Alexander Beilinson

Alexander Beilinson
Born	(1957-06-13) June 13, 1957
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

In 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 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. However, the proof of Beilinson–Bernstein introduced a method of localization. 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 paper Higher Regulators and values of L-functions, in which 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 conjecture 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.

From the early 1990s onwards, Beilinson worked with Vladimir Drinfeld to rebuild the theory of vertex algebras. After some informal circulation, this research was 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. He was a visiting scholar at the Institute for Advanced Study in the fall of 1994 and again from 1996 to 1998.

See also

• Beilinson–Parshin conjecture

External links

• Alexander Beilinson at the Mathematics Genealogy Project
• Letter from Beilinson to Soule containing his conjectures on motivic cohomology
• Citation for the 1999 Ostrowski Prize