American Philosophical Society
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104. Mathematics[X]
1Name:  Dr. Robert P. Langlands
 Institution:  Institute for Advanced Study
 Year Elected:  2004
 Class:  1. Mathematical and Physical Sciences
 Subdivision:  104. Mathematics
 Residency:  Resident
 Living? :   Living
 Birth Date:  1936
The "Langlands Philosophy" is widely recognized as the most far-reaching dream that mathematicians currently have for the future development of mathematics. For the past three centuries, the subject of modular forms has been a major strand of mathematics, treated by such great mathematicians as Euler and Gauss. But it had the character of a bag of tricks and special results. Then, in 1967, Dr. Langlands announced the "Langlands conjectures," which displayed for the first time the underlying patterns at work. In the 35 years since then, these conjectures have become increasingly important. Guided by them, an underlying unity has been found, with deep consequences for many branches of mathematics. These include number theory (where Langlands' work played a role in Wiles' proof of Fermat's conjecture), algebraic geometry (where 30 of the best young geometers work in what they call "geometric Langlands theory"), and representation theory (where the Langlands conjectures lead to a classification of the representations that come up in the study of quantum mechanics). Today, the Langlands conjectures provide the basic motivation and guidance for the work of many mathematicians working in diverse fields. Dr. Langlands has also written extensively on mathematical physics, and he has a strong interest in history. A graduate of Yale University (Ph.D., 1960), he has been a professor at the Institute for Advanced Study since 1972. He is currently Professor of Mathematics Emeritus. He is a member of the American Academy of Arts & Sciences, a foreign member of the Russian Academy of Sciences, and has been awarded the Lester R. Ford Prize from the Mathematical Association of America. In 2018 he was awarded the Abel Prize.
2Name:  Dr. Benoit B. Mandelbrot
 Institution:  Yale University
 Year Elected:  2004
 Class:  1. Mathematical and Physical Sciences
 Subdivision:  104. Mathematics
 Residency:  Resident
 Living? :   Deceased
 Birth Date:  1924
 Death Date:  October 14, 2010
His Wolf Prize citation hails Benoit Mandelbrot for having "changed our view of nature", and IBM had cited him earlier in words that have been repeatedly confirmed: "Few contemporary scholars have made such penetrating contributions to as many fields of physical and social science. . . His success, where others have faltered, has been due to a combination of command of mathematical tools, extraordinary breadth, and even rarer intellectual courage." Fractal geometry, which he pioneered and named, also changed the way students and the world at large view mathematics and science. In pure mathematics, examination of masterful computer graphics led him to conjectures of great taste and difficulty that brought several slowly moving fields to intense activity. His observations revived iteration theory after a half century of forced inactivity; but his MLC conjecture (that the "Mandelbrot set is locally connected") is still unsolved after more than a quarter century. In probability theory, his conjecture that the boundary of a segment of Brownian notion is of dimension 4/3 was only proved after 18 years. He broadened the scope of physics by quantifying for the first time a holdover basic sensation, showing that the roughness of typical surfaces can actually be measured by a fractal dimension or Hölder exponent that turned out to be a new "universal." He showed how the support of intermittent turbulence can be measured and how the physics of diverse clusters is determined by their fractal geometry. In economics he enunciated the scaling principle in the 1960s, and his models for price variation, including his later notion of variable (fractal) trading time, are central to current developments in finance. A native of Poland, Benoit Mandelbrot became Docteur d'Etat ès Sciences Mathématiques in Paris in 1952. He was IBM Fellow Emeritus in Physics and Sterling Professor Emeritus of Mathematical Sciences at Yale University at the time of his death on October 14,2010, at the age of 85.
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