Alexander Beilinson

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Alexander Beilinson
Born (1957-06-13) June 13, 1957 (age 57)
Nationality Russian
Fields Mathematics
Institutions University of Chicago
Doctoral advisor Yuri I. Manin
Doctoral students Eitan Bachmat
Lorenzo Ramero
Andrés Rodríguez
Ilya Shapiro
Known for Contributions to representation theory, algebraic geometry and mathematical physics
Notable awards Ostrowski Prize (1999)

Alexander A. Beilinson (born 1957) is the David and Mary Winton Green University Professor at the University of Chicago and works on mathematics. His research has spanned representation theory, algebraic geometry and mathematical physics. In 1999 Beilinson was awarded the Ostrowski Prize with Helmut Hofer.

Work[edit]

As early as 1978, Beilinson published a paper on coherent sheaves and several problems in linear algebra. His two page note in the journal Functional Analysis and Its Applications was one of the more important papers on the study of derived categories of coherent sheaves.

In 1981 Beilinson announced a proof of the Kazhdan–Lusztig conjectures and Jantzen conjectures with Joseph Bernstein. Independent of Beilinson and Bernstein, Brylinski and Kashiwara obtained a proof of the Kazhdan–Lusztig conjectures.[1] However, the proof of Beilinson–Bernstein introduced a method of localization[disambiguation needed]. This established a geometric description of the entire category of representations of the Lie algebra, by "spreading out" representations as geometric objects living on the flag variety. These geometric objects naturally have an intrinsic notion of parallel transport: they are D-modules.

In 1982 Beilinson stated his own, seemingly profound conjectures about the existence of motivic cohomology groups for schemes, provided as hypercohomology groups of a complex of abelian groups and related to algebraic K-theory by a motivic spectral sequence, analogous to the Atiyah–Hirzebruch spectral sequence in algebraic topology. These conjectures have since been dubbed the Beilinson-Soulé conjectures; they are intertwined with Vladimir Voevodsky's program to develop a homotopy theory for schemes.

In 1984, Beilinson published the landmark paper Higher Regulators and values of L-functions where he related higher regulators for K-theory and their relationship to L-functions. The paper also provided a generalization to arithmetic varieties of the Lichtenbaum conjectures for K-groups of number rings, the Hodge conjecture, the Tate conjecture about algebraic cycles, the Birch and Swinnerton-Dyer conjecture about elliptic curves, and Bloch's conjecture about K2 of elliptic curves.

Beilinson continued to work on algebraic K-theory throughout the mid-1980s. He collaborated with Pierre Deligne on the developing a motivic interpretation of Don Zagier's polylogarithm conjectures that proved to be very influential.

From the early 1990s onwards, Beilinson worked with Vladimir Drinfeld to totally rebuild the theory of vertex algebras. After many years of informal circulation, this research was finally published in 2004 in a form of a monograph on chiral algebras. This has led to new advances in conformal field theory, string theory and the geometric Langlands program. He was elected a Fellow of the American Academy of Arts and Sciences in 2008.[2] He was a visiting scholar at the Institute for Advanced Study in the fall of 1994 and again from 1996 to 1998.[3]

Selected publications[edit]

  • Beilinson, A. A.; Ginzburg, V. and Soergel, W. (1996). "Koszul duality patterns in representation theory". Journal of the American Mathematical Society 9 (2): 473–527. doi:10.1090/S0894-0347-96-00192-0. ISSN: 0894-0347. 
  • Beilinson, A. A.; Lusztig, G. and MacPherson, R. (1990). "A geometric setting for the quantum deformation of GLn". Duke Mathematical Journal 61 (2): 655–677. doi:10.1215/S0012-7094-90-06124-1. ISSN: 0012-7094. 
  • Beilinson, A. A.; Ginzburg, V. and Schechtman, V. (1988). "Koszul duality". Journal of Geometry and Physics 5 (3): 317–350. doi:10.1016/0393-0440(88)90028-9. ISSN: 0393-0440. 
  • Beilinson, A. A. (1987). "K-theory, arithmetic and geometry (Manin seminar, Moscow, 1984--1986) in Lecture Notes in Math." 1289. Springer-Verlag. pp. 42–51.  |chapter= ignored (help)
  • Beilinson, A. A. (1987). "K-theory, arithmetic and geometry (Manin seminar, Moscow, 1984--1986) in Lecture Notes in Math." 1289. Springer-Verlag. pp. 27–41.  |chapter= ignored (help)
  • Beilinson, A. A.; MacPherson, R. and Schechtman, V. (1987). "Notes on motivic cohomology". Duke Mathematical Journal 54 (2): 679–710. doi:10.1215/S0012-7094-87-05430-5. ISSN: 0012-7094. 
  • Beilinson, A. A. (1986). "Applications of algebraic K-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), Contemporary Mathematics" 55. American Mathematical Society. pp. 35–68.  |chapter= ignored (help)
  • Beilinson, A. A. (1984). "Itogi Nauki i Tekhniki, Current problems in mathematics" 24. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow. pp. 181–238.  |chapter= ignored (help)
  • Beilinson, A. A.; Bernstein, J. and Deligne, P. (1982). "Analysis and topology on singular spaces, I (Luminy, 1981), Astèrisque" 100. Soc. Math. France, Paris. pp. 5–171.  |chapter= ignored (help)
  • Beilinson, A. A. (1980). "Residues and adèles". Funktsional. Anal. I Prilozhen. 14 (1): 44–45. ISSN: 0374-1990. 
  • Beilinson, A. A. (1978). "Coherent sheaves on Pn and problems in linear algebra". Funktsional. Anal. I Prilozhen. 12 (3): 68–69. doi:10.1007/BF01681436. ISSN: 0374-1990. 

See also[edit]

References[edit]

  1. ^ Brylinski, Jean-Luc; Kashiwara, Masaki (October 1981). "Kazhdan-Lusztig conjecture and holonomic systems". Inventiones Mathematicae (Springer-Verlag) 64 (3): 387–410. doi:10.1007/BF01389272. ISSN 0020-9910. 
  2. ^ "Book of Members, 1780-2010: Chapter B". American Academy of Arts and Sciences. Retrieved May 30, 2011. 
  3. ^ Institute for Advanced Study: A Community of Scholars

External links[edit]