Alexander (or Sasha) Beĭlinson is currently a professor at University of Chicago. He was student of Yuri Manin at Moscow State University, with main works in algebraic geometry. He has made visionary contributions to the study of algebraic cycles, automorphic forms and L-functions, algebraic K-theory, Hodge theory, motives and motivic cohomology. He conjectured the category of motivic sheaves with remarkable cohomological properties, what is often called the Beĭlinson dream. Some of his works, especially those in collaboration with Vladimir Drinfel'd are of large importance to mathematical physics, especially on their concept of chiral algebras which are an approach to a chiral part of the conformal field theory on a curve, which is a geometric counterpart of the theory of vertex operator algebras. In late 1980s Beĭlinson proposed a geometric analogue of the Langlands program, now called geometric Langlands program, which has been continued in his collaboration with Drinfel’d and also by Ed Frenkel, Dennis Gaitsgory, Ivan Mirković, Kari Vilonen, and more recently taken up by mathematical physics community lead by Witten.
Beĭlinson has substantial contributions to geometric representation theory, which has been revolutionized after two discoveries: his proof (with input from Bernstein) of Kazhdan-Lusztig’s connjectures in 1980, and his paper with Bernstein on what is now called Beĭlinson-Bernstein localization theorem, which lead to influx of the algebraic methods involving algebraic D-modules to representation theory. With Kazhdan, Beĭlinson has used D-modules in the proof of Jantzen’s conjecture, where he introduced the noton of D-affinity and the geometric viewpoint via D-schemes. In the work on Hitchin fibration and Hitchin integrable system (with Drinfel’d) much of technique of algebraic geometry on ind-schemes, including study of D-modules is developed and used. In similar spirit to D-modules, he was also using perverse modules; with Ofer Gabber, Bernstein and Deligne he developed their basic theory including deep and extremely powerful theorem for usage in representation theory, the decomposition theorem (see a survey: pdf). On technical side, he also described appropriate gluing procedure for the derived categories of perverse sheaves, involving t-structures.
Beĭlinson has shown a remarkable structure of the bounded derived category of coherent sheaves on projective spaces, and its connections to quivers. This work, together with subsequent work with Joseph Bernstein and also later works of Mikhail Kapranov and Alexei Bondal marked the birth of the derived noncommutative algebraic geometry. With Victor Ginzburg, Manin, Wolfgang Soergel and others, Beĭlinson introduced a wide picture of “Koszul duality patterns” in representation theory.
A. Beilinson, J. Bernstein, Localisations de –modules, C. R. Acad. Sci. Paris 292 (1981), 15–18.
A. A. Beilinson, V. Ginzburg, W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (2): 473–527 (1996).
A. A. Beĭlinson, V. A. Ginsburg, V. V. Schechtman, Koszul duality, J. Geom. Phys. 5 (1988), no. 3, 317–350.
A. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Astérisque 100 (1980).