Tag - Homotopy theory
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.
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.
We provide a calculational method for rational stable equivariant homotopy theory for a torus G based on the homology of the Borel construction on fixed points. More precisely we define an abelian torsion model, 𝒜t(G) of finite injective dimension, a homology theory π∗𝒜t taking values in 𝒜t(G) based on the homology of the Borel construction, and a finite Adams spectral sequence
Ext𝒜t(G)∗ , ∗ (π∗𝒜t(X), π∗𝒜t(Y)) → [X,Y]∗G
for rational G-spectra X and Y.
This approach should be viewed as an analogue of the Cousin complex in algebraic geometry. It is expected that a similar method will apply to other tensor triangulated categories with finite-dimensional Noetherian Balmer spectra.
The honest answer to the question is that I actually do not know. I will therefore rather talk about several famous examples that are widely called 'h-principle results' and try to explain some of the ideas behind the ones I am most familiar with.
I am going to talk about the group-theoretic aspects of the Andrews-Curtis conjecture, some recent results, and some old. From my viewpoint the Andrews-Curtis conjecture is not just a hard stand-alone question, coming from topology, but a host of very interesting problems in group theory.
Mackey functors play a central role in equivariant homotopy theory, where homotopy groups are replaced by homotopy Mackey functors. In this talk, I will discuss joint work with Dan Dugger and Christy Hazel classifying perfect chain complexes of constant Mackey functors for G=ℤ/2. Our decomposition leads to a computation of the Balmer spectrum of the derived category. We extend these results to classify all finite modules over the equivariant Eilenberg-MacLane spectrum Hℤ/2.
Factorization systems (both weak and strong) are commonly defined as consisting of two classes of maps satisfying a certain orthogonality relation and a factorization axiom. The standard definition of algebraic weak factorization system, involving comonads and monads, is rather different. The goal of this talk will be to describe an equivalent definition of algebraic weak factorization system emphasising orthogonality and factorization.
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