Satisfiability and group rings

Informazioni relative al progetto

SATURN

ID dell’accordo di sovvenzione: 101076148

DOI

10.3030/101076148

Data della firma CE 16 Luglio 2023

Data di avvio 1 Ottobre 2023

Data di completamento 30 Settembre 2028

Finanziato da

• European Research Council (ERC)

Costo totale

€ 1 207 875,00

Contributo UE

€ 1 207 875,00

Coordinato da

RHEINISCHE FRIEDRICH-WILHELMS-UNIVERSITAT BONN
Germany

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

Periodo di rendicontazione: 2023-10-01 al 2026-03-31

Sintesi del contesto e degli obiettivi generali del progetto

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.

Lavoro eseguito dall’inizio del progetto fino alla fine del periodo coperto dalla relazione e principali risultati finora ottenuti

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.

Progressi oltre lo stato dell’arte e potenziale impatto previsto (incluso l’impatto socioeconomico e le implicazioni sociali più ampie del progetto fino ad ora)

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.