Log Rank Conjecture Collaborative Approach Recently a post was made in Mathoverflow seeking possible avenues for collaborative projects.

I made a proposal for Log Rank conjecture in https://mathoverflow.net/questions/219638/proposals-for-polymath-projects/219639#219639 which has been well received so far

I feel that this is a good project for collaborative approach. 

I know this is a fundamental open problem in CS.

1. Is there a list of possible avenues of research that could open up if progress comes on this topic? For example, I know there are connections to Matrix Rigidity.

2. Could we have a list of possible impact of Log Rank Conjecture and related research in CS?
it seems researchers might be cautious/ unlikely to conjecture/ speculate solution to this problem would likely/ definitely open up other areas (eg one you list of matrix rigidity is provably nearly equivalent to the extremely hard challenge of proving nontrivial lower bounds on eg circuits etc) but it definitely provably touches many areas. here is a nice recent survey (Jan 2015) by Lovett that lists "many connections" sec 6 along with two other refs that show crosscutting connections/ formulations


Recent advances on the log-rank conjecture in
communication complexity / Lovett


6.1 Randomized protocols vs approximate rank
6.2 Quantum protocols for low-rank matrices
6.3 The structure of low-rank sparse matrices, and matrix rigidity

En Route to the log-rank Conjecture:
New Reductions and Equivalent Formulations / Gavinsky, Lovett
On the "log rank"-Conjecture in Communication Complexity / Raz, Spieker