Excerpt

One of the most striking features of the development of fundamental theory in the sciences during the past decade has been the convergence of its focal interests with major themes in mathematical research. Mathematical concepts and tools that have arisen in an apparently autonomous way in relatively recent research have turned out to be important as major components of the description of nature. At the same time, this use of novel mathematical tools in the sciences has reacted back upon the development of mathematical subject matter having no obvious connection with the scientific subject matter to yield new and surprising mathematical consequences. It is this theme of strong reciprocal interaction that I propose to present in the present discussion. We must ask why this has been so, whether this kind of interaction is a major trend that will continue in a serious way into the foreseeable future and, if so, what the consequences will be for the future development of mathematics and the sciences.

Mathematics and the Natural Sciences

Let us begin our analysis by examining the different ways in which novel and relatively sophisticated mathematical tools have been applied in recent scientific developments. We may classify them into five relatively broad modes of attack.

1. The use of sophisticated mathematical concepts in the formulation of new basic physical theories on the most fundamental level. At the present moment, this takes the form of the superstring theory, which has as its objective the total unification of all the basic physical forces and interactions: electromagnetic, weak, strong, and gravitational. This new phase of physical theory, which is the culmination of the earlier development of gauge field theories and of theories of supersymmetry, exhibits the use of a wide variety of relatively new mathematical tools developed in the past two decades such as Kac–Moody algebras and their representations, the existence of Einstein metrics on compact Kahlerian manifolds satisfying simple topological restrictions, and representations of exceptional Lie groups. The body of techniques and mathematical arguments embodied here includes the theory of Lie groups and algebras, their generalizations, and their representation theory, differential geometry in its modern global form in terms of vector bundles, the study of the existence of solutions of on manifolds of highly nonlinear, partial differential equations, differential and algebraic topology, and the whole mélange of analysis, algebra, and geometry on manifolds, which has been called global analysis. The implementation of this program involves still other major directions of mathematical research, most particularly problems in algebraic geometry.

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