Spectral Geometry of Higher Categories

Introduction

The overarching goal of this project is to reveal and systematically study the geometry of important categories in homotopy theory, algebraic geometry, and representation theory. To this end, we will introduce and develop the framework of higher Zariski geometry in which commutative rings are replaced by ring-like categories as the fundamental objects.

Theory Development

The resulting theory simultaneously generalizes modern algebraic geometry, derived algebraic geometry as studied by Lurie and Toën--Vezzosi, as well as Balmer's tensor triangular geometry, while introducing entirely new global objects and methods.

Canonical Spectral Decomposition

In particular, it provides a canonical spectral decomposition of a large class of higher categories over their Balmer spectrum, which is then used to produce powerful new tools to tackle some of the most important conjectures in their respective fields:

1. Higher Analogue of Étale Topology
   Firstly, we will construct a higher analogue of the étale topology and étale homotopy types for such categories, giving rise to refined computational tools via descent. This machinery will be applied to modular representation theory to give a complete description of the group of endotrivial modules for any finite group and any field, extending the celebrated work of Carlson--Thévenaz and completing a program that began about 50 years ago.

2. Categorical Analogue of Beilinson--Parshin Adèles
   Secondly, we will introduce a categorical analogue of the Beilinson--Parshin adèles, in particular bringing to bear techniques from the point-set topology of spectral spaces. Applications include significant progress on Greenlees' conjecture on algebraic models for G-equivariant cohomology for a general compact Lie group G, which has remained open for more than 20 years.

3. Compactifications of Categories
   Thirdly, building on our earlier work on higher ultraproducts, we will study compactifications of categories and plan to combine these with recent advances in arithmetic geometry to make progress on the rational part of Hopkins' chromatic splitting conjecture, one of the most important open problems in homotopy theory.