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.

翻译：若干通信复杂度中的定理与猜想表明或推测，在给定通信模型下，矩阵的复杂度受相关解析或代数矩阵参数（如秩、符号秩、偏差等）控制。正向方向通常较为简单，因为低复杂度蕴含的结构性结论往往能推导出对某矩阵参数的约束。难点在于建立反向方向，这需要理解给定矩阵参数取值较小或较大时布尔矩阵的结构。本文将讨论与这一总体主题相关的若干研究方向。