Series

Test

Test

Niko Naumann: Quillen stratification in equivariant homotopy theory

31st January, 2023

We prove a variant of Quillen's stratification theorem in equivariant homotopy theory for a finite group working with arbitrary commutative equivariant ring spectra as coefficients, and suitably categorifying it. We then apply our methods to the case of Borel-equivariant Lubin-Tate E-theory. In particular, this provides a computation of the Balmer spectrum as well as a cohomological parametrization of all localizing tensor-ideals of the category of equivariant modules over Lubin-Tate theory, thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory.

Toni Mikael Annala: Stable Homotopy without Homotopy

30th January, 2023

Many cohomology theories in algebraic geometry, such as crystalline and syntomic cohomology, are not homotopy invariant. This is a shame, because it means that the stable motivic homotopy theory of Morel-Voevodsky cannot be employed in studying the deeper aspects of such theories, such as cohomology operations that act on the cohomology groups. In this talk, I will discuss ongoing efforts, joint with Ryomei Iwasa and Marc Hoyois, to set up a workable theory of non-homotopy invariant stable motivic homotopy theory, with the goal of providing effective tools of studying cohomology theories in algebraic geometry by geometric means.

Henrik Holm: The Q-shaped derived category of a ring

24th January, 2023

The derived category, D(A), of the category Mod(A) of modules over a ring A is an important example of a triangulated category in algebra. It can be obtained as the homotopy category of the category Ch(A) of chain complexes of A-modules equipped with its standard model structure. One can view Ch(A) as the category Fun(Q, Mod(A)) of additive functors from a certain small preadditive category Q to Mod(A). The model structure on Ch(A) = Fun(Q, Mod(A)) is not inherited from a model structure on Mod(A) but arises instead from the "self-injectivity" of the special category Q. We will show that the functor category Fun(Q, Mod(A)) has two interesting model structures for many other self-injective small preadditive categories Q. These two model structures have the same weak equivalences, and the associated homotopy category is what we call the Q-shaped derived category of A. We will also show that it is possible to generalize the homology functors on Ch(A) to homology functors on Fun(Q, Mod(A)) for most self-injective small preadditive categories Q.

Ayelet Lindenstrauss: Algebraic K-theory and the Cyclotomic Trace

14th November, 2022

Projective modules over rings are the algebraic analogues of vector bundles; more precisely, they are direct summands of free modules. Some rings have non-free projective modules. For instance, the ideals of a number ring are projective, and for some number rings they need not be free. Even for rings like ℤ, over which all finitely generated projective modules are free, the category of such modules contains a wealth of interesting information. In this talk I will introduce algebraic K-theory, which encodes this information. I will also explain why one would try to use some kind of trace from K-theory to simpler invariants, outline the cyclotomic trace and briefly show how it is used in calculations.

Laurent Bartholdi: Dimension series and homotopy groups of spheres

28th October, 2022

The lower central series of a group G is defined by γ1=G and γn = [G,γn-1]. The 'dimension series', introduced by Magnus, is defined using the group algebra over the integers:

δn = { g : g-1 belongs to the nth power of the augmentation ideal }.

It has been, for the last 80 years, a fundamental problem of group theory to relate these two series. One always has δn ≥ γn, and a conjecture by Magnus, with false proofs by Cohn, Losey, etc., claims that they coincide; but Rips constructed an example with δ4 / γ4 cyclic of order 2. On the positive side, Sjogren showed that δn / γn is always a torsion group, of exponent bounded by a function of n. Furthermore, it was believed (and falsely proved by Gupta) that only 2-torsion may occur.

In joint work with Roman Mikhailov, we prove however that every torsion abelian group may occur as a quotient δn / γn; this proves that Sjogren's result is essentially optimal.

Even more interestingly, we show that this problem is intimately connected to the homotopy groups πn(Sm) of spheres; more precisely, the quotient δn / γn is related to the difference between homotopy and homology. We may explicitly produce p-torsion elements starting from the order-p element in the homotopy group π2p(S2) due to Serre.

Elizabeth Tatum: On a Spectrum-level Splitting of the BP⟨2⟩-Cooperations Algebra

9th August, 2022

In the 1980s, Mahowald and Kane used Brown-Gitler spectra to construct splittings of bo ⋀ bo and BP⟨1⟩ ⋀ BP⟨1⟩. These splittings helped make it feasible to do computations using the bo- and BP⟨1⟩-based Adams spectral sequences. In this talk, we will discuss an analogous splitting for BP⟨2⟩ ⋀ BP⟨2⟩ at primes larger than 3.

Mark Behrens: tmf-resolutions  

19th July, 2022

I'll talk about the tmf-based Adams spectral sequence, and how it detects most of the v2-periodic elements in the known range of the 2-primary stable stems.  Parts of the material I will discuss are joint with Dominic Culver, Prasit Bhattacharya, JD Quigley, and Mark Mahowald. 

David Barnes: Sheaf models for rational stable equivariant homotopy theory

28th June, 2022

Sheaves sit at an interface of algebra and geometry. Equivariant sheaves offer even more structure, allowing for different group actions at different stalks. We are interested in the case where both the base space and group of equivariance are profinite (that is, compact, Hausdorff and totally disconnected). This combination provides many useful consequences, such as a good notion of equivariant presheaves and an explicit construction of infinite products. 

The 2019 PhD thesis of Sugrue used equivariant sheaves to give an algebraic model for rational G-equivariant stable homotopy theory, where G is profinite. In this talk I will explain the model and related results, such as the equivalence between equivariant sheaves and rational Mackey functors (for profinite G).

Hans-Werner Henn: On the Brown Comenetz dual of the K(2)-local sphere at the prime 2  

21st June, 2022

The Brown Comenetz dual I of the sphere represents the functor which on a spectrum X is given by the Pontryagin dual of the 0-th homotopy group of X. For a prime p and a chromatic level n there is a K(n)-local version In of I. For a type n-complex X, this is given by the Pontryagin dual of the 0-th homotopy group of the K(n)-localization of X. By work of Hopkins and Gross, the homotopy type of the spectra In for a prime p is determined by its Morava module if p is sufficiently large. For small primes, the result of Hopkins and Gross determines In modulo an "error term". For n=1 every odd prime is sufficiently large and the case of the prime 2 has been understood for almost 30 years. For n>2 very little is known if the prime is small. For n=2 every prime bigger than 3 is sufficiently large. The case p=3 has been settled in joint work with Paul Goerss. This talk is a report on work in progress with Paul Goerss on the case p=2. The "error term" is given by an element in the exotic Picard group which in this case is an explicitly known abelian group of order 29. We use chromatic splitting in order to get information on the error term.     

Jesper Grodal: A guided tour to the Picard group of the stable module category

24th May, 2022

The Picard group Tk(G) of the stable module category of a finite group has been an important object of study in modular representation theory, starting with work Dade in the 1970s. Its elements are equivalence classes of so-called endotrivial modules, i.e., modules M such that End(M) is isomorphic to a trivial module direct sum a projective kG-module. 1-dimensional characters, and their shifts, are examples of such modules, but often exotic elements exist as well. My talk will be a guided tour of how to calculate Tk(G), using methods from homotopy theory. The tour will visit joint work with Tobias Barthel and Joshua Hunt, with Jon Carlson, Nadia Mazza and Dan Nakano, and with Achim Krause.

Jordan Williamson: Duality and definability in triangulated categories

17th May, 2022

In the category of modules over a ring, purity may be viewed as a weakening of splitting - a short exact sequence is pure if and only if it is split exact after applying the character dual. The notion of purity in triangulated categories was introduced by Krause, and it has since been seen to be intimately related to many questions of interest in representation theory and homotopy theory. However, in general, it can be hard to check whether a class is closed under purity operations. In this talk, I will explain a framework of duality pairs in triangulated categories which provides an elementary way to check pure closure properties, and illustrate this with a range of examples, often from the tensor-triangular perspective. I will also discuss an application to the study of definable subcategories of triangulated categories.

Martin Frankland: On good morphisms of exact triangles

3rd May, 2022

When studying the Adams spectral sequence in triangulated categories, one runs into the issue of choosing suitably coherent cofibers in an Adams resolution. Motivated by this, in joint work with Dan Christensen, we develop tools to deal with the limited coherence afforded by the triangulated structure. We use and expand Neeman's work on good morphisms of exact triangles. The talk will include examples from stable module categories of group algebras. 

Neil Strickland: Questions around the Chromatic Splitting Conjecture

19th April, 2022

The chromatic splitting conjecture (CSC) is an important open problem in stable homotopy theory. Although Beaudry has shown that the strongest version fails when n=p=2, one can still hope that the conjecture is valid at large primes, or up to a filtration that is nearly split in some appropriate sense.

The CSC gives a description of Ln-1LK(n)(S), but from that one can deduce similar descriptions of many other spectra, including those of the form LK(n1)LK(n2)...LK(nr)(S) with n1 < ... < nr.  The spectrum Ln(S) can be expressed as the homotopy inverse limit of a diagram of spectra of that form, and one can ask whether that, and various similar phenomena, are consistent with the CSC. We will explain a conjecture about how all this fits together, which involves some interesting algebra and combinatorics.  We will also explain how Morava K-theory Euler characteristics can be used to do some basic consistency checks.

Beren Sanders: Stratification in tensor triangular geometry with applications to spectral Mackey functors

22nd February, 2022

In a recent paper, joint with Tobias Barthel and Drew Heard, we develop a theory of stratification in the context of tensor triangular geometry and apply it to classify the localizing tensor-ideals of certain categories of spectral G-Mackey functors for all finite groups G. In this talk, I will provide an introduction to the problem of classifying thick and localizing tensor-ideals via theories of support, describe in broad strokes some of the highlights of our theory (which builds on the work of Balmer-Favi, Stevenson, and Benson-Iyengar-Krause) and, time-permitting, discuss our applications in equivariant homotopy theory. The starting point for these equivariant applications is a recent computation (joint with Irakli Patchkoria and Christian Wimmer) of the Balmer spectrum of the category of derived Mackey functors. We similarly study the Balmer spectrum of the category of E(n)-local spectral Mackey functors, and harness our geometric theory of stratification to classify the localizing tensor-ideals of both categories. 

James Cameron: Homological residue fields for tensor triangulated categories and cooperations

15th February, 2022

Many tensor triangulated categories admit 'residue field functors' that control their large-scale structure. The derived category of a ring is controlled by the residue fields of the ring, the structure of the stable homotopy category is controlled by the Morava K-theories, and in modular representation theory there are the pi-points. Unfortunately, it is not known if every tensor triangulated category has a notion of tensor triangulated residue fields. Homological residue fields were introduced by Balmer, Krause, and Stevenson as an abelian avatar of the putative tensor triangulated residue fields. They exist in complete generality, but they are hard to understand and compute with in general. I will discuss how to connect homological residue fields with the tensor triangulated residue fields that exist in examples. I will show that for the derived category of a ring, homological residue fields are closely related to usual residue fields, and in stable homotopy theory they are closely related to Morava K-theories. In fact, the homological residue fields have even more structure, and can be identified with comodules for a Tor coalgebra which in the case of the stable homotopy category is the coalgebra of coooperations for a Morava K-theory. I will introduce homological residue fields, give some examples, and mention some open problems. This is joint work with Paul Balmer and with Greg Stevenson. 

Kai Cieliebak: Poincaré duality for loop spaces

7th February, 2022

This talk is about ongoing joint work with Nancy Hingston and Alexandru Oancea. I will explain how various puzzles in string topology get resolved in terms of symplectic geometry: Loop space homology and cohomology are merged into a larger space, Rabinowitz Floer homology, which carries a product and coproduct extending those from string topology and satisfies Poincaré duality.

Niall Taggart: Homological localizations of orthogonal calculus

1st February, 2022

Orthogonal calculus is a version of functor calculus that sits at the interface between geometry and homotopy theory; the calculus takes as input functors defined on Euclidean spaces and outputs a Taylor tower of functors reminiscent of a Taylor series of functions from differential calculus. The interplay between the geometric nature of the functors and the homotopical constructions produces a calculus in which computations are incredibly complex. These complexities ultimately result in orthogonal calculus being an underexplored variant of functor calculus.

On the other hand, homological localizations are ubiquitous in homotopy theory. They are employed to split ‘integral' information into ‘prime' pieces, typically simplifying both computation and theory.

In this talk, I will describe a 'local' version of orthogonal calculus for homological localizations, and survey several immediate applications.

Doug Ravenel: String cobordism at the prime 3

18th January, 2022

String cobordism refers to the Thom spectrum for the 7-connected cover of BO, the classifying space for real vector bundles. I will describe progress toward a description of its 3-primary homotopy type in joint work with Vitaly Lorman and Carl McTague. It supports a map to tmf (the spectrum associated with topological modular forms) which is surjective in homotopy groups.