Project description
A new perspective on knot homology theories
Knot theory has made significant progress in recent decades with the introduction of homological knot invariants. These invariants extend beyond low-dimensional topology, linking knot theory to fields such as algebraic geometry, representation theory, Floer theory and physics. The ERC-funded CAPCAM project uses multicurve invariants to offer a fresh perspective on knot homology theories. Multicurves exhibit exceptional geometric and glueing properties, making them well-suited for implementing the divide-and-conquer approach to tackling challenging open problems. The project delves into problems in low-dimensional topology, investigates the topological properties of the new invariants, and applies their general principles to other contexts.
Objective
Knot theory has seen extraordinary developments over the past decades. The arrival of modern homological knot invariants has had far-reaching implications beyond low-dimensional topology, giving insight into old problems through deep ties between knot theory, algebraic geometry, representation theory, Floer theory, and physics.
My ERC project aims to establish a new perspective on knot homology theories using a new type of invariants, so-called multicurves. As objects of Fukaya categories of simple surfaces, these multicurve invariants make local versions of knot homology theories amenable to essentially combinatorial techniques. Thanks to their exceptional geometric and gluing properties, multicurves are ideally suited to implement the divide-and-conquer principle for attacking hard open problems. In fact, I have not only been directly involved in the definition of three of these invariants, but I have also applied them to resolve several open conjectures in the field already.
The purpose of my research programme is to investigate fundamental open problems in low-dimensional topology that require a deeper understanding of the new technology of multicurves. To this end, I will pursue the following four lines of basic research: I will investigate the topological properties of the new invariants and their relation to classical invariants. I will explore the existence of local versions of various spectral sequences that are known to relate knot homology theories. I will make the invariants more computable. Finally, I will apply the generic principles that underlie the definition of multicurves to other settings.
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 knot theory
- 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)
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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
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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-2023-STG
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44801 Bochum
Germany
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