The
Beauty of Duality and Triality in Mathematics and Sciences
David Gao
gao@vt.edu
Department of Mathematics
Virginia Polytechnic Institute and State University
U.S.A.
Abstract
Duality is a beautiful, fundamental concept that underlies
almost all natural phenomena. In modern mathematics, science, economics,
physics, system theory, optimization, numerical methods and scientific
computation, duality principles and methods are playing more and more
important roles. Triality is a newly proposed concept which reveals an
intrinsic duality pattern in general systems.
Beginning with dualities in the Garden of Eden, motivated by many very
interesting problems in natural science, the speaker will present a unified
structure and a splendid beauty through mathematical physics to game theory,
fine art, linguistic, and philosophy. By use of a very simple nonconvex
minimization problem, a powerful canonical duality theory will be briefly
introduced. The speaker will show that by this theory, many difficult
problems in algebra, geometry, differential equations, and chaotic dynamics
can be converted into certain simple dual problems. Therefore, closed
form solutions can be obtained for a large class of problems. In addition
to the traditional saddleLagrange duality in convex systems, a nice biduality
and an interesting triduality will be presented in two person game theory
and general nonconvex systems. This triduality plays fundamental roles
in mathematical modeling, global optimization, chaotic dynamics, and computational
science.
Finally, the speaker will present complete solutions to some very interesting
problems including minimum distance between two nonconvex surfaces, polynomial
minimization, Boolean least squares, and nonlinear algebraic/differential
equations.
