Vortex Dynamics, Combinatorics and Statistics

This report represents an overview of the interconnections between the dynamics of large vortex systems, combinatorics, n-body problems and statistical mechanics. The combinatorial perturbation method for the 2D vortex problem is discussed; the essential combinatorial sym-plectic transformations to Jacobi-type variables which are based on a binary tree algorithm, is introduced and extended to the 3D vortex problem. Combinatorial and graph-theoretic results which are motivated by the computational needs of the vortex problem, are mentioned. They include new results on sign-nonsingular patterns and noneven digraphs. A simpliied singular limit of the 3D Hamiltonian for vortex dynamics is derived and its basic properties discussed. The 2-and 3-body problems in this simple model is studied.