The geometry of chromatic categories

Información del proyecto

ChromoCats

Identificador del acuerdo de subvención: 751794

DOI

10.3030/751794

Proyecto finalizado el 31 Mayo 2019

Fecha de la firma de la CE 14 Noviembre 2017

Fecha de inicio 1 Enero 2018

Fecha de finalización 31 Diciembre 2019

Financiado con arreglo a

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Coste total

€ 200 194,80

Aportación de la UE

€ 200 194,80

Coordinado por

KOBENHAVNS UNIVERSITET
Denmark

Objetivo

This project studies the local and global structure of fundamental categories in topology, algebra, and algebraic geometry from a geometric point of view. Deep structural results have been proven in special cases, but the lack of a unified theory has prevented progress on several key conjectures, for example pertaining to local-to-global principles.

In a first step, we introduce the concept of chromatic category, which axiomatizes certain properties found on the derived category of quasi-coherent sheaves on a scheme or stack. Important examples of chromatic categories include the category of spectra in stable homotopy theory and the stable module category for a finite group. The resulting framework allows us to transfer tools and questions from one context to another, thereby shedding light on three key aspects of the geometry of a chromatic category: Its local structure, local-to-global principles, and compactifications.

In a second step, we study these three interrelated aspects in detail. The local structure of a chromatic category is controlled by its local Picard groups, which give new and subtle invariants in modular representation theory. We then gain new insights about the structure of these groups via local duality and a profinite extension of the theory of ambidexterity due to Hopkins and Lurie. Moreover, local-to-global principles like the chromatic splitting conjecture, blueshift, or redshift are shown to be governed by a generalization of Tate cohomology, for which we introduce powerful new tools of computation with applications to various Balmer spectra. Finally, we construct compactifications of chromatic categories via a categorification of ultraproducts from mathematical logic. This solves the algebraization problem in chromatic homotopy.

In conclusion, the outcome of this project is a framework that systematically describes the geometry of chromatic categories, leading to substantial progress on outstanding conjectures in algebra and topology.

Ámbito científico (EuroSciVoc)

• ingeniería y tecnología › ingeniería de materiales › colores
• ciencias naturales › matemáticas › matemáticas puras › matemáticas discretas › lógica matemática
• ciencias naturales › matemáticas › matemáticas puras › topología
• ciencias naturales › matemáticas › matemáticas puras › geometría
• ciencias naturales › matemáticas › matemáticas puras › álgebra › geometría algebraica

Palabras clave

• Chromatic homotopy theory
• stable module category
• Balmer spectrum
• Picard groups
• chromatic splitting conjecture
• blueshift
• local duality
• Tate construction
• compact groups
• ultraproducts

Programa(s)

• H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions PROGRAMA PRINCIPAL
  
• H2020-EU.1.3.2. - Nurturing excellence by means of cross-border and cross-sector mobility
  

Tema(s)

• MSCA-IF-2016 - Individual Fellowships
  

Régimen de financiación

MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)

Convocatoria de propuestas

H2020-MSCA-IF-2016

Coordinador

KOBENHAVNS UNIVERSITET

• Aportación neta de la UEn: € 200 194,80
• Dirección: NORREGADE 10 ; 1165 KOBENHAVN ; Dinamarca
• Región: Danmark › Hovedstaden › Byen København
• Tipo de actividad: Higher or Secondary Education Establishments
• Coste total: € 200 194,80