Alexander Beilinson

Alexander A. "Sasha" Beilinson (born June 13, 1957 ) is a Russian mathematician who deals with arithmetic-algebraic geometry , representation theory and mathematical physics .

life and work

Beilinson studied in Moscow , where he received his doctorate from Yuri Manin . 1980 to 1988 he was a research mathematician at a cardiology center in Moscow. From 1989 to 1998 he taught at the Massachusetts Institute of Technology (MIT) in the fall and was also a scientist at the Landau Institute for Theoretical Physics in Chernogolowka, Russia. Since 1998, he has "David and Mary Winton Green" -Professor of mathematics at the University of Chicago , where Spencer Bloch taught, with whom he worked closely.

In representation theory , he and Joseph Bernstein proved the Kazhdan-Lusztig conjectures in 1981 , with the methods used in the proof being influential in the development of geometric representation theory. With Bernstein, Pierre Deligne and Ofer Gabber he introduced perverse sheaves and proved for them around 1982 the decomposition theorem , the difficult Lefschetz theorem and a semi-simplicity theorem (for positive characteristics and existence of a Galois group effect).

In 1982, in a letter to Soulé, he published conjectures about the existence of a motivic cohomology for schemes or algebraic varieties. The conjectures were partially realized in the 1990s in the program of Vladimir Wojewodski and Andrei Suslin for the construction of such motivic cohomology theories.

In 1984 he formulated in Higher regulators and values of -Functions ${\displaystyle L}$ the groundbreaking for the further development of arithmetic algebraic geometry Beilinson conjectures that the leading terms of the Taylor expansion of the functions of algebraic varieties at integer points with the groups and the Deligne-Beilinson- Connect the cohomology of the variety. They also include a number of well known older assumptions as the Tate conjecture on algebraic cycles and the Birch and Swinnerton-Dyer conjecture for elliptic curves . ${\displaystyle L}$ ${\displaystyle K}$

With Vladimir Drinfeld , who is also at the University of Chicago and whom he still knew from Moscow, he worked on a reformulation of the theory of vertex algebras , published in 2004 as the book Chiral Algebras (AMS), which have applications in string theory and conformal field theory. Both also made fundamental contributions to the geometric Langland program (they recognized, following a suggestion by Edward Witten , the importance of Hitchin fibers ).

In 1983 he was invited speaker at the International Congress of Mathematicians in Warsaw ( Localization of representations of reductive Lie algebras ). In 1984 he received the Mathematics Prize of the Moscow Mathematical Society, and in 1999 the Ostrowski Prize . He has been an external member of the Academia Europaea since 2000 . In 2008 he was accepted into the American Academy of Arts and Sciences . In 2018 he received the Wolf Prize for Mathematics, in 2020 the Shaw Prize for Mathematics.

literature

• Michael Rapoport , Norbert Schappacher , Peter Schneider (Eds.): Beilinson's conjectures and special values ​​of functions ${\displaystyle L}$ (= Perspectives in Mathematics. 4). Academic Press, Boston MA et al. 1988, ISBN 0-12-581120-9 (from an Oberwolfach seminar 1986).

Fonts

• with J. Bernstein: Localization de -modules. ${\displaystyle {\mathfrak {g}}}$In: Comptes Rendus des Séances de l'Académie des Sciences. Series 1: Sciences Mathématique. Volume 292, No. 1, 1981, pp. 15-18 .
• with J. Bernstein, P. Deligne: Faisceaux pervers. In: Analyze et topologie sur les espaces singuliers. CIRM, 6-10 July 1981 (= Astérisque. 100). Volume 1. Société Mathématique de France, Paris 1982, pp. 5-171.
• Высшие регуляторы и значения -функций. ${\displaystyle L}$In: Итоги Науки и Техники. Volume 24, 1984, pp. 181-238 , (English: Higher regulators and values ​​of functions. ${\displaystyle L}$ In: Journal of Soviet Mathematics. Volume 30, No. 2, 1985, pp. 2036-2070, doi : 10.1007 / BF02105861 ) .
• with J. Bernstein: A proof of Jantzen conjectures. In: Sergei Gelfand, Simon Gindikin (eds.): IM Gelfand Seminar (= Advances in Soviet Mathematics. 16, 1). American Mathematical Society, Providence RI 1993, ISBN 0-8218-4118-1 , pp. 1-50, ( digitized ).
• with Victor Ginzburg , Wolfgang Soergel : Koszul duality patterns in representation theory. In: Journal of the American Mathematical Society . Volume 9, No. 2, 1996, pp. 473-527, doi : 10.1090 / S0894-0347-96-00192-0 .
• with V. Drinfeld: Chiral algebras (= American Mathematical Society. Colloquium Publications. 51). American Mathematical Society, Providence RI 2004, ISBN 0-8218-3528-9 .

Web links

• For the appointment of Beilinson u. a. in Chicago
• Spencer Bloch: Algebraic cycles and values ​​of functions. ${\displaystyle L}$In: Journal for pure and applied mathematics . Volume 350, 1984, pp. 94-108 .

Individual evidence

1. ↑ Jean-Luc Brylinski and Masaki Kashiwara independently gave evidence in 1981
2. ^ Mark Andrea A. de Cataldo , Luca Migliorini : The Decomposition theorem, perverse sheaves and the topology of algebraic maps. In: Bulletin of the American Mathematical Society . Volume 46, No. 4, 2009, pp. 535-633, doi : 10.1090 / S0273-0979-09-01260-9 .
3. ^ Letter from Beilinson to Soulé . Also Beilinson: Height pairing between algebraic cycles . In: Yuri I. Manin (Ed.): -Theory, Arithmetic and Geometry. Seminar, Moscow University, 1984–1986 (= Lecture Notes in Mathematics . 1289). Springer, Berlin et al. 1987, ISBN 3-540-18571-2 , pp. 1–26. ${\displaystyle K}$
4. ↑ Высшие регуляторы и значения -функций. ${\displaystyle L}$In: Итоги Науки и Техники. Volume 24, 1984, pp. 181-238, (English: Higher regulators and values ​​of functions. ${\displaystyle L}$ In: Journal of Soviet Mathematics. Volume 30, No. 2, 1985, pp. 2036-2070).
5. ^ Preprint of Chiral Algebras
6. ^ Beilinson, Drinfeld: Quantization of Hitchin's integrable system and Hecke eigensheaves. Preprint 1991, ( pdf ).