Title pretalk: Prime ideal spectra for tensor categories
Abstract pretalk: Given a non-semisimple abelian tensor category C (e.g. representations of a finite group taken over a field of characteristic p, which divides the order of the group) a basic problem is to describe the indecomposable objects up to a reasonable notion of equivalence. In even basic examples, describing the indecomposables up to isomorphism is impossible. In order to understand indecomposable modules up to a rougher notion of equivalence, we will define a topological space, called the Balmer spectrum, associated to C via prime ideals. This was accomplished in the symmetric case by Balmer, and was developed in the nonsymmetric case by Buan—Krause—Solberg and by Nakano—V.--Yakimov. We will describe topological properties of the Balmer spectrum and give illustrations using semidirect products.
Title: Support varieties and the Nerves of Steel Conjecture
Abstract: Support varieties, in many different forms, have been a major tool in representation theory, and include Quillen's cohomological support varieties for finite groups, Carlson's rank varieties for elementary abelian p-groups, and the Pi-point supports of Friedlander--Pevtsova for finite group schemes. The Balmer spectrum is universal space for supports for any monoidal triangulated category. The Nerves of Steel Conjecture, proposed by Balmer in 2020, posits that for any tensor triangulated category, every point of the Balmer spectrum is witnessed by a unique tensor functor to a sufficiently simple tensor category. We prove that Nerves of Steel holds for coordinate rings of finite group schemes. This talk will focus on joint work with Dan Nakano and Milen Yakimov.