Stability and Testability: Groups and Codes

Projektinformationen

TeStability

ID Finanzhilfevereinbarung: 882751

DOI

10.3030/882751

EK-Unterschriftsdatum 3 April 2020

Startdatum 1 August 2021

Enddatum 31 Juli 2027

Finanziert unter

• EXCELLENT SCIENCE - European Research Council (ERC)

Gesamtkosten

€ 1 628 750,00

EU-Beitrag

€ 1 628 750,00

Koordiniert durch

WEIZMANN INSTITUTE OF SCIENCE
Israel

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

Berichtszeitraum: 2023-04-01 bis 2024-09-30

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

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.

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

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.

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)

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.