Project description
Using homotopical tools to study moduli spaces of low-dimensional manifolds
The study of moduli spaces of low-dimensional manifolds lies in the intersection of algebra, geometry and topology. The ERC-funded ModuLow project will focus on studying moduli spaces of three-dimensional manifolds and knots or links within them. Moduli spaces are ‘spaces of spaces’: they capture all possible variations of these geometric objects in a single space. The ModuLow project aims to extend discrete, object-level techniques to space-level homotopical techniques on moduli spaces of low-dimensional manifolds, with applications to their symmetries and configurations.
Objective
The topological properties of low dimensional moduli spaces play a fundamental role across algebra, geometry, and topology. My research programme will build homotopical tools for moduli spaces of 3- and 4-manifolds, and moduli spaces of links in 3-manifolds, with my main applications being to manifold symmetries and configurations. The upgrading of discrete techniques for ‘an object’ to space-level techniques on ‘the moduli space of all such objects’ is a central theme.
Moduli spaces of manifolds, or, equivalently, classifying spaces of manifold diffeomorphism groups, are foundational objects which classify smooth manifold bundles. The diffeomorphism group of a manifold—the group of symmetries—is a topological group which ought to be studied through a homotopical lens. My recent joint work solved a conjecture of Kontsevich on the homotopy type of moduli spaces of 3-manifolds. This breakthrough lays the groundwork for some of my goals: I will use it to compute rational cohomology rings of salient 3-manifold moduli spaces, yielding characteristic classes of 3-manifold bundles. Furthermore, I will classify the existence of sections for natural maps between 3-manifold and 4-manifold moduli spaces.
The topological properties of moduli spaces of unparametrised links in 3-manifolds will play a central role: these can be viewed as configuration spaces of 1-manifolds in 3-manifolds. These spaces are yet to be thoroughly understood, and it is a fundamental problem to describe their homotopy type. I will prove a finiteness theorem, and develop a framework to compute motion groups of configurations of links in 3-manifolds. An underlying theme is homological stability, and I will show families of these spaces satisfy (higher) homological stability.
One of my main aims is to combine my work on all three moduli spaces, by introducing an innovative method to represent families of 4-manifold diffeomorphisms via motions of Kirby diagrams (decorated links) in a 3-manifold.
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.
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-2025-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.
G12 8QQ Glasgow
United Kingdom
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