Spectral Geometry of Higher Categories

Project Information

GeoCats

Grant agreement ID: 101042990

DOI

10.3030/101042990

EC signature date 27 January 2022

Start date 1 September 2022

End date 31 August 2027

Funded under

• European Research Council (ERC)

Total cost

€ 1 500 000,00

EU contribution

€ 1 500 000,00

Coordinated by

MAX-PLANCK-GESELLSCHAFT ZUR FORDERUNG DER WISSENSCHAFTEN EV
Germany

Periodic Reporting for period 1 - GeoCats (Spectral Geometry of Higher Categories)

Reporting period: 2022-09-01 to 2025-02-28

Summary of the context and overall objectives of the project

Categories provide a powerful organizational tool to transport ideas, methods, and results across different fields of mathematics. The underlying perspective taken in this project is that a large class of categories carry an intrinsic geometric structure which offers new insights on various outstanding open problems. Combining and extending ideas from modern algebraic geometric as well as tensor-triangular geometry, our first aim is to systematically develop a theory that uncovers, describes, and studies the hidden geometry of important categories. Building on this foundational work, we will then utilize the newfound geometric structures to tackle fundamental questions and conjectures in both homotopy theory and representation theory.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The team members and their collaborators have made significant progress on several of the projects of the proposal, both pertaining to the foundational aspects as well as their applications. One highlight is the resolution of conjectures of Morava from the early 1970s and of Hopkins from the early 1990s in chromatic homotopy theory. These results are based on deep connections between homotopy theory and p-adic geometry. Other advances include far-reaching generalizations of Quillen's work on stratification to equivariant homotopy theory as well as progress on Greenlees' program aiming to construct algebraic models of rational equivariant homotopy theory of compact Lie groups.

In the first part of the funding period, the members of the project have published over 20 research articles, all submitted for publication in peer-reviewed journals or already accepted or published.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The computation of the rational homotopy groups of K(n)-local spheres as well as Hopkins' Picard groups at all heights and primes have been seen as breakthroughs. Particularly interesting are the connections between homotopy theory and p-adic geometry that go into the proofs, and these are ripe for further investigations. There has also been partial progress on the remaining projects of the proposal, which are being actively pursued by the PI and the other team members.