Alexander Beilinson
Encyclopedia
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.
. 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. However, the proof of Beilinson–Bernstein introduced a method of . 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 Drinfel'd
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 algebra
s. 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.
Professor
A professor is a scholarly teacher; the precise meaning of the term varies by country. Literally, professor derives from Latin as a "person who professes" being usually an expert in arts or sciences; a teacher of high rank...
at the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
and works on mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. His research has spanned representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
, algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
and mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
. In 1999 Beilinson was awarded the Ostrowski Prize
Ostrowski Prize
The Ostrowski Prize is a mathematics award given every other year by an international jury from the universities of Basel, Jerusalem, Waterloo and the academies of Denmark and the Netherlands...
with Helmut Hofer.
Work
As early as 1978, Beilinson published a paper on coherent sheaves and several problems in linear algebraLinear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
. 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
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
.
In 1981 Beilinson announced a proof of the Kazhdan–Lusztig conjectures and Jantzen conjectures with Joseph Bernstein
Joseph Bernstein
Joseph Bernstein is an Israeli mathematician working at Tel Aviv University. He works in algebraic geometry, representation theory, and number theory....
. Independent of Beilinson and Bernstein, Brylinski
Jean-Luc Brylinski
Jean-Luc Brylinski is a French-American mathematician. Educated at the Lycée Pasteur and the Ecole Normale Supérieure in Paris, after an appointment as researcher with the C. N. R. S., he became a Professor of Mathematics at Pennsylvania State University. He proved the Kazhdan–Lusztig conjectures...
and Kashiwara
Masaki Kashiwara
is a Japanese mathematician. He was a student of Mikio Sato at the University of Tokyo. Sato and Kashiwara have collaborated on algebraic analysis and D-module theory.He is a member of the French Academy of Sciences.- Concepts named after Kashiwara :...
obtained a proof of the Kazhdan–Lusztig conjectures. However, the proof of Beilinson–Bernstein introduced a method of . This established a geometric description of the entire category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of representations of the Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
, by "spreading out" representations as geometric objects living on the flag variety. These geometric objects naturally have an intrinsic notion of parallel transport
Parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
: they are D-modules.
In 1982 Beilinson stated his own, seemingly profound conjectures about the existence of motivic cohomology
Motivic cohomology
Motivic cohomology is a cohomological theory in mathematics, the existence of which was first conjectured by Alexander Grothendieck during the 1960s. At that time, it was conceived as a theory constructed on the basis of the so-called standard conjectures on algebraic cycles, in algebraic geometry...
groups for schemes
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
, provided as hypercohomology groups of a complex of abelian groups and related to algebraic K-theory
Algebraic K-theory
In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....
by a motivic spectral sequence, analogous to the Atiyah–Hirzebruch spectral sequence in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
. These conjectures have since been dubbed the Beilinson-Soulé conjectures; they are intertwined with Vladimir Voevodsky
Vladimir Voevodsky
Vladimir Voevodsky is a Russian American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002.- Biography :...
's program to develop a homotopy theory for schemes
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
.
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
Arithmetic variety
In mathematics, an arithmetic variety is the quotient space of a Hermitian symmetric space by an arithmetic subgroup of the associated algebraic Lie group.-Further reading:*"Introduction to modern number theory", By Yu I. Manin, Alekseĭ A. Panchishkin...
of the Lichtenbaum conjectures for K-groups of number rings
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
, the Hodge conjecture
Hodge conjecture
The Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they...
, the Tate conjecture
Tate conjecture
In mathematics, the Tate conjecture is a 1963 conjecture of John Tate linking algebraic geometry, and more specifically the identification of algebraic cycles, with Galois modules coming from étale cohomology...
about algebraic cycles, the Birch and Swinnerton-Dyer conjecture
Birch and Swinnerton-Dyer conjecture
In mathematics, the Birch and Swinnerton-Dyer conjecture is an open problem in the field of number theory. Its status as one of the most challenging mathematical questions has become widely recognized; the conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay...
about elliptic curves, and Bloch's conjecture about K2 of elliptic curves.
Beilinson continued to work on algebraic K-theory
Algebraic K-theory
In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....
throughout the mid 1980s. He collaborated with Pierre Deligne
Pierre Deligne
- See also :* Deligne conjecture* Deligne–Mumford moduli space of curves* Deligne–Mumford stacks* Deligne cohomology* Fourier–Deligne transform* Langlands–Deligne local constant- External links :...
on the developing a motivic
Motive (algebraic geometry)
In algebraic geometry, a motive denotes 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples , where X is a smooth projective variety, p : X ⊢ X is an idempotent correspondence, and m an integer...
interpretation of Don Zagier
Don Zagier
Don Bernard Zagier is an American mathematician whose main area of work is number theory. He is currently one of the directors of the Max Planck Institute for Mathematics in Bonn, Germany, and a professor at the Collège de France in Paris, France.He was born in Heidelberg, Germany...
's polylogarithm conjectures that proved to be very influential.
From the early 1990s onwards, Beilinson worked with Vladimir Drinfel'd
Vladimir Drinfel'd
Vladimir Gershonovich Drinfel'd is a Ukrainian and Soviet mathematician at the University of Chicago.The work of Drinfeld related algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the...
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 algebra
Chiral algebra
In mathematics, a chiral algebra is an algebraic structure introduced by as a rigorous version of the rather vague concept of a chiral algebra in physics....
s. This has led to new advances in conformal field theory
Conformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations...
, string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
and the geometric Langlands program
Langlands program
The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
. He was elected a Fellow of the American Academy of Arts and Sciences
American Academy of Arts and Sciences
The American Academy of Arts and Sciences is an independent policy research center that conducts multidisciplinary studies of complex and emerging problems. The Academy’s elected members are leaders in the academic disciplines, the arts, business, and public affairs.James Bowdoin, John Adams, and...
in 2008.