Geometry and nonlinear analysis

TL;DR

This paper reviews recent advances in nonlinear partial differential equations and their applications in geometry and topology, highlighting developments like Donaldson and Seiberg-Witten invariants and Gromov-Witten theory.

Contribution

It summarizes recent progress and applications of nonlinear PDEs in geometry, emphasizing new results and ongoing research directions.

Findings

Advances in Yang-Mills and Seiberg-Witten invariants for 4-manifolds

Development of Gromov-Witten invariants via pseudo-holomorphic maps

Ongoing progress in nonlinear PDE applications in geometric analysis

Abstract

Nonlinear analysis has played a prominent role in the recent developments in geometry and topology. The study of the Yang-Mills equation and its cousins gave rise to the Donaldson invariants and more recently, the Seiberg-Witten invariants. Those invariants have enabled us to prove a number of striking results for low dimensional manifolds, particularly, 4-manifolds. The theory of Gromov-Witten invariants was established by using solutions of the Cauchy-Riemann equation. These solutions are often refered as pseudo-holomorphic maps which are special minimal surfaces studied long in geometry. It is certainly not the end of applications of nonlinear partial differential equations to geometry. In this talk, we will discuss some recent progress on nonlinear partial differential equations in geometry. We will be selective, partly because of my own interest and partly because of recent applications of nonlinear equations. There are also talks in this ICM to cover some other topics of geometric analysis by R. Bartnik, B. Andrew, P. Li and X.X. Chen, etc.