Tag - Algebraic topology
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.
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.
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.
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.
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.
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.
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.
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.
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