Elias M. Stein, 1931-2018

Elias Menachem Stein, one of the most influential mathematicians of the post-World War II era, died on December 23 at the age of 87.

During a career that spanned over six decades, Stein revolutionized the field of harmonic analysis through first-rate research, exposition, and mentoring. Stein’s research work was an embodiment of what Zygmund called “mathematical imperialism,” pushing the boundaries of harmonic analysis far beyond its initial focus on the Fourier series and establishing it as an essential tool in the study of partial differential equations, representation theory, several complex variables, ergodic theory, functional analysis, geometric measure theory, and number theory.

Not only was Stein a key innovator in harmonic analysis, he also fostered a large community of harmonic analysts with his tireless mentoring. By his retirement in 2012, Stein had over fifty doctoral students, many of whom went on to become leaders of the field. Stein, together with over five hundred academic descendants, has fundamentally changed the way mathematical analysis is practiced, establishing the influential Calderón–Zygmund–Stein school of analysis. [1]

Beyond his research activities, Stein sought to enrich the mathematics community at large through his legendary expository works, providing students and non-experts access to techniques previously available only to a small group of experts. In the 2000s, he pushed to share the core ideas of the field with an even wider audience by penning a now-famous four-volume series on mathematical analysis, the first-ever widely-used series of textbooks at the advanced undergraduate and beginning graduate level that presents the many subdisciplines of mathematical analysis as a unified whole. [2]

Although I have met Stein only a few times in person [3], I can say without hesitation that he has been the single most influential person in my professional life thus far. It was a student of Stein who first encouraged me to study mathematics seriously, and it was Stein’s summer program that helped me decide the overarching direction of my graduate studies. I learned much of the mathematics I know from Stein’s books [4], and it is through Stein’s writing that I developed my interest in science and technical writing. Even computer science was a subject I first encountered while attempting to study Stein’s works more broadly. Stein’s influence is pervasive in everything I have done professionally, and what I have learned from Stein and his works will no doubt continue to serve as a profound inspiration throughout my career.

[1] Zygmund was Stein’s advisor, and Calderón was a student of Zygmund. The name “Calderón–Zygmund–Stein theory” typically refers to the collection of mathematical techniques centered around singular integrals, maximal functions, pseudodifferential operators, and oscillatory integrals.

[2] Bourbaki de-emphasized analytic approaches to mathematics, Dieudonné was never widely adopted, and Simon will likely be regarded as references rather than expository works suitable for beginning students.

[3] Once, Stein told a story of having to read the Harry Potter series to hang out with his grandchildren. I asked how he could possibly have time to read such a massive series with all the work he is doing. He grinned and said, well, I read the last volume and Wikipedia-ed the rest.

[4] I thought I had eight books by Stein, but I can only find seven in my bookshelf. Bonus points for guessing what they are!