Project description
Modular forms unifying K3 structures and Hilbert spaces
Enumerative geometry counts geometric objects defined by polynomial equations. It was revolutionised a few decades ago by string theory (a physical framework describing fundamental particles as tiny vibrating strings) and serves as a fundamental bridge between mathematics and physics. The ERC-funded K3Mod project will probe structural links between different counting frameworks, focusing on two special classes of mathematical space – K3 surfaces and Hilbert schemes – whose geometric properties remain incompletely understood. By demonstrating that these frameworks are unified through structures known as modular forms, K3Mod will resolve several long-open conjectures in enumerative geometry. In addition, the project will generate new insights across number theory, representation theory, and mathematical physics.
Objective
Enumerative geometry is concerned with counting geometric objects on spaces defined by polynomial equations. The subject, which has roots going back to the ancient Greeks, was revolutionized by string theory in the 90s and has since become a fundamental link between algebraic geometry, representation theory, number theory and physics. With K3Mod I propose to establish a wide range of new correspondences in enumerative geometry. These link together different enumerative theories and open new perspectives to attack long-standing problems concerning the quantum cohomology of the Hilbert scheme of points on surfaces, modular properties of invariants of K3 surfaces, string partition functions of Calabi-Yau threefolds with links to Conway Moonshine, and a major case of the Crepant Resolution Conjecture.
The geometry of the Hilbert scheme of points on a surface will play a central role. I aim to prove a correspondence between its Gromov-Witten theory, and the Donaldson-Thomas theory of certain threefold families. Correspondences for moduli spaces of Higgs bundles and the orbifold theory of the symmetric product of surfaces will be considered as well. This provides methods to prove that Gromov-Witten invariants of Hilbert schemes of points on K3 surfaces are Fourier coefficients of quasi-Jacobi forms, possibly leading to a complete solution of their enumerative geometry. After elliptic curves, K3 surfaces form the simplest Calabi-Yau geometry for which a complete understanding of the Gromov-Witten theory is in reach. For elliptic threefolds, I will study the relationship of their Donaldson-Thomas invariants with quasi-Jacobi forms, using both degeneration techniques and wallcrossing formulae.
The research goals of this proposal will lead to exciting new connections between geometry, modular forms, and representation theory. The results will provide a clear understanding of the interplay between Hilbert schemes, K3 surfaces, and modularity in enumerative geometry.
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.
This project's classification has been human-validated.
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.
This project's classification has been human-validated.
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)
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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.
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.
100 44 STOCKHOLM
Sweden
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.