Polyhedral Dynamics: I. The relevance of Algebraic Topology to Human Affairs

Casti, J.L. (1975). Polyhedral Dynamics: I. The relevance of Algebraic Topology to Human Affairs. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-75-030.pdf

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Abstract

The purpose of the current note is to explore a recent algebro-geometric approach to the structural analysis of large-scale systems. This approach, based upon ideas of algebraic topology, was introduced by Atkin in a recent series of works which, unfortunately, have not yet received the circulation they deserve. By a very ingenious coupling of classical ideas in combinational topology and new notions of connectivity, patterns, and obstructions, this work presents a mathematical framework within which an extremely broad class of global systems questions can be precisely analyzed.

The objective of this work is two-fold: to present the basic theory of what we have chosen to call "polyhedral dynamics" as quickly as possible. This presentation includes the basic ideas of Atkin, plus extensions of our own which extend and broaden the original work. The second goal is to illustrate the concepts involved on a variety of problems relevant to ongoing IIASA activities.

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