# Jean Bourgain, 2017 Breakthrough Prize in Mathematics winner, Dies at 64

On December 22, Jean Bourgain passed away after a three-year battle with cancer. Among many other honors, Bourgain won the 2017 Breakthrough Prize in Mathematics “for multiple transformative contributions to analysis, combinatorics, partial differential equations, high-dimensional geometry and number theory.”

The above citation reflects the remarkable range of mathematical fields across which Bourgain made important discoveries. These include high-dimensional geometry, where he solved major problems in vector spaces; ergodic theory, where he proved one of the two most fundamental theorems and developed new techniques for the study of randomness; partial differential equations, where among other things he provided new insights into the mathematical extension of the Schrödinger equation from physics – the fundamental equation governing interactions at the quantum level; number theory, where he proved a conjecture of importance to the study of a class of problems extending the famous Lagrange four-square theorem; and combinatorics, where he solved a long-standing problem with extender graphs and developed the sum-product phenomenon – which he then applied to a completely different branch of math, harmonic analysis.

A 2016 conference celebrating and exploring Bourgain’s work can be found online here.

In a blog post, fellow Breakthrough Prize winner Terry Tao described Bourgain as "a truly remarkable person and mathematician.” Everyone who knew him and his work would agree. The Breakthrough Prize offers deep condolences to his family and his friends, his colleagues at the Institute for Advanced Study and the mathematical community worldwide.