Work
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. 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 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. He was a visiting scholar at the Institute for Advanced Study in the fall of 1994 and again from 1996 to 1998.
Read more about this topic: Alexander Beilinson
Famous quotes containing the word work:
“When I consider thy heavens, the work of thy fingers, the moon and the stars, which thou hast ordained; what is man, that thou art mindful of him? and the son of man, that thou visitest him? For thou hast made him a little lower than the angels, and hast crowned him with glory and honour.”
—Bible: Hebrew Psalms, 8:2.
“Man was kreated a little lower than the angells and has bin gittin a little lower ever sinse.” (Josh Billings, His Sayings, ch. 28, 1865)
“So is the English Parliament provincial. Mere country bumpkins, they betray themselves, when any more important question arises for them to settle, the Irish question, for instance,—the English question why did I not say? Their natures are subdued to what they work in. Their “good breeding” respects only secondary objects.”
—Henry David Thoreau (1817–1862)
“The work of adult life is not easy. As in childhood, each step presents not only new tasks of development but requires a letting go of the techniques that worked before. With each passage some magic must be given up, some cherished illusion of safety and comfortably familiar sense of self must be cast off, to allow for the greater expansion of our distinctiveness.”
—Gail Sheehy (20th century)