Satisfiability and group rings

Projektinformationen

SATURN

ID Finanzhilfevereinbarung: 101076148

DOI

10.3030/101076148

EK-Unterschriftsdatum 16 Juli 2023

Startdatum 1 Oktober 2023

Enddatum 30 September 2028

Finanziert unter

• European Research Council (ERC)

Gesamtkosten

€ 1 207 875,00

EU-Beitrag

€ 1 207 875,00

Koordiniert durch

RHEINISCHE FRIEDRICH-WILHELMS-UNIVERSITAT BONN
Germany

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

Berichtszeitraum: 2023-10-01 bis 2026-03-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

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.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

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.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

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.