Satisfiability and group rings

Información del proyecto

SATURN

Identificador del acuerdo de subvención: 101076148

DOI

10.3030/101076148

Fecha de la firma de la CE 16 Julio 2023

Fecha de inicio 1 Octubre 2023

Fecha de finalización 30 Septiembre 2028

Financiado con arreglo a

• European Research Council (ERC)

Coste total

€ 1 207 875,00

Aportación de la UE

€ 1 207 875,00

Coordinado por

RHEINISCHE FRIEDRICH-WILHELMS-UNIVERSITAT BONN
Germany

Periodic Reporting for period 1 - SATURN (Satisfiability and group rings)

Período documentado: 2023-10-01 hasta 2026-03-31

Resumen del contexto y de los objetivos generales del proyecto

This project is concerned with algebraic objects called group rings. They are important in many different mathematical subjects, such as algebraic topology, representation theory, and operator algebras. There is a series of fundamental conjectures about group rings attributed to Kaplansky which claim to characterize certain elements in group rings, such as their units. They can be thought of as conjectures of tameness. The aim of this project is to produce counterexamples to these conjectures, indicating that group rings contains surprisingly wild elements. As the Kaplansky conjectures connect to other questions about groups, these counterexamples could lead to resolving other major open problems.

Trabajo realizado desde el comienzo del proyecto hasta el final del período abarcado por el informe y los principales resultados hasta la fecha

We have both investigated which groups are potential counterexamples and attempted to advance our techniques for constructing counterexamples in a given candidate group. So far the main achievement has been producing the first example of a non-trivial unit in the group ring of a torsion-free group over a field of characteristic zero, which was one of the three main objectives of this project.

Avances que van más allá del estado de la técnica e impacto potencial esperado (incluida la repercusión socioeconómica y las implicaciones sociales más amplias del proyecto hasta la fecha)

The main achievement, namely constructing non-trivial units in characteristic zero, opens up the possibility that zero divisors could also exist in characteristic zero, which would disprove the so-called Atiyah conjecture.