Statistics Postdoc Tames Decades-Old Geometry Problem

Erica Klarreich:

In the mid-1980s, the mathematician Jean Bourgain thought up a simple question about high-dimensional shapes. And then he remained stuck on it for the rest of his life.

Bourgain, who died in 2018, was one of the preeminent mathematicians of the modern era. A winner of the Fields Medal, mathematics’ highest honor, he was known as a problem-solver extraordinaire — the kind of person you might talk to about a problem you’d been working on for months, only to have him solve it on the spot. Yet Bourgain could not answer his own question about high-dimensional shapes.

“I was told once by Jean that he had spent more time on this problem and had dedicated more efforts to it than to any other problem he had ever worked on,” wrote Vitali Milman of Tel Aviv University earlier this year.

In the years since Bourgain formulated his problem, it has become what Milman and Bo’az Klartag of the Weizmann Institute of Science in Israel calledthe “opening gate” to understanding a wide range of questions about high-dimensional convex shapes — shapes that always contain the entire line segment connecting any two of their points. High-dimensional convex shapes are a central object of study not just for pure mathematicians but also for statisticians, machine learning researchers and other computer scientists working with high-dimensional data sets.