Objective
We will use modern techniques in algebraic geometry, originating from string theory and mirror symmetry, to study fundamental problems of classical flavour. More concretely, we apply wall-crossing in the derived category to the birational geometry of moduli spaces.
Bridgeland stability is a notion of stability for complexes in the derived category. Wall-crossing describes how moduli spaces of stable complexes change under deformation of the stability condition, often via a birational surgery occurring in its minimal model program (MMP). This relates wall-crossing to the most basic question of algebraic geometry, the classification of algebraic varieties.
Our previous results additionally provide a very direct connection between Bridgeland stability conditions and positivity of divisors, the main tool of modern birational geometry. This makes the above link significantly more effective, precise and useful. We will exploit this in the following long-term projects:
1. Prove a Bogomolov-Gieseker type inequality for threefolds that we conjectured previously. This would provide a solution in dimension three to well-known open problems of seemingly completely different nature: the existence of Bridgeland stability conditions, Fujita's conjecture on very ampleness of adjoint line bundles, and projective normality of toric varieties.
2. Study the birational geometry of moduli space of sheaves via wall-crossing, adding more geometric meaning to their MMP.
3. Prove that the MMP for local Calabi-Yau threefolds is completely induced by deformation of Bridgeland stability conditions. The motivation is a derived version of the Kawamata-Morrison cone conjecture, classical questions on Chern classes of stable bundles, and mirror symmetry.
4. Answer major open questions on the birational geometry of the moduli space of genus zero curves (for example, the F-conjecture) using exceptional collections in the derived category and wall-crossing.
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: https://op.europa.eu/en/web/eu-vocabularies/euroscivoc.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: https://op.europa.eu/en/web/eu-vocabularies/euroscivoc.
- medical and health sciences clinical medicine surgery
- natural sciences physical sciences theoretical physics string theory
- natural sciences mathematics pure mathematics geometry
- natural sciences mathematics pure mathematics algebra algebraic geometry
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Programme(s)
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Multi-annual funding programmes that define the EU’s priorities for research and innovation.
Topic(s)
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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.
Call for proposal
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Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
ERC-2013-StG
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Funding Scheme
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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.
Host institution
EH8 9YL Edinburgh
United Kingdom
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