Project description
Cartan subalgebra could drive forward our understanding of important mathematical problems
Cartan subalgebras serve as a bridge between C*-algebras, topological dynamics and geometric group theory. This concept has recently come into the spotlight by the scientists working on the EU-funded IGOC project. Their goal is now to shed more light on Cartan subalgebras and related notions to tackle four major open questions: the universal coefficient theorem, the Baum-Connes conjecture, the conjugacy problem for topological shifts and quasi-isometry rigidity for polycyclic groups. The project will initiate new interactions between several mathematical areas. Any progress in any of the four addressed problems should result in major breakthroughs.
Objective
Recently, we discovered that the notion of Cartan subalgebras builds bridges between C*-algebras, topological dynamics, and geometric group theory. The goal of this research project is to develop our understanding of this concept in order to attack the following major open questions:
I. The UCT question
II. The Baum-Connes conjecture
III. The conjugacy problem for topological shifts
IV. Quasi-isometry rigidity for polycyclic groups
UCT stands for Universal Coefficient Theorem and is a crucial ingredient in classification. I want to make progress on the open question whether sufficiently regular C*-algebras satisfy the UCT, taking my joint work with Barlak as a starting point.
The Baum-Connes conjecture predicts a K-theory formula for group C*-algebras which has far-reaching applications in geometry and algebra as it implies open conjectures of Novikov and Kaplansky. My new approach to II will be based on Cartan subalgebras and the notion of independent resolutions due to Norling and myself.
Problem III asks for algorithms deciding which shifts are topologically conjugate. It has driven a lot of research in symbolic dynamics.
Conjecture IV asserts that every group quasi-isometric to a polycyclic group must already be virtually polycyclic. A solution would be a milestone in our understanding of solvable Lie groups.
To attack III and IV, I want to develop the new notion of continuous orbit equivalence which (as I recently showed) is closely related to Cartan subalgebras.
Problems I to IV address important challenges, so that any progress will result in a major breakthrough. On top of that, my project will initiate new interactions between several mathematical areas. It is exactly the right time to develop the proposed research programme as it takes up recent breakthroughs in classification of C*-algebras, orbit equivalence for Cantor minimal systems, and measured group theory, where measure-theoretic analogues of our key concepts have been highly successful.
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 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)
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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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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-COG - Consolidator 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-2018-COG
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G12 8QQ Glasgow
United Kingdom
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