Project description
Uniformisation of fractal metric surfaces and rigidity in the complex plane
The study of quasiconformal geometry extends classical geometry to more complicated settings, providing tools to study shapes that are not smooth or Euclidean. This framework enables mathematicians to analyse shapes with fractal, non-smooth properties and understand the relationships between different geometries. The ERC-funded GRComPaS project will develop novel methods for the study of quasiconformal geometry in the context of metric surfaces and subsets of the complex plane. It will emphasise uniformisation – the transformation of a non-smooth space into a simple, standard shape in a controlled manner, making it easier to study the initial space. GRComPaS will investigate the uniformisation of fractal metric surfaces and subsets of the complex plane, as well as problems associated with rigidity and conformal removability.
Objective
The main objective of the proposal is to develop novel methods for the study of the quasiconformal geometry of metric surfaces and of subsets of the complex plane.
Geometry of metric surfaces: The uniformization problem asks for conditions on a fractal metric space so that it can be transformed to a smooth space with a well-behaved transformation that preserves the geometry. In joint work with Romney the PI has resolved a major open problem and proved a general quasiconformal uniformization result for 2-dimensional spheres of finite area. The current project expects to exploit this recent breakthrough and develop an analytic theory for 2-dimensional surfaces of locally finite area under no other assumption; the classical approaches in the field require instead several geometric assumptions. In particular, the PI proposes the study of the following problems on fractal surfaces: quasiconformal classification, embedding in Euclidean space, uniformization of surfaces of infinite area, and connections between quasiconformal geometry and rectifiability.
Uniformization and rigidity in the plane: A long-standing conjecture of Koebe asserts that every domain in the plane can be conformally transformed to a circle domain. This proposal introduces a wide class of domains to test the conjecture, using techniques recently developed by the PI and collaborators, in combination with the transboundary modulus of Schramm. The PI will also study the problem of uniqueness of this conformal transformation and connections to the problem of conformal removability. The latter is a rigidity problem asking whether a given set in the plane is negligible from the domain of a conformal map. The PI has recently displayed several examples of (non)removable sets and has identified a new general class of sets that are conjectured to provide a characterization of removable sets. The PI will study this conjecture and several related deep open problems.
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
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-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.
546 36 THESSALONIKI
Greece
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