Follow-up WS to TP "Spectral Methods in Algebra, Geometry and Topology"

Monday

Greg Stevenson

Beren Sanders

Martin Gallauer

Scott Balchin

Tuesday

Dave Benson

Emmy van Rooy

Sam Miller

Luca Pol

Wednesday

Drew Heard

Jordan Williamson

Thursday

Jan Stovicek

Janina Letz

Jun Maillard

Juan Omar Gomez

Friday

Jesper Grodal

Tom Peirce

Stefan Schwede

Beren Sanders (University of California, Santa Cruz): An overview of the theory of stratification

We will review the theory of stratification for classifying localizing ideals of tensortriangulated categories and discuss how two complementary approaches together yield strong descent results. Time permitting, we will discuss recent results on the classification of homological and cohomological Bousfield classes.

Dave Benson (University of Aberdeen): The Balmer spectrum of the Schur algebra

This is a preliminary report of work in progress, joint with Iyengar, Krause and Pevtsova.

The aim is to classify the thick subcategories of the bounded derived category of the Schur algebra in prime characteristic. The methods involve translating through Schur-Weyl duality and adapting methods developed by Balmer and Gallauer in a closely related context. Their tensor induced complexes don’t work in our context, and are replaced by Troesch complexes.

Emmy van Rooy (University of California, Los Angeles): Residual regularity for tt-categories

In this talk, we introduce the new notion of ’residual regularity’ for rigidly-compactly generated tt-categories, generalising the classical concept of regularity from algebraic geometry. The approach is based on the theory of homological residue fields, as developed by Balmer-Krause-Stevenson. Analysing residual regularity in the context of modular representation theory, we obtain a complete classification of finite groups G whose derived category of permutation kG-modules is residually regular. A core part of the proof is a technical result concerning finite separable extensions of tt-categories,

which are generalisations of finite étale extensions in algebraic geometry. This technical result asserts that, under some mild additional conditions, a finite separable extension preserves and reflects residual regularity.

Luca Pol (MPIM Bonn): On the tt-geometry of global representations

A global representation is a compatible collection of representations of the outer automorphism groups of the finite groups belonging to a family U. These arise in classical representation theory, in the study of representation stability, as well as in global homotopy theory. In this talk, I will discuss how to use tensor-triangular geometry to study the derived category of global representations over fields k of characteristic zero. In particular, I will present some calculations of Balmer spectra for various infinite families of finite groups. Depending on time I will also discuss the classification of localizing tensor ideals in global representation theory.

Drew Heard (NTNU): A non-spatial frame of smashing ideals

The smashing tensor ideals of a rigidly-compactly generated tensor-triangulated category

form a frame, but it has remained unclear whether this frame must be spatial. I will give a negative answer, already for the unbounded derived category of a (necessarily non-noetherian) commutative ring, via an embedding of the frame of idempotent ideals into the frame of smashing ideals. This is joint with Tobias Barthel.

Sam Miller (University of Georgia): Some updates on permutation twisted cohomology

Twisted cohomology for permutation modules, as first developed by Balmer—Gallauer then extended by the speaker, plays a crucial role in the tensor-triangualr geometry of the derived category of permutation modules. We give a few updates on the status of developing a twisted cohomology theory for all finite groups and showing that it completely recovers the Balmer spectrum of the category. We also say some words on orientability of invertible objects and representation spheres. Parts of this talk are joint work in progress with J. O. Gómez.