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\section*{Alexander Beilinson}

\textbf{Alexander (or Sasha) Belinson} 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, [[motive]]s and [[motivic cohomology]]. He conjectured the category of motivic sheaves with remarkable cohomological properties, what is often called the Belinson dream. Some of his works, especially those in collaboration with [[Vladimir Drinfeld|Vladimir Drinfel'd]] are of large importance to mathematical physics, especially on their concept of \textbf{[[chiral algebra]]s} 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 algebra]]s. In late 1980s Belinson 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 [[Edward Witten|Witten]].

Belinson 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, Belinson 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 \textbf{decomposition theorem} (see a survey: \href{http://www.ams.org/bull/2009-46-04/S0273-0979-09-01260-9/S0273-0979-09-01260-9.pdf}{pdf}). On technical side, he also described appropriate gluing procedure for the derived categories of perverse sheaves, involving t-structures.

Belinson 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]], [[Yuri Manin|Manin]], [[Wolfgang Soergel]] and others, Belinson introduced a wide picture of ``Koszul duality patterns'' in [[representation theory]].

His other works concentrated on [[motives]], [[higher regulators]], [[epsilon-factors]], and so on.

\begin{itemize}%
\item \href{http://www.math.uchicago.edu/~mitya/langlands.html}{geometric Langlands homepage}


\item wikipedia: \href{http://en.wikipedia.org/wiki/Alexander_Beilinson}{Beilinson}


\item [[Alexander Beilinson]] \emph{Higher regulators and values of L-functions}, Journal of Soviet Mathematics 30 (1985), 2036-2070, (\href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intd&paperid=73&option_lang=eng}{mathnet (Russian)}, \href{http://dx.doi.org/10.1007%2FBF02105861}{DOI})


\item [[Alexander Beilinson]], \emph{Higher regulators of curves}, Funct. Anal. Appl. 14 (1980), 116-118, \href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=faa&paperid=1800&option_lang=eng}{mathnet (Russian)}.


\item [[Alexander Beilinson]], \emph{Height pairing between algebraic cycles}, in \emph{K-Theory, Arithmetic and Geometry}, Lecture Notes in Mathematics Volume 1289, 1987, pp 1-26, \href{http://dx.doi.org/10.1007/BFb0078364}{DOI}.


\item A. Beilinson, J. Bernstein, \emph{Localisations de $\mathfrak{g}$--modules}, C. R. Acad. Sci. Paris \textbf{292} (1981), 15--18.


\item A. A. Beilinson, [[V. Drinfeld]], \emph{[[Chiral Algebras]]}, AMS 2004 (a preprint in various forms since around 1995, cf. \href{http://www.math.uchicago.edu/~mitya/langlands.html}{here}).


\item A. A. Beilinson, V. Ginzburg, W. Soergel, \emph{Koszul duality patterns in representation theory}, J. Amer. Math. Soc. \textbf{9} (2): 473--527 (1996).


\item A. A. Belinson, V. A. Ginsburg, V. V. Schechtman, \emph{Koszul duality}, J. Geom. Phys. \textbf{5} (1988), no. 3, 317--350.


\item A. A. Beilinson, J. Bernstein, [[P. Deligne]], \emph{Faisceaux pervers}, Ast\'e{}risque \textbf{100} (1980).


\item A. Beilinson, [[Victor Ginzburg|V. Ginzburg]], \emph{[[wall crossing|Wall-crossing]] functors and $D$-modules}, Representation Theory \textbf{3} (electronic), 1--31 (1999)


\item A. Belinson, [[J. Bernstein]], \emph{A proof of Jantzen conjectures}, I. M. Gelfand Seminar, 1--50, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc. 1993, \href{http://www.math.harvard.edu/~gaitsgde/grad_2009/BB%20-%20Jantzen.pdf}{pdf}



\end{itemize}
\hypertarget{related_lab_entries}{}\subsection*{{Related $n$Lab entries}}\label{related_lab_entries}

\begin{itemize}%
\item [[Deligne-Beilinson cohomology]]


\item [[Beilinson regulator]]


\item [[Beilinson conjecture]]


\item [[derived noncommutative algebraic geometry]]


\item [[p-adic Hodge theory]]



\end{itemize}
[[!redirects Beilinson]] [[!redirects A. A. Beilinson]] [[!redirects A.A. Beilinson]] [[!redirects A. Beilinson]] [[!redirects Sasha Beilinson]]

[[!redirects Beĭlinson]] [[!redirects A. A. Beĭlinson]] [[!redirects A. Beĭlinson]]



\end{document}