The current state and prospects of geometry and nonlinear differential equations

Abstract

In the development of mathematics in the twentieth century, multidimensional manifolds have been one of the central subjects of research. In the last ten years, theoretical physicists have needed some results on infinite-dimensional manifolds, which appear in modeling certain problems; for example, a system with an infinite number of particles in statistical mechanics. Some of these results could be translated to finite-dimensional manifolds. Perhaps some day even old classical problems can be solved by applying some of these infinite-dimensional theories directly to universal moduli space.

A second area of important future development is the study of singularities, which occur practically in every field. For instance, the algebraic theory of singularities has been developed in the last forty years, but the application of this theory to natural systems related to partial differential equations will be more than welcome.

Finally, modern computers generate many interesting questions. It is conceivable that deep mathematics may arise from understanding computer calculations in the future.

Fields Medal 1982 for this contributions in differential equations, also to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge-Ampère equations.