GEOMETRICAL AND TOPOLOGICAL DUALITY IN CROWD DYNAMICS

The purpose of this paper is to establish strong theoretical basis for solving practical problems in modeling the behavior of crowds. Based on the previously developed (entropic) geometrical model of crowd behavior dynamics, in this paper we formulate two duality theorems related to the crowd manifold. Firstly, we formulate the geometrical crowd-duality theorem and prove it using Lie-functorial and Riemannian proofs. Secondly, we formulate the topological crowd-duality theorem and prove it using cohomological and homological proofs. After that we discuss the related question of the connection between Lagrangian and Hamiltonian crowd-duality, and finally establish the globally dual structure of crowd dynamics. All used terms from algebraic topology are defined in Appendix A.