  Several theorems and conjectures in communication complexity state or
speculate that the complexity of a matrix in a given communication model is
controlled by a related analytic or algebraic matrix parameter, e.g., rank,
sign-rank, discrepancy, etc. The forward direction is typically easy as the
structural implications of small complexity often imply a bound on some matrix
parameter. The challenge lies in establishing the reverse direction, which
requires understanding the structure of Boolean matrices for which a given
matrix parameter is small or large. We will discuss several research directions
that align with this overarching theme.
