-modules, for R a Banach ring. In this talk, I will present a representability theorem which holds in a very general context, encompassing both the derived algebraic geometry context of Toën and Vezzosi and these new derived analytic geometry contexts.

Daniel Graves: Reflexive homology and beyond

Reflexive homology is a homology theory for involutive algebras. In this talk I will introduce reflexive homology, explain how it fits into the framework of crossed simplicial groups and present some recent results and calculations. Time permitting, I will discuss some recent work (joint with Sarah Whitehouse) where we build an action of a group into these constructions.

Jack Davies: Geometric norms on equivariant elliptic cohomology

A notion of multiplicative structure on equivariant cohomology theories will be called a ``geometric norm structure''. We will then introduce equivariant elliptic cohomology, as defined by Lurie and Gepner--Meier, and discuss how it naturally comes with a geometric norm structure. We will also touch on a few applications of these structures: constructions of C_p-normed algebra structures and integral models for Behrens' Q(N) spectra. This is joint work-in-progress with William Balderrama and Sil Linskens.

Ramón Flores: Baum-Connes isomorphism for certain groups of matrices

In this talk, we present an analysis of the Baum-Connes isomorphism for general and special affine groups. Specifically, after introducing the structure of these groups and certain crystallographic groups involved in our calculations, and providing a brief review of the Baum-Connes conjecture, we will describe in detail the computation of the K-homology groups from which the assembly map originates, constituting the left-hand side of the isomorphism. The talk will conclude with an analysis of how the conjecture can be proven ad hoc for the general group, and why it has not been possible to do so in the special case. This is joint work with Sanaz Pooya (Potsdam) and Alain Valette (Neuchâtel).

Term 2

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

Date	Speaker	Affiliation	Title	Notes
Jan 9	Doosung Park	University of Wuppertal	Syntomic cohomology and real topological cyclic homology
Jan 16	Ian Leary	University of Southampton	The spectrum of the Chern subring for a fusion system
Jan 23	Juan Omar Gomez Rodriguez	Bielefeld University	Decompositions of the stable module category for infinite groups
Jan 30	Alexey Elagin	University of Edinburgh	Thick subcategories on smooth curves and orbifold curves
Feb 6	Stephen Theriault	University of Southampton	Homotopy theoretic properties of Poincare Duality complexes	Talk cancelled
Feb 13	Adam Thomas	University of Warwick	Symmetric spaces for real compact Lie groups: an algebraist's viewpoint
Feb 20	Magdalena Kedziorek	Radboud University Nijmegen	An algebraic model for rational SO(2)-spectra
Feb 27	Marco Linton	University of Oxford	The relation gap conjecture for groups with cyclic relation modules
Mar 5	Isambard Goodbody	University of Glasgow	Reflexivity and Hochschild cohomology
Mar 12	Matthew Pressland	University of Glasgow	Positroid varieties via triangulated categories

Abstracts

Doosung Park: Syntomic cohomology and real topological cyclic homology

In this talk, I will show that real topological cyclic homology admits a complete exhaustive filtration whose graded pieces are equivariant suspensions of syntomic cohomology. Combined with the announced results of Antieau-Krause-Nikolaus and Harpaz-Nikolaus-Shah, this would lead to the computation of the equivariant slices of the real K-theory of Z/p^n after a certain suspension. The key ingredients of the proof are a real refinement of the Hochschild-Kostant-Rosenberg filtration and the computation of real topological Hochschild homology of perfectoid rings in my joint work with Hornbostel.

Ian Leary: The spectrum of the Chern subring for a fusion system

(Joint work with Jason Semeraro) Around 1970 Quillen gave a complete description of the prime ideal spectrum for the mod-p cohomology of a finite group $G$ in terms of its elementary abelian subgroups. Around 1995 David Green and I gave an analogous result for the subring of cohomology generated by Chern classes of representations of $G$.

A (saturated) fusion system is a sort of `finite group at a single prime $p$'. An analogue of Quillen's theorem has been proved for fusion systems. Jason Semeraro and I give an analogues of my result with David Green and of some related results.

I intend that the talk will be accessible to someone who has seen neither Quillen's theorem nor fusion systems previously.

Juan Omar Gomez Rodriguez: Decompositions of the stable module category for infinite groups

In this talk, we will present a homotopy-theoretic interpretation of the stable module category for infinite groups over a commutative noetherian ring. For groups acting on trees with finite isotropy, we use this approach to exhibit a decomposition of the stable category in terms of their finite subgroups. As an application, we give a complete description of the Picard group of the stable category for discrete p-toral groups over a field.

Alexey Elagin: Thick subcategories on smooth curves and orbifold curves

I will talk about the classification problem for thick triangulated subcategories in triangulated categories, such as derived categories of coherent sheaves on algebraic varieties, or derived categories of modules over algebras. In some cases such classification is possible, but in general thick subcategories are abundant and hardly controllable. Nevertheless, one can classify thick subcategories in the derived category of a smooth projective curve _up to an equivalence_: every thick subcategory has a simple explicit description in terms of some quiver. Moreover, one can describe all quivers that arise in this way. If time permits, I will explain how these results generalise to weighted projective curves.

Stephen Theriault: Homotopy theoretic properties of Poincare Duality complexes

Let M be a simply-connected closed Poincare Duality complex of dimension n. The goal is to gain insight into the homotopy theory of M by determining properties of its based loop space. An approach will be outlined that has had some success for particular families of Poincare Duality complexes.

We will go on to discuss a new development. Rationally, it is known that if X is the (n-1)-skeleton of M then the based loops on M retracts off the based loops on X, provided the rational cohomology ring of M is not generated by a single element. We will show that under certain conditions this is true integrally. Families for which the integral statement holds include moment-angle manifolds and quasi-toric manifolds.

Adam Thomas: Symmetric spaces for real compact Lie groups: an algebraist's viewpoint

I have been fortunate enough to be introduced to some topics in algebraic topology by working with John Jones and Dmitriy Rumynin since arriving at Warwick. This talk will be my, heavily algebraic, take on our joint work. My punchline will be a uniform approach to calculating the rational cohomology ring of Rosenfeld Projective Planes. In fact, the approach is applicable to all homogenous spaces of Lie groups G/H where H has the same rank as G. I hope to make it accessible (and enjoyable) for algebraists, topologists and everyone in between!

Magdalena Kedziorek: An algebraic model for rational SO(2)-spectra

The category of G-spectra, for any compact Lie group G is very interesting, but at the same time very complicated. A big part of the interesting information comes from the internally encoded group action while one of the main complications comes from working over the integers. The first step in our understanding is to simplify this category by working over rationals. This removes the complexity coming from ordinary stable homotopy theory while leaving much of the information about group G still in the picture.

In this talk, I will discuss what we know about algebraic models for rational SO(2)-spectra, concentrating on our understanding of the most naive and the most genuine levels of commutativity in this setting. This talk is based on joint work with Dave Barnes and John Greenlees and ongoing work with John Greenlees.

Marco Linton: The relation gap conjecture for groups with cyclic relation modules

If F is a free group and F/N is a presentation of the group G, there is a natural way to turn the abelianisation of N into a left ZG-module, known as the relation module of the presentation. A presentation F/N is said to have relation gap if its relation module has strictly fewer ZG-module generators than N has normal generators. Infinite relation gaps were found by Bestvina--Brady, but the existence of finite relation gaps remains an open problem, closely connected to questions on the homotopy types of finite 2-complexes such as Wall's D(2) problem. I will first motivate the problem and survey what is known and what is not known. Then I will present a solution to the relation gap and relation lifting problems for certain groups with cyclic relation module.

Isambard Goodbody: Reflexivity and Hochschild cohomology

Smooth and proper DG-categories are noncommutative models for smooth and proper schemes. They also include finite dimensional algebras of finite global dimension. Kuznetsov and Shinder defined reflexive DG-categories as a (vast) generalisation; they include all projective schemes and all finite dimensional algebras. Smooth and proper DG-categories can also be characterised as the dualizable objects in the monoidal category of DG-categories localised at Morita equivalences. I’ll explain how by using a monoidal characterisation of reflexive DG-categories one can prove that the Hochschild cohomology of a reflexive DG-category is isomorphic to that of its derived category of cohomologically finite modules.

Matthew Pressland: Positroid varieties via triangulated categories

The totally non-negative Grassmannian is an important object in several stories, including Lusztig's total positivity, and the calculation of scattering amplitudes via the amplituhedron. It has a cell decomposition, described by Postnikov, in which each cell is obtained by intersecting the totally non-negative Grassmannian with a particular subvariety of the full Grassmannian: a so-called open positroid variety.

A useful tool in studying totally positive spaces is Fomin–Zelevinsky's theory of cluster algebras. A recent result of Galashin and Lam is that the coordinate ring of (the cone on) an open positroid variety has the structure of such an algebra, in two different but closely related ways, confirming a long-standing expectation. Muller and Speyer conjectured a precise relationship (quasi-coincidence) between these two cluster structures, which makes them equivalent from the point of view of total positivity. In this talk, I will explain how to prove their conjecture. Perhaps surprisingly, the proof is highly dependent on additive categorification, or in other words, on homological algebra in exact and triangulated categories.

Term 1

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

Date	Speaker	Affiliation	Title	Notes
Oct 3	Daniel Kasprowski	University of Southampton		postponed, train strikes
Oct 10	Özgür Bayındır	City University of London	Algebraic K-theory of the two-periodic first Morava K-theory
Oct 17	Daniel Kasprowski	University of Southampton	Stable equivalence relations of 4-manifolds
Oct 24	Itamar Mor	Queen Mary University of London	Profinite Galois descent in K(n)-local homotopy theory
Oct 31	Bastiaan Cnossen	University of Regensburg	Genuine sheaves on differentiable stacks
Nov 7	Scott Balchin	Queen's University Belfast	A jaunt through the tensor-triangular geometry of rational G spectra for G profinite or compact Lie
Nov 14	Kaif Hilman	Max Planck Institute Bonn	Equivariant hermitian K-theory and a periodicity result for C_2-L-theory
Nov 21	Emel Yavuz	Queen's University Belfast	C_2-Equivariant Orthogonal Calculus
Nov 28	Eric Finster	University of Birmingham	A Topos Theoretic View of Goodwillie Calculus
Dec 5	Irakli Patchkoria	University of Aberdeen	Posets of finite abelian subgroups and Morava K-theory
Dec 12	Anna Marie Bohmann	Vanderbilt University	Scissors congruence and the K-theory of covers

Abstracts

Özgür Bayındır: Algebraic K-theory of the two-periodic first Morava K-theory

Using a root adjunction formalism developed in an earlier work and logarithmic THH, we obtain a simplified computation of the algebraic K-theory of the complex K-theory spectrum. Furthermore, our computational methods also provide the algebraic K-theory of the two-periodic Morava K-theory spectrum of height 1.

Daniel Kasprowski: Stable equivalence relations of 4-manifolds

Kreck’s modified surgery gives an approach to classify 2n-manifolds up to stable diffeomorphism, i.e., up to a connected sum with copies of S^n x S^n. In dimension 4, we use a combination of modified and classical surgery to compare the stable diffeomorphism classification with other stable equivalence relations. Most importantly, we consider homotopy equivalence up to connected sum with copies of S^n x S^n. This is joint work with John Nicholson and Simona Veselá.

Itamar Mor: Profinite Galois descent in K(n)-local homotopy theory

Using condensed mathematics, I give a construction of the K(n)-local E_n-Adams spectral sequence as a HFPSS for the continuous action of the Morava stabiliser group. A modified version gives a spectral sequence computing the Picard and Brauer groups of K(n)-local spectra.

Bastiaan Cnossen: Genuine sheaves on differentiable stacks

Cohomology theories for equivariant spaces typically only depend on the associated quotient stacks X//G. It would thus be desirable to have a flexible framework for cohomology theories for stacks. In this talk, I will present such a framework, following ideas from motivic homotopy theory. The main result is a version of relative Poincaré duality for differentiable stacks, which generalizes Poincaré duality for smooth manifolds, Atyah duality for equivariant manifolds, and the Wirthmüller isomorphism in equivariant stable homotopy theory.

Scott Balchin: A jaunt through the tensor-triangular geometry of rational G spectra for G profinite or compact Lie

In this talk, I will report on joint work with Barnes--Barthel and Barthel--Greenlees which analyses the category of rational G equivariant spectra for G a profinite group or compact Lie group respectively. In particular, I will focus on a series of results regarding the Balmer spectra of these categories, and how the topology of these topological spaces informs structural results regarding the category.

Kaif Hilman: Equivariant hermitian K-theory and a periodicity result for C_2-L-theory

Building upon the recent foundational advances in hermitian K-theory by a set of nine authors and the formalism of parametrised higher categories by a set of five authors, I will introduce in this talk a G-equivariant enhancement of the hermitian theory via the notion of a G-Poincare category for finite groups G. As an application, I will indicate how one can use said enhancement to deduce a new periodicity result for L-theory when G is the cyclic group of order 2.

Emel Yavuz: C_2-Equivariant Orthogonal Calculus

Orthogonal homotopy calculus is the branch of functor calculus involving the study of functors from the category of finite dimensional real vector spaces to the category of pointed topological spaces. Using it, one can construct a Taylor tower of approximations to such functors, consisting of polynomial functors, and the layers of the tower are characterised by orthogonal spectra, making them much easier to compute.

A natural question is; what happens when the functors come with a group action? Such functors are of great interest, as they arise naturally within algebraic topology, for example the functor V \mapsto BO(V) where V is a G-representation. After an introduction to orthogonal calculus, I will discuss the main constructions and theorems of a C_2-equivariant orthogonal calculus, that works for functors from finite dimensional C_2-inner product spaces to C_2-spaces.

Eric Finster: A Topos Theoretic View of Goodwillie Calculus

I will describe a framework for understanding the unstable version of Goodwillie’s calculus of functors from a topos-theoretic perspective which builds on an analogy between higher topos theory and commutative algebra. In particular, I will describe how both Goodwilile’s original ”homotopy calculus” as well as the ”orthogonal calculus” of Michael Weiss can be understood in this framework. Along the way, we will see emerge a picture of the topos of n-excisive functors as classifying ”n-nilpotent” objects. This is joint with with M. Anel, G. Biedermann and A. Joyal.

Irakli Patchkoria: Posets of finite abelian subgroups and Morava K-theory

A result of K. Brown says that for a nice enough discrete group G, the orbifold Euler characteristic of G and the equivariant Euler characteristic of the poset of its non-trivial finite subgroups have the same fractional parts. We will present an analogous result for the poset of non-trivial finite abelian subgroups for which we will use Morava K-theory. After presenting some computations, we will discuss potential applications in number theory analogous to Brown’s results on denominators of special values of zeta functions.

Anna Marie Bohmann: Scissors congruence and the K-theory of covers

Scissors congruence, the subject of Hilbert's Third Problem, asks for invariants of polytopes under cutting and pasting operations. One such invariant is obvious: two polytopes that are scissors congruent must have the same volume, but Dehn showed in 1901 that volume is not a complete invariant. Trying to understand these invariants leads to the notion of the scissors congruence group of polytopes, first defined the 1970s. Elegant recent work of Zakharevich allows us to view this as the zeroth level of a series of higher scissors congruence groups.

In this talk, I'll discuss some of the classical story of scissors congruence and then describe a way to build the higher scissors congruence groups via K-theory of covers, a new framework for such constructions. We'll also see how to relate coinvariants and K-theory to produce concrete nontrivial elements in the higher scissors congruence groups. This work is joint with Gerhardt, Malkiewich, Merling and Zakharevich.