Project description
New approach to study geometric shapes effectively
Mathematics often seeks to answer fundamental questions, such as how to count curves in manifolds or find rational solutions to polynomial equations. The ERC-funded EAGL project seeks to further understand the double ramification cycle – a key geometric structure to advance these areas. The research will focus on Gromov-Witten invariants – objects essential in physics and integrable systems. Traditionally, mathematicians either break complex shapes into simpler pieces or simplify the spaces where these shapes exist. EAGL will combine these two approaches for the first time. Furthermore, the project will study how many special solutions called rational torsion points exist on abelian surfaces – objects central to cryptography, coding theory and modular forms. Project results could lead to significant progress in geometry and arithmetic.
Objective
The PI has recently developed new techniques to understand the structure of the double ramification cycle, a geometric object playing a central role in the degeneration of curves and jacobians. In this project we will use these tools and results to count algebraic curves in manifolds, and to count solutions in the rational numbers to polynomial equations.
Our counts of curves will be algebraic Gromov-Witten invariants, which play a role in diverse areas including physics (where they are one of the two faces of mirror symmetry) and integrable systems (where their generating functions solve important hierarchies of PDEs). The strongest techniques currently available to understand Gromov-Witten invariants are to break the curve into simpler pieces (a cohomological field theory structure), or break the target into simpler pieces (enhancing to logarithmic Gromov-Witten invariants). This project will build the theoretical foundations needed to combine these two techniques, and explore the delicate combinatorial structures of the resulting invariants.
The rational solutions we count will be torsion points on abelian varieties. Abelian varieties are algebraic analogues of compact Lie groups, and play a pivotal role in diverse areas such as cryptography, coding theory, and modular and automorphic forms. The size of the torsion subgroup is one of the key invariants for the conjecture of Birch and Swinnerton-Dyer. We will prove a formal immersion property in dimension 2. This is a major step towards the Torsion Conjecture in dimension 2, which predicts that the number of rational torsion points on abelian surfaces is uniformly bounded.
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 computer and information sciences computer security cryptography
- natural sciences mathematics pure mathematics geometry
- natural sciences mathematics pure mathematics mathematical analysis differential equations partial differential equations
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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)
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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)
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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.
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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.
HORIZON-ERC - HORIZON ERC Grants
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Call for proposal
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Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2024-COG
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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.
2311 EZ Leiden
Netherlands
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