The 2022 Wolf Prize in Mathematics has been awarded to Prof. George Lustig, Massachusetts Institute of Technology (MIT) for his pioneering contributions to modern representation theory and related fields
The 2022 Wolf Prize in Mathematics is awarded to Professor Lustig "for his pioneering contributions to representation theory and related fields."
Lustig, an American mathematician born in Romania, who was known for his contributions to the theory of geometric representations, especially of finite bundles
reductions (reductive) and algebraic groups. Lustig's work is characterized by a very high level of originality, enormous breadth
of subjects, impressive technical virtuosity and great depth in getting to the heart of the problems involved. Lustig's groundbreaking contributions mark him as one of the great mathematicians of our time.
Lustig's passion for mathematics began at a young age. In fact, it was math competitions at school that made him realize he was gifted in math. When he was in the 1962th grade, Lustig represented Romania in the International Mathematics Olympiad in 1963 and XNUMX, and won a silver medal in both competitions.
Lustig graduated from the University of Bucharest in 1968. His master's degree and doctorate were awarded to him by Princeton University in 1971, under the direction of Michael Atiyeh and William Browder. He joined the MIT mathematics faculty in 1978, after serving as a professor at the University of Warwick in 1974-77. He was appointed to the Norbert Wiener Chair at MIT in 1999-2009.
Lustig is known for his work on representation theory, in particular on the objects closely related to algebraic bundles, such as finite reductive bundles, Hecke algebras, P-ed bundles, quantum bundles and Weyl bundles. Lustig actually paved the way for modern performance theory. It included new basic concepts, including sheaves (character sheaves, "Deligne-Lusztig" sheaves and the "Kazhdan-Lusztig" polynomials
Lustig's first breakthrough was with Deligne around 1975, with the construction of the "Deligne-Lusztig" hypothesis. Lustig then obtained a complete description of the nondecomposable representations of reductive bundles over finite fields. Lustig's description of the character table of finite reductive groups is considered one of the extraordinary achievements of a single mathematician in the 20th century. To achieve his goal, Lustig developed a variety of techniques, which are used today by hundreds of mathematicians. The notable ones include the use of étale cohomology; the role played by the dual group; The use of cut cohomology, followed by the theory of character beams and quasi-characters, and the non-commutative Fourier transform.
In 1979, Kashdan and Lustig defined the "Kashdan-Lustig" basis for the Hecke algebra of the Coxeter group and formulated the "Kashdan-Lustig" conjecture. The "Kashdan-Lustig" conjecture led directly to the "Bilinson-Bernstein" localization theorem, which four decades later, remains our most powerful tool for understanding representations of reductive Lie algebras. Lustig's work with Vogan later introduced a version of the "Kashdan-Lustig" algorithm for generating "Lusztig-Vogan" polynomials. These polynomials are fundamental to our understanding of real reductive bundles and their unitary representations.
In the 90s, Lustig made a pioneering contribution to the theory of quantum bunches. His contributions include the introduction of the canonical base; the introduction of the Lustig template (which allows specialization to a single root and links to modular presentations); Quantum Probenius Copy and Small Quantum Bunch; and connections to the representation theory of effinic Lie algebras. Lustig's canonical basis theory (and Kashiwara's corresponding theory of crystal bases) led to profound results in combinatorics and representation theory. Recently, there has been significant progress in representation theory and low-dimensional topology through "categorization"; The roots of this work lie in Lustig's geometric classification of quantum bunches using perverted beams on quiver modules.
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