Alexander Beilinson

Alexander A. Beilinson 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.

Work

In 1981 Beilinson announced a proof of the Kazhdan-Lusztig conjectures and Jantzen conjectures with Joseph Bernstein. 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 a landmark paper entitled "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 "K"₂ of elliptic curves.

Beilinson has also worked with 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.

Selected publications

*cite book
author = Beilinson, A. A.; Drinfeld, V.
year = 2004
title = Chiral Algebras
publisher = American Mathematical Society
id = ISBN 978-0821835289

*cite journal
author = Beilinson, A. A.
coauthors = Ginzburg, V. and Soergel, W.
title = Koszul duality patterns in representation theory
journal = Journal of the American Mathematical Society
volume = 9
year = 1996
issue = 2
pages = 473–527
id = ISSN: 0894-0347
doi = 10.1090/S0894-0347-96-00192-0

*cite journal
author = Beilinson, A. A.
coauthors = Lusztig, G. and MacPherson, R.
title = A geometric setting for the quantum deformation of GL_n
journal = Duke Mathematical Journal
volume = 61
year = 1990
issue = 2
pages = 655--677
id = ISSN: 0012-7094
doi = 10.1215/S0012-7094-90-06124-1

*cite journal
author = Beilinson, A. A.
coauthors = Ginzburg, V. and Schechtman, V.
title = Koszul duality
journal = Journal of Geometry and Physics
volume = 5
year = 1988
issue = 3
pages = 317–350
id = ISSN: 0393-0440
doi = 10.1016/0393-0440(88)90028-9

*cite conference
author = Beilinson, A. A.
title = How to glue perverse sheaves
booktitle = K-theory, arithmetic and geometry (Manin seminar, Moscow, 1984--1986) in Lecture Notes in Math.
publisher = Springer-Verlag
volume = 1289
pages = 42-51
year = 1987

*cite conference
author = Beilinson, A. A.
title = On the derived category of perverse sheaves
booktitle = K-theory, arithmetic and geometry (Manin seminar, Moscow, 1984--1986) in Lecture Notes in Math.
publisher = Springer-Verlag
volume = 1289
pages = 27-41
year = 1987
*cite journal
author = Beilinson, A. A.
coauthors = MacPherson, R. and Schechtman, V.
title = Notes on motivic cohomology
journal = Duke Mathematical Journal
volume = 54
year = 1987
issue = 2
pages = 679--710
id = ISSN: 0012-7094
doi = 10.1215/S0012-7094-87-05430-5

*cite conference
author = Beilinson, A. A.
title = Notes on absolute Hodge cohomology
booktitle = Applications of algebraic K-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), Contemporary Mathematics
volume = 55
pages = 35-68
year = 1986
publisher = American Mathematical Society

*cite conference
author = Beilinson, A. A.
title = Higher regulators and values of L-functions
publisher = Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow
booktitle = Itogi Nauki i Tekhniki, Current problems in mathematics
volume = 24
pages = 181-238
year = 1984

*cite conference
author = Beilinson, A. A.
coauthors = Bernstein, J. and Deligne, P.
title = Faisceaux pervers
booktitle = Analysis and topology on singular spaces, I (Luminy, 1981), Astèrisque
volume = 100
pages = 5-171
publisher = Soc. Math. France, Paris
year = 1982

*cite journal
author = Beilinson, A. A.
title = Residues and adèles
journal = Funktsional. Anal. i Prilozhen.
volume = 14
year = 1980
issue = 1
pages = 44–45
id = ISSN: 0374-1990

*cite journal
author = Beilinson, A. A.
title = Coherent sheaves on "P"ⁿ and problems in linear algebra
journal = Funktsional. Anal. i Prilozhen.
volume = 12
year = 1978
issue = 3
pages = 68–69
id = ISSN: 0374-1990
doi = 10.1007/BF01681436

References

*MathGenealogy|id=44354

External links

* [http://www.math.uiuc.edu/K-theory/0694/ Letter from Beilinson to Soule containing his conjectures on motivic cohomology]