On John Mather's Work

TL;DR

John Mather's work spans foundational advances in singularity theory, Hamiltonian dynamics, and characteristic classes, addressing how maps behave near singularities and how variational structures govern complex dynamics. The study develops core tools such as $k$-jet analyses, Malgrange's preparation, and stability criteria, culminating in the Mather-Yau holomorphic germ classification and robust Aubry-Mather theory for area-preserving systems. It also connects singularity theory to geometry via Nash transformations and Mather-Chern/Wu classes, and extends foliation theory through the Mather-Thurston theorem. Collectively, these contributions establish deep, broadly influential frameworks that shape modern analysis, geometry, and dynamical systems, with lasting impact on both theory and applications.

Abstract

John Mather is a great scholar who was dedicated to mathematics in his whole life. His works in mathematics can be characterized as original and foundational. He laid out the foundation of singularity theory while he was a graduate student. He also laid out the foundation of modern Hamiltonian dynamical systems. Those fields became main stream in mathematics and it attracts many talents to pursue. His other works on characteristic classes, foliations, celestial mechanics, prime ends of conformal mappings are of the same quality with great influence in mathematics.