Project description
Algebraic solutions to enumerative problems in real and complex geometry
Enumerative geometry, the mathematics of counting numbers of solutions to geometry problems, analyses geometric problems by computing numerical invariants. This branch of algebraic geometry has successfully provided solutions to counting problems in geometry over the complex numbers. The EU-funded QUADAG project is using algebraic geometry and motivic homotopy theory to develop new, purely algebraic methods for handling enumerative problems over the real numbers, rational numbers or finite fields. The project will build on successful previous work by the researcher that has led to the development of a purely algebraic approach to tackling enumerative geometry problems, shedding light on both the complex and real solutions in a unified way.
Objective
Enumerative geometry, the mathematics of counting numbers of solutions to geometric problems, and its modern descendents, Gromov-Witten theory, Donaldson-Thomas theory, quantum cohomology and many other related fields, analyze geometric problems by computing numerical invariants, such as intersection numbers or degrees of characteristic classes. This essentially algebraic approach has been successful mainly in the study of problems over the complex numbers and other algebraically closed fields. There has been progress in attacking enumerative problems over the real numbers; the methods are mainly non-algebraic. Arithmetic content underlying the numerical invariants is hidden when analyzed by these non-algebraic methods. Recent work by the PI and others has opened the door to a new, purely algebraic approach to enumerative geometry that recovers results in both the complex and real cases in one package and reveals this arithmetic content over arbitrary fields. Building on these new developments, the goals of this proposal are, firstly, to use motivic homotopy theory, algebraic geometry and symplectic geometry to develop new purely algebraic methods for handling enumerative problems over an arbitrary field, secondly, to apply these methods to central enumerative problems, recovering and unifying known results over both C and R and thirdly, to use this new approach to reveal the hidden arithmetic nature of enumerative problems. In 2009 R. Pandharipande and I applied algebraic cobordism to prove the degree zero MNOP conjecture in Donaldson-Thomas theory. More recently, I have developed several aspects of the theory of quadratic invariants using motivic 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: 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.
- natural sciences mathematics pure mathematics arithmetics
- 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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H2020-EU.1.1. - EXCELLENT SCIENCE - 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.
ERC-ADG - Advanced Grant
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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-2018-ADG
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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.
45141 ESSEN
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.