Stability and Testability: Groups and Codes

Informations projet

TeStability

N° de convention de subvention: 882751

DOI

10.3030/882751

Date de signature de la CE 3 Avril 2020

Date de début 1 Août 2021

Date de fin 31 Juillet 2027

Financé au titre de

• EXCELLENT SCIENCE - European Research Council (ERC)

Coût total

€ 1 628 750,00

Contribution de l’UE

€ 1 628 750,00

Coordonné par

WEIZMANN INSTITUTE OF SCIENCE
Israel

Periodic Reporting for period 3 - TeStability (Stability and Testability: Groups and Codes)

Période du rapport: 2023-04-01 au 2024-09-30

Résumé du contexte et des objectifs généraux du projet

The proposal dealt with two long-standing open problems: one in the theory of error-correcting codes and one in group theory. The main novel of our proposal is the discovery that these problems, despite looking so different, are related.

The first problem is: Are there locally testable good codes? This has been a major ( if not, the major) open problem in the area. This problem is of great importance in theory and even in practice. It enable a receiver of a message to know if the message is correct by reading only a small part of it!
We ( this is joint work with Irit Dinur, Shai Evra, Ron Livne, and Shahar Mozes) solved this completely! This was by far more than our expectations. This breakthrough has already received a lot of attention and we have already got two prizes for this paper.

The other problem is in group theory: Is every group sofic? Again a major long-standing problem in group theory. We believe the answer is NO. So far we have not solved it but we made progress in two directions: We proposed a concrete counter example and a plan to resolve it. We are also working ( with the hope to solve soon) the very much related " Aldous-Lyon conjecture"- also a long standing problem.

Added Nov. 24: In the last year and half we ( with Bowen, Chapman and Vidick ) indeed managed to repute the Aldous Lyon - Conjecture. This is a major a chivvement as this conjecture has been spead around for many years in probability, limit graphs theory, group theory and more. The proof is a combination of groups, quantum information theory, model theory and more.

Travail effectué depuis le début du projet jusqu’à la fin de la période considérée dans le rapport et principaux résultats atteints jusqu’à présent

As mentioned, one of the major problems we solved completely ( this was faster than we expected as we found a surprising short-cut).
We are still working on the other. We are also developing systematically the connection between groups and codes- which is the main novelty of our proposal.
We ( with Bowen, Chapman and Vidick) reputed the Aldous-Lyon conjecture. \With Chapman and Irit is found a far reaching generalization of the classical theorem of Shannon and error correcting codes. The generalization if for all groups: finite or infinite, commutative or not. It open a new direction of research.

Progrès au-delà de l’état des connaissances et impact potentiel prévu (y compris l’impact socio-économique et les conséquences sociétales plus larges du projet jusqu’à présent)

Well, as mentioned we have constructed "good locally testable codes" this by itself is a big breakthrough.
We hope to resolve soon the Aldous lyon conjecture.
We are still working in an effort to find a non-sofic group.