Periodic Reporting for period 3 - TeStability (Stability and Testability: Groups and Codes)
Reporting period: 2023-04-01 to 2024-09-30
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.
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.
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.
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.
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.
We hope to resolve soon the Aldous lyon conjecture.
We are still working in an effort to find a non-sofic group.