"A Physicist's Approach to Complex Mathematics Problems"

 Based on work with V.Gudkov, J. Johnson, & Z. Nussinov
 Presented by S. Nussinov from Tel-Aviv University

ABSTRACT

"Complex Problems"  involving a large  number ,n, of elements
and for which  any general method of solution {Presumeably!}
requires a large number (growing faster than any finite power of n)
of elementary steps, and the P=(?)NP issue are briefly described.
This is done in the context of a "proverbial" n students in a dorm
example - a layman's description of the largest "clique" problem.
We suggest a simple physical analog model which is easy to simulate.
The n students or n vertices in a graph are represented by
n points in d=n-1 dimensions, initially residing at the n vertices of
symmetric n simplex and which move due to (constant) attractive/repulsive
forces introduced between compatible/incompatible students or between
connected/disconnected vertices in the graph.
The deterministic evolution of the n points is free from local minima traps,
easy to simulate, and appears(?!) to solve in polynomial time the:

1) the Heuristic problem of finding" clusters" in a network
{i.e in graphs or in  communication,commercial,biological,etc systems}
by physically ( geometrically) clustering the representative points.

2) The Graph isomorphism problem by evolving independently via
identical dynamics the simplexes corresponding to the two graphs and
checking the ( Geometric) congruence of the later.

3) the largest clique problem.
These problems are of increasing intrinsic difficulty and this
reflects in our "Solutions".
We briefly speculate on possible extensins to other "Complex Problems"
such as the Traveling- salesman or Hamiltonian circuit problem and on
possible implications in sociology,bilogy, neural-nets and other
areas.
No previous background beyond the most elementary geometry and
physics is required.