Log rank conjecture is one of the most famous open problems in the area of communication compleixty.
Lets consider the two party cdommunication complexity. Alice and Bob have $n$ bit strings $a,b$ , respectively, and they wish to compute arbitrary boolean function $f(a,b)$.
Communication matrix of $f$ is a $2^{n}\times 2^{n}$ matrix whose indices of row (column) corresponds to the inputs of Alice (Bob) and each entry $M_f (a,b)$ correspond to function value $f(a,b)$.
Log rank conjecture asserts that $CC(f) = (\log rank (M_{f}))^{O(1)}$
On the other hand, rank is a crucial role in the linea algebra and similar concepts of rank are considered.
QUESTION:I am looking for similar conjectures in the area of communication compleixty and related compleixty theoretic areas, similar-seeming conjectures without the communication complexity machinery, similar-seeming conjectures in linear algebra.Is there an appropriate reference ?