Spectral Geometry of Higher Categories

Informazioni relative al progetto

GeoCats

ID dell’accordo di sovvenzione: 101042990

DOI

10.3030/101042990

Data della firma CE 27 Gennaio 2022

Data di avvio 1 Settembre 2022

Data di completamento 31 Agosto 2027

Finanziato da

• European Research Council (ERC)

Costo totale

€ 1 500 000,00

Contributo UE

€ 1 500 000,00

Coordinato da

MAX-PLANCK-GESELLSCHAFT ZUR FORDERUNG DER WISSENSCHAFTEN EV
Germany

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

Periodo di rendicontazione: 2022-09-01 al 2025-02-28

Sintesi del contesto e degli obiettivi generali del progetto

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.

Lavoro eseguito dall’inizio del progetto fino alla fine del periodo coperto dalla relazione e principali risultati finora ottenuti

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.

Progressi oltre lo stato dell’arte e potenziale impatto previsto (incluso l’impatto socioeconomico e le implicazioni sociali più ampie del progetto fino ad ora)

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.