Project description
Advanced computational approaches could help reshape understanding of group rings
Group rings are central to many mathematical fields like algebra, topology and representation theory, yet fundamental mysteries remain unresolved, including Kaplansky’s conjectures. These conjectures are a set of ideas about how group rings behave. For example, there is a prediction that zero divisors (where two non-zero elements multiply to zero) do not exist, and another suggests that only 0 and 1 act as idempotents (unchanged when multiplied by themselves). These questions have puzzled mathematicians for over 80 years. The ERC-funded SATURN project aims to find examples that disprove some of these ideas, using modern computational tools to tackle problems that once seemed impossible. Project work could reveal new insights about group rings and also help solve other major mathematical problems.
Objective
Group rings are key objects in many fields of mathematics including algebra, topology, operator algebras and representation theory. Fundamental questions about them remain unanswered, in particular several conjectures attributed to Kaplansky. For torsion-free groups and field coefficients, the zero divisor conjecture predicts the absence of zero divisors and the idempotent conjecture predicts that 0 and 1 are the only idempotents in the group ring. The direct finiteness conjecture says that left-invertible elements are invertible in group rings of arbitrary groups over fields. These conjectures have a history spanning more than 80 years. Although special cases are known, resolving any of the conjectures in full generality seemed intractable until the recent disproof of the closely related unit conjecture.
The goal of this project is to construct counterexamples to the zero divisor and direct finiteness conjectures. The latter will give the first example of a non-sofic group. We also seek to resolve the unit conjecture in characteristic zero. Key to our approach is the application of modern solvers for Boolean satisfiability. This paradigm shift, which was successful against the unit conjecture, shows that these problems are substantially more vulnerable to computational techniques than previously thought. Constructing our counterexamples will require both developing our understanding of candidate groups and their properties and building a toolkit for the effective application of existing computational machinery. The unique product property obstructs the existence of counterexamples to these conjectures and is thus of great interest. We will answer fundamental questions about this property.
Although we focus on the positive characteristic case, this project will lay serious groundwork towards the construction of counterexamples in characteristic zero to the zero divisor and idempotent conjectures and thus to the Atiyah, Baum-Connes and Farrell-Jones conjectures.
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.
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- natural sciences mathematics pure mathematics topology
- social sciences sociology social issues social inequalities
- natural sciences mathematics pure mathematics algebra
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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
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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.
53113 BONN
Germany
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.