Project description
‘Polish cover’ uncovers new connections between logic and other mathematical fields
Algebraic topology applies algebraic methods to problems in topology, the study of mathematical structures whose properties are preserved (invariant) under continuous deformation. The ERC-funded DAT project will develop a unique approach to algebraic topology harnessing mathematical logic, specifically descriptive set theory. Algebraic objects will be enriched with additional information provided by a so-called Polish cover, providing invariants that are finer, richer and more rigid than purely algebraic ones. These invariants will enable access to classification problems that were previously out of reach, opening the door to comprehensive study and enhanced understanding of such problems and new areas of research at the interface between logic and other areas of mathematics.
Objective
This project addresses fundamental issues in the development of algebraic topology, coarse geometry, and other areas of mathematics, related to the problem of doing algebra when the structures under considerations also have a topology. A number of other approaches have been proposed recently, showing the current importance of these issues for the mathematical community. The approach followed in this project is unique, in harnessing powerful tools from mathematical logic, and especially descriptive set theory.
The fundamental idea is to enrich an algebraic object with additional information provided by a Polish cover, which is an explicit presentation of the given object as a suitable quotient of a structure endowed with a compatible Polish topology. The goal of this project is to show that fundamental invariants from homological algebra, algebraic topology, operator algebras, and coarse geometry, such as Ext, Cech cohomology, KK-theory, and coarse K-homology, can be seen as functors to the category of groups with a Polish cover. Furthermore, doing so provides invariants that are finer, richer, and more rigid than the purely algebraic ones.
These invariants will allow us to tackle classifications problems for topological spaces, coarse spaces, C*-algebras, and maps, that had been so far out of reach. Furthermore, we will use these invariants to calibrate the complexity of such classification problems from the perspective of Borel complexity theory. In turn, this will enable us to isolate complexity-theoretic consequences of the Universal Coefficient Theorem for C*-algebras and of the coarse Baum-Connes Conjecture for coarse spaces, and to construct examples of strong failure of such results.
Ultimately, the completion of this project will lead to the development of entirely new fields of research at the interface between logic and other areas of mathematics (algebraic topology, coarse geometry, operator algebras).
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
- natural sciences mathematics pure mathematics algebra
- 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
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-2022-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.
40126 Bologna
Italy
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.