Stability and Testability: Groups and Codes

Project Information

TeStability

Grant agreement ID: 882751

DOI

10.3030/882751

EC signature date 3 April 2020

Start date 1 August 2021

End date 31 July 2027

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 628 750,00

EU contribution

€ 1 628 750,00

Coordinated by

WEIZMANN INSTITUTE OF SCIENCE
Israel

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

Reporting period: 2023-04-01 to 2024-09-30

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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.