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Cut-and-paste conjectures and multicurves

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.

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Topic(s)

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HORIZON-ERC - HORIZON ERC Grants

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Call for proposal

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(opens in new window) ERC-2023-STG

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Host institution

RUHR-UNIVERSITAET BOCHUM
Net EU contribution

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.

€ 1 444 860,00
Address
UNIVERSITAETSSTRASSE 150
44801 Bochum
Germany

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Region
Nordrhein-Westfalen Arnsberg Bochum, Kreisfreie Stadt
Activity type
Higher or Secondary Education Establishments
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Total cost

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.

€ 1 444 860,00

Beneficiaries (1)

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