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ChromoCats

The geometry of chromatic categories

Total Cost €

0

EC-Contrib. €

0

Partnership

0

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 ChromoCats project word cloud

Explore the words cloud of the ChromoCats project. It provides you a very rough idea of what is the project "ChromoCats" about.

ing    progress    ultraproducts    finite    duality    conjecture    redshift    framework    global    mathematical    blueshift    insights    questions    powerful    structure    proven    principles    give    describes    gain    introduce    spectra    pertaining    topology    cohomology    representation    compactifications    modular    governed    examples    tools    found    conjectures    groups    generalization    splitting    extension    group    point    categories    light    geometric    algebraic    algebra    picard    thereby    transfer    prevented    outstanding    tate    shedding    context    lurie    substantial    balmer    unified    geometry    computation    categorification    fundamental    ambidexterity    subtle    interrelated    quasi    algebraization    hopkins    theory    construct    special    module    stack    deep    lack    chromatic    invariants    outcome    scheme    homotopy    systematically    solves    category    logic    axiomatizes    coherent    first    sheaves    structural    profinite    shown    stable    view    local    certain   

Project "ChromoCats" data sheet

The following table provides information about the project.

Coordinator
KOBENHAVNS UNIVERSITET 

Organization address
address: NORREGADE 10
city: KOBENHAVN
postcode: 1165
website: www.ku.dk

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country Denmark [DK]
 Project website https://www.math.ku.dk/english/about/news/marie-curie-to-two-sym-postdocs/
 Total cost 200˙194 €
 EC max contribution 200˙194 € (100%)
 Programme 1. H2020-EU.1.3.2. (Nurturing excellence by means of cross-border and cross-sector mobility)
 Code Call H2020-MSCA-IF-2016
 Funding Scheme MSCA-IF-EF-ST
 Starting year 2018
 Duration (year-month-day) from 2018-01-01   to  2019-12-31

 Partnership

Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 
1    KOBENHAVNS UNIVERSITET DK (KOBENHAVN) coordinator 200˙194.00

Map

 Project objective

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.

 Publications

year authors and title journal last update
List of publications.
2019 Tobias Barthel, Natalia Castellana, Drew Heard, and Gabriel Valenzuela
On stratification for spaces with Noetherian mod p cohomology
published pages: , ISSN: , DOI:
2019-10-10
2019 Barthel, Tobias; Heuts, Gijs; Meier, Lennart
A Whitehead theorem for periodic homotopy groups
published pages: , ISSN: , DOI:
2019-10-10
2019 Barthel, Tobias; Schlank, Tomer M.; Stapleton, Nathaniel
Monochromatic homotopy theory is asymptotically algebraic
published pages: , ISSN: , DOI:
2019-10-10
2019 Barthel, Tobias; Greenlees, J. P. C.; Hausmann, Markus
On the Balmer spectrum for compact Lie groups
published pages: , ISSN: 0010-437X, DOI:
Compositio Mathematica 2019-10-10
2018 Tobias Barthel, Bernhard Keller, and Henning Krause
Completing perfect complexes
published pages: , ISSN: , DOI:
2019-10-10
2018 Barthel, Tobias; Beaudry, Agnès; Goerss, Paul G.; Stojanoska, Vesna
Constructing the determinant sphere using a Tate twist
published pages: , ISSN: , DOI:
2019-10-10
2019 Barthel, Tobias
A short introduction to the telescope and chromatic splitting conjectures
published pages: , ISSN: , DOI:
Surveys around Ohkawa\'s theorem on Bousfield classes 2019-10-10
2019 Barthel, Tobias; Beaudry, Agnès
Chromatic structures in stable homotopy theory
published pages: , ISSN: , DOI:
Handbook of Homotopy Theory 2019-10-10
2019 Tobias Barthel, Drew Heard, Gabriel Valenzuela
Derived completion for comodules
published pages: , ISSN: 0025-2611, DOI: 10.1007/s00229-018-1094-0
manuscripta mathematica 2019-10-10

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