# Spectral Methods in Algebra, Geometry, and Topology

Trimester Program

September 12 - December 16, 2022

Organizers: Paul Balmer, Tobias Barthel, John Greenlees, Henning Krause, Julia Pevtsova

Spectra are ubiquitous throughout modern mathematics: The Zariski spectrum of a commutative ring, the topological spectrum representing a generalized cohomology theory, and the Balmer spectrum of a tensor-triangulated category are important instances. In each case, the spectral representation of a familiar object reveals its hidden geometry and symmetries. Amplified by modern homotopical and representation-theoretic techniques, recent years have seen a whirlwind of activity and groundbreaking progress in the development and application of spectral techniques, which may be loosely organized in the following themes:

**(1) Global classification problems:**

Classification of thick tensor ideals and localizing tensor ideals as the key to capturing the global structure of tensor categories; construction of novel support theories.**(2) Local-to-global principles:**

Assembly and disassembly in homotopy theory and modular representation theory; adelic techniques in rational equivariant homotopy theory; reconstruction theorems in (non-)commutative algebraic geometry.**(3) Invariants, duality, and descent:**

The computation of Picard groups and higher invariants like Brauer groups via descent techniques; local and global dualities.

The goal of this program is to bring together experts in the broad range of areas involving spectral methods in order to understand the most recent developments, exploit their interactions, foster future collaborations, while also providing a platform for junior mathematicians to enter this vibrant area of research.

**Activities during the trimester program:**

The program will include one school primarily addressed to PhD students and postdocs and two workshops

- School: Spectral methods in algebra, geometry, and topology (September 19-23, 2022)
- Workshop I: Spectral methods in equivariant mathematics (October 24-28, 2022)
- Workshop II: Spectra, triangles, and higher structures (December 5-9, 2022)