Elias Stein, leader of harmonic analysis, passes away | Science

Elias Stein, leader of harmonic analysis, passes away | Science



Elias Menachen Stein, Eli for his many friends, has passed away this December 23 at the age of 87 years. Stein He can be considered one of the most profound and original contemporary mathematicians and, in particular, his numerous and fundamental contributions to the branch of harmonic analysis have made him a recognized leader for many years of the field.

Born in Antwerp (Belgium) on January 13, 1931, the German invasion of 1940 forced the emigration of his family to the United States, embarking on Lisbon after a hazardous journey in which he crossed half of Spain. There he stood out as a brilliant young man, which allowed him to enter the baccalaureate of the famous Stuyvesant school in New York for gifted students. Then it coincided with Paul Cohen, who years later would receive the Fields Medal for the resolution of one of Hilbert's famous problems, the independence of the continuum hypothesis.

Both Stein and Cohen completed their PhD studies at the University of Chicago under the direction of Antoni Zygmund. In those years Zygmund was a young Polish professor who had come to the prestigious department of mathematics in Chicago as a result, too, of the disasters of a war that had crushed the brilliant Polish mathematical generation that emerged in the thirties of the last century.

Upon completing his thesis, he taught at MIT and the University of Chicago, until in 1963 he moved to Princeton University as a professor, where he helped to turn that place into the world center of mathematics. Stein has left behind important theorems in all the areas in which he has worked. Among his many scientific contributions stands out the now known as Stein's complex interpolation method; his theorem characterizing the sequences of operators that converge at almost every point in terms of estimates for the call maximal function; the restriction theorem Stein-Tomas illustrating the interesting non-linear properties of the Fourier transform; the call slogan of Cotlar-Stein about the sums of almost-orthogonal operators and the theory of spaces Hp, developed together with his student Charles Fefferman, which is perhaps his most cited work.

Legend has it that the great mathematician Leonhard Euler, who was also long-lived, never stopped creating mathematics. Stein has not stopped doing it either

However, he also owes his great fame to his gifts as a lecturer, with a precise and passionate style as a romantic symphony, and as a writer of monographs. In particular, his treaty Singular integrals and differentiability properties of functions It is a masterpiece, a kind of bible for all harmonic analysts. As its name suggests, the book published by Stein contains an exposition of the theory of singular integrals developed by the Chicago school (formed by Alberto Calderón, Zygmund, Stein himself and others). This theory offers fundamental analytical tools to study diverse problems of the models of science, from mathematical physics (differential equations), to the theory of numbers (Waring problems Y of Goldbach) and differential geometry (index theorem).

Prior to the contributions of the Chicago school, only the one-dimensional case had been understood, with ingenious methods of complex variables difficult to extend to larger dimensions. Calderón-Zygmund-Stein's real variable theory, much more flexible, allowed it to be done in any dimension. And that opened the way for numerous applications, including the important case of problems whose differential equations are linear. Stein's monograph brilliantly systematized the theoretical corpus generated by the harmonic analysts of the Chicago school and that was scattered in a multitude of articles.

Legend has it that the great mathematician Leonhard Euler, who was also long-lived, never stopped creating mathematics. Stein has not stopped doing it either, as shown by the fact that in 2018 he has signed six research articles with several collaborators. In addition to being a distinguished Princeton professor and member of the US Academy of Sciences, Stein has received numerous awards and honors throughout his career, including the Steeles Prize, the Schock Prize, the Wolf Prize, the Prize Bergman and the National Medal of Science. But his legacy goes much further: he has had more than 52 doctoral students, two of them winners of the Fields Medal (Charles Fefferman and Terence Tao) and more than 500 academic descendants, among whom is the author of these lines. We will miss him a lot.

Antonio Córdoba he is director of the ICMAT and professor of Analysis of the Autonomous University of Madrid

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