To Be Presented in the 11th Asian Technology Conference in Mathematics
December 12-16, 2006, Hong Kong SAR, China

The Beauty of Duality and Triality in Mathematics and Sciences

David Gao
Department of Mathematics
Virginia Polytechnic Institute and State University


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 saddle-Lagrange duality in convex systems, a nice bi-duality and an interesting tri-duality will be presented in two person game theory and general nonconvex systems. This tri-duality 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.

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