This project concentrates on the study of communication complexity, an important aspect of distributed computing, using a continuous model of H. Abelson. It focuses on the development of a general method for establishing lower bound results on communication complexity as well as to obtain lower bounds for some important concrete problems. For that purpose, a study is conducted to understand the geometric structure of continuous distributed computations and to seek a mathematical characterization of communication complexity. Relations of the mathematical structure of the computational problem to its complexity are explored. Connections between the characterization of communication complexity and the classical Pfaff's problem and between the design of distributed algorithms and Hilbert's 13th problem will also be investigated. Several conjectures about lower bounds are tested with the hope that a proof or disproof will result. The tools and techniques used in this study come mainly from several branches of continuous mathematics such as analysis, differential geometry, and topology. To aid the computation of lower bounds on communication complexity, a Mathematica symbolic computing software package is expanded and refined. This package is important for performing experiments in order to formulate and help prove conjectures about the lower bounds for various problems. An important part of this project is the summer research experience for undergraduate students under the NSF Research Undergraduate Institutions Program, who are directly involved in various aspects of this research.
