Objective
The overarching goal of this project is to reveal and systematically study the geometry of important categories in homotopy theory, algebraic geometry, and representation theory. To this end, we will introduce and develop the framework of higher Zariski geometry in which commutative rings are replaced by ring-like categories as the fundamental objects. The resulting theory simultaneously generalizes modern algebraic geometry, derived algebraic geometry as studied by Lurie and Toën--Vezzosi, as well as Balmer's tensor triangular geometry, while introducing entirely new global objects and methods. In particular, it provides a canonical spectral decomposition of a large class of higher categories over their Balmer spectrum, which is then used to produce powerful new tools to tackle some of the most important conjectures in their respective fields: Firstly, we will construct a higher analogue of the étale topology and étale homotopy types for such categories, giving rise to refined computational tools via descent. This machinery will be applied to modular representation theory to give a complete description of the group of endotrivial modules for any finite group and any field, extending the celebrated work of Carlson--Thévenaz and completing a program that began about 50 years ago. Secondly, we will introduce a categorical analogue of the Beilinson--Parshin adèles, in particular bringing to bear techniques from the point-set topology of spectral spaces. Applications include significant progress on Greenlees' conjecture on algebraic models for G-equivariant cohomology for a general compact Lie group G, which has remained open for more than 20 years. Thirdly, building on our earlier work on higher ultraproducts, we will study compactifications of categories and plan to combine these with recent advances in arithmetic geometry to make progress on the rational part of Hopkins' chromatic splitting conjecture, one of the most important open problems in homotopy theory.
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
- natural sciences mathematics pure mathematics topology
- natural sciences mathematics pure mathematics geometry
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Keywords
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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HORIZON.1.1 - European Research Council (ERC)
MAIN PROGRAMME
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Topic(s)
Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Funding Scheme
Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
HORIZON-ERC - HORIZON ERC Grants
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Call for proposal
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2021-STG
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Net EU financial contribution. The sum of money that the participant receives, deducted by the EU contribution to its linked third party. It considers the distribution of the EU financial contribution between direct beneficiaries of the project and other types of participants, like third-party participants.
80539 MUNCHEN
Germany
The total costs incurred by this organisation to participate in the project, including direct and indirect costs. This amount is a subset of the overall project budget.