Project description
Identifying the ways in which a circle is linked to a non-sphere is not as easy as it seems
Topology is the mathematical study of properties preserved through deformation, twisting, and stretching of objects. For example, a circle is topologically equivalent to an ellipse into which it can be stretched. Manifolds are objects that can be mapped in this way. Fibres, and fibring, describe the mathematical way in which a point on the base manifold is projected to a neighbouring space. Numerous important questions remain unanswered in relation to the fibring of manifolds and groups. The EU-funded FIBRING project is tackling several of them with the goal of achieving a complete description of all possible fibrings over the circle for aspherical manifolds and groups in high dimensions.
Objective
The study of manifolds that fibre over the circle has a long and exciting history at the core of modern manifold topology, starting with Farrell's work on the problem in high ('surgery') dimensions, and running through the celebrated work of Stallings and Thurston in dimension 3, to Agol's 2013 solution of Thurston's virtual fibring conjecture. Parallel developments in group theory have placed the study of Bieri-Neumann-Strebel (BNS) invariants, which emerged in the 1980s, at the heart of the subject; these invariants describe when a group fibres, i.e. admits a map onto Z with finitely generated kernel. In the research outlined here a powerful new set of algebraic invariants - agrarian polytopes - will be used to establish a new frontier in the theory of fibring. The main goal is to achieve a complete description of all possible fibrings over the circle for aspherical manifolds in surgery dimensions.
An agrarian polytope is a subset of the vector space H_1(X;R), where X is a group or a manifold. It is defined in the novel framework of agrarian invariants that I am developing, a theory that has already borne remarkable fruit. The theory provides algebraic counterparts to the (analytic) L2-invariants that have proved so powerful in geometric topology, group theory and global analysis over the last four decades.
The primary focus of the research proposed here lies in establishing new deep connections between the algebra of group rings and their completions, and global properties of aspherical manifolds and groups. Three further goals of the proposal are: to develop the theory of agrarian invariants in positive characteristic; to show that agrarian invariants are profinitely rigid; to apply the new technology to the study of dynamical zeta functions. Each of these goals promises a breakthrough in its respective domain.
Fields of science (EuroSciVoc)
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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.
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- natural sciences mathematics pure mathematics topology
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Keywords
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Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
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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)
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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.
ERC-STG - Starting Grant
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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-2019-STG
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OX1 2JD Oxford
United Kingdom
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