# Alexander Alekseevich Beilinson

### Quick Info

Moscow, USSR (now Russia)

**Alexander Beilinson**is a Russian-American mathematician who has won major prizes for his work on representation theory, algebraic geometry and mathematical physics. He has won the Ostrowski Prize, the Wolf Prize and the Shaw Prize.

### Biography

**Alexander Beilinson**, the son of the mathematician Alexander Alekseevich Beilinson, was born into a Jewish family in Moscow. He writes in [6]:-

The city was much smaller then and still retained some rural character: small wooden houses with gardens, an occasional horse-driven cart. After the joy of early childhood, going to school was a setback. After 7th grade I went to mathematical school no. 2. It was a true change: lectures and seminars on advanced mathematical topics taught by professors and students from the University, the shining classes on literature and on history.The family had a summer house and near there was a neighbour who was a mathematician. As soon as he started reading mathematics books, Alexander started talking to him about mathematics. By 1972 he was a senior at the 2nd mathematical school and, in the autumn of that year, was introduced by Alesha Parshin to Israel Moiseevich Gelfand's mathematical seminar which began at the beginning of September and was held in the large auditorium on the 14th floor of Moscow University's main building. It was an evening event, with 6 p.m. to 7 p.m. being preseminar discussion time, then Gelfand arrived and the seminar went on until 10 p.m. Beilinson described Gelfand's mathematical seminar in [12]:-

During the preseminar dozens of people congregated near the auditorium entrance, chatting and exchanging books and texts of all kinds. The seminar typically began with Gelfand telling some anecdotes and mathematical news, after which would come a talk by an invited speaker. Often there was not enough time to finish, and the talk continued serially, each time beginning from scratch and covering about half of the material from the week before, the speaker gradually fading away and being replaced by a student assigned by Gelfand to explain what the talk was, or should have been, about. Any speaker deemed not to have understood the subject, or to have explained it badly (or if the writing was too small and the voice not clear) was harshly reprimanded.Beilinson graduated from the 2nd mathematical school and sat the entrance examinations for the University of Moscow. Given that he was already far ahead of others sitting the entrance examination, one might expect this to be a formality. This, however, was not the case since, as we mentioned above, he was Jewish. At this time the University of Moscow was highly anti-Semitic and made sure that almost all Jewish students failed the entrance examinations. You can read in Jay Egenhoff's article [15] how the examiners went about failing talented Jewish students. An example is given of a Jewish student being asked "What is the definition of a circle?" and being failed for giving the answer "It is the set of points in a plane, equidistant from a fixed point." The examiners said the correct answer was "the set of

*all*points in a plane, equidistant from a fixed point."

Having failed the University of Moscow entrance examination, Beilinson went to the Pedagogical Institute which he described as a blessing. He attended classes and seminars but often missed these to take a train to the woods where he loved to walk. In 1977 Beilinson moved to the University of Moscow where he undertook research advised by Yuri Ivanovich Manin. The first result that he obtained was published in his 1978 paper

*Coherent sheaves on $\mathbb{P}^{n}$ and problems in linear algebra*(Russian). By a coincidence Israel Moiseevich Gelfand, working with Joseph N Bernstein and Sergei I Gelfand (I M Gelfand's son), had found a similar result near the end of 1977. It was published in the paper J N Berstein, I M Gelfand and S I Gelfand,

*Algebraic vector bundles on $\mathbb{P}^{n}$ and problems of linear algebra*(Russian). The two papers appear consecutively in the journal

*Akademiya Nauk SSSR. Funktsional'nyi Analiz i ego Prilozheniya*. Beilinson explained that before either work was published [12]:-

Gelfand gave a talk about his work, mentioning that I had obtained a similar theorem. After the talk I approached Gelfand, and he at once ordered me to leave Yuri Ivanovich Manin, who was my supervisor, and be his student. I refused. When I told Manin about the accolade, he said this had happened to many, e.g., to himself and to Shafarevich. Thereafter I stayed in an outer orbit of Gelfand's influence, and our relationship was excellent.The authors of [16] write:-

Sasha's very first mathematical paper - a note on coherent sheaves on $\mathbb{P}^{n}$, written in 1977, opened a new direction in algebraic geometry. Together with the paper of Bernstein-Gelfand-Gelfand on the same topic (which appeared simultaneously with Sasha's), it revealed the role of the derived category of coherent sheaves as an invariant of an algebraic variety, and started the theory of exceptional collections, which now became an indispensable part of algebraic geometry, especially in connection with homological mirror symmetry. One of the significant impacts of these papers, amplified by the subsequent one of Beilinson-Bernstein-Deligne on perverse sheaves, was that they brought derived categories within the field of vision of mathematicians working in many different fields, and they started to play an increasingly pivotal role in areas well beyond algebraic geometry: representation theory, non-commutative geometry, symplectic topology, mathematical physics, etc. The fact that the two approaches to coherent sheaves on $\mathbb{P}^{n}$ (by Beilinson and by Bernstein-Gelfand-Gelfand) appeared at the same time was providential for the birth of the theory of Koszul duality. Even though Koszul algebras had been known for some time, the connection of the symmetric and exterior algebras to derived categories of sheaves on $\mathbb{P}^{n}$ showed that Koszul duality can be stated as an equivalence of derived categories of modules. This picture was developed in Beilinson's later works.Beilinson graduated from the University of Moscow in 1980 and was employed in the mathematical laboratory of the Moscow Cardiological Centre. The head of the laboratory was Vladimir Mikhailovich Alexeev who appears to have been happy that Beilinson spent much time working on his own mathematics. For example Beilinson published three papers in 1980, two single author papers and one with S I Gelfand and Ju I Manin. In 1981 he published a joint paper with Joseph Bernstein. Alexeev died in December 1980 and his successor was not happy for Beilinson to be undertaking his own work and wanted to dismiss him. Gelfand stepped in and had Beilinson transferred to the biological sector of the Moscow Cardiological Centre whose director was happy to employ Beilinson as an engineer with no responsibilities.

Beilinson married the biologist Irene Ogievetskaya; they had two children Helen Alexander Beilinson and Vera Beilinson.

The Six-Day Arab-Israeli War in 1967 had led to increased anti-Semitism in the Soviet Union. In 1971 the ban which had been imposed on emigration of Soviet Jews was lifted and many considered emigrating to Israel while some preferred emigrating to the United States. Dmitry Aleksandrovich Kazhdan, a member of Gelfand's school, emigrated to the United States in 1975 taking up a position at Harvard and changing his name to David Kazhdan. In fact he would share the 2020 Shaw Prize with Beilinson. Ilya Iosifovich Piatetski-Shapiro, a professor at Moscow University, was dismissed from his professorship after signing a letter of support for a mathematician sent to a mental institution. After a few very difficult years he managed to emigrate to Israel in 1976. Joseph Bernstein, who was working closely with Beilinson, emigrated to the United States in 1981 and became a professor at Harvard. Beilinson writes [12]:-

No one at the seminar could replace them.Collaboration between Beilinson and Joseph Bernstein had led to them announcing a proof of the Kazhdan-Lusztig conjectures and the Jantzen conjectures in 1981. They published the paper

*Localisation de g-modules*in

*Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique*in 1981. Floyd Williams writes in the review [26]:-

The authors show how to construct a localization functor for certain noncommutative rings related to a reductive Lie algebra g. This functor identifies g-modules with sheaves of modules over a ring of (twisted) differential operators on a suitable flag manifold. As an application a new classification of irreducible Harish-Chandra modules and a proof of the Kazhdan-Lusztig conjecture are sketched. Other proofs of the latter conjecture (in one form or another) have been given by Brylinski and Kashiwara and by Vogan.In 1982 Beilinson produced conjectures which quickly became known as the 'Beilinson Conjectures'. Spencer Bloch gave a talk on these conjectures at a conference on algebraic geometry, algebraic topology and differential equations held in Mexico City in December 1984. A special session at the Oberwolfach Mathematical Research Institute on the Beilinson Conjectures led to the publication of the book

*Beilinson's conjectures on special values of L-functions*(1988). As of November 2023, MathSciNet list over 40 papers with Beilinson Conjectures in the title.

Beilinson was awarded the Moscow Mathematical Society Prize in 1985. Anatole Katok explained in [19]:-

The Moscow Mathematical Society Prize for young mathematicians carried a considerable prestige, especially with the mathematical community at-large, as opposed to the official authorities. It was awarded by the elected Society Council which represented the cream-of-the-crop of the community in terms of research achievements and international reputation. The prize was awarded for a speciﬁc body of, work, jointly if the recognised work was joint, and was subject to the upper age restriction of thirty years for all nominees. The prize usually was given for really outstanding work which produced a strong and lasting impact and was also a good predictor of the winner's long-term success. Usually two prizes were awarded every year.He received his PhD in 1988 from the Landau Institute of Theoretical Physics and was a researcher at the Landau Institute from 1987 to 1993. He writes [6]:-

At the end of the 80s "perestroika" brought onto Moscow streets immense crowds calling for changes. These arrived: the country was split and pillaged by the robber barons, the life losses on a par with those in the Civil War 74 years earlier.He split his time between the United States and Russia, being a professor of mathematics at the Massachusetts Institute of Technology from 1988 to 1998. In 1998 Beilinson was appointed to the University of Chicago [21]:-

At 41, Beilinson no longer is eligible for the Fields Medal. Nevertheless, "his mathematical achievements are on the level of those of the most renowned Fields Medalists," Manin said. The influence of Beilinson's work extends into representation theory, arithmetical geometry and modern mathematical physics, said Manin. Beilinson holds the prestigious first David and Mary Winton Green University Professorship in Mathematics. Since 1989, Beilinson largely has spent fall semesters teaching at the Massachusetts Institute of Technology as a professor of mathematics and working the rest of the year as a researcher at the Landau Institute of Theoretical Physics in Chernogolovka, Russia.Helen Alexander Beilinson, Beilinson's daughter, grew up during the years Beilinson spent settling in the United States [18]:-

"There are several people here whose research is very close to mine and who inspired it in a sense, and so I wish to work with them," Beilinson said. He was referring to Spencer Bloch, the Robert Maynard Hutchins Distinguished Service Professor in Mathematics, and Victor Ginzburg, Robert Kottwitz and Madhav Nori, Professors in Mathematics. "People here would like to create something new," Beilinson said. "It's very nice to come to a place that hopefully will create something wonderful when everything is moving." Beilinson also collaborates with Drinfeld, whom he has known for more than two decades.

Science comes naturally to Helen Beilinson. She was born in Moscow and grew up in academia; her father is a mathematician and her mother is a biologist. Her family left Russia when she was a young child and led a nomadic academic life until settling at the University of Chicago. Beilinson's baby-sitters were postdocs and grad students and she spent her childhood roaming her mom's biology lab.In 1999 Beilinson shared the Ostrowski Prize with Helmut Hofer of the Courant Institute. Beilinson was awarded the prize [8]:-

... for achievements in the areas of representation theory, arithmetic geometry, and modern mathematical physics.For more details about Beilinson's work which led to the award of the 1999 Ostrowski Prize, see THIS LINK.

In 2018 Beilinson and Vladimir Drinfeld were awarded the Wolf Prize [27]:-

Alexander Beilinson's outstanding achievements include proofs of the Kazhdan-Lusztig and Jantzen conjectures, which play a key role in the representation theory, the development of important conjectures ("Beilinson's Conjectures") in algebraic geometry, and a significant contribution to the interface between geometry and mathematical physics. The joint work of Beilinson and Vladimir Drinfeld on the Langlands Program - a woven fabric of theorems and conjectures designed to link key areas of mathematics - has led to impressive progress in implementing the program in important areas of physics, such as quantum field theory and string theory.For more details about Beilinson's work which led to the award of the 2018 Wolf Prize, see THIS LINK.

In 2020 Beilinson and David Kazhdan shared the Shaw Prize [14]:-

Alexander Beilinson and David Kazhdan are two mathematicians who have made profound contributions to the branch of mathematics known as representation theory, but who are also famous for the fundamental influence they have had on many other areas, such as arithmetic geometry, K-theory, conformal field theory, number theory, algebraic and complex geometry, group theory, and algebra more generally. As well as proving remarkable theorems themselves, they have created conceptual tools that have been essential to many breakthroughs of other mathematicians. Thanks to their work and its exceptionally broad reach, large areas of mathematics are significantly more advanced than they would otherwise have been.For more details about Beilinson's work which led to the award of the 2020 Shaw Prize, see THIS LINK.

Group theory is intimately related to the notion of symmetry and one can think of a representation of a group as a "description" of it as a group of transformations, or symmetries, of some mathematical object, usually linear transformations of a vector space. Representations of groups are important as they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system and the representations also make the symmetry group better understood. In loose terms, representation theory is the study of the basic symmetries of mathematics and physics. Symmetry groups are of many different kinds: finite groups, Lie groups, algebraic groups, p-adic groups, loop groups, adelic groups. This may partly explain how Beilinson and Kazhdan have been able to contribute to so many different fields.

In 2013 Beilinson wrote a letter to the Editor of the

*Notices*of the American Mathematical Society suggesting that the American Mathematical Society sever all ties with the NSA (the National Security Agency). Here is a short extract from that letter [11]:-

... NSA destroyed the security of the Internet and privacy of communications for the whole planet. But if any healing is possible, it would probably start with making the NSA and its ilk socially unacceptable - just as, in the days of my youth, working for the KGB was socially unacceptable for many in the Soviet Union. ... NSA needs mathematicians for its tasks, and the AMS has an interest in increasing research funding. But any relationship with an organisation whose activity is so harmful for the fabric of human society is unhealthy. For the sake of integrity, the AMS should shun all contacts with the NSA.His letter started a good discussion on the topic with, as one would expect, people arguing both for and against.

Let us end with two quotes. The first one is from [16] which begins as follows:-

Sasha Beilinson has been a source of light and inspiration for all of us around him - in mathematics and in life in general. As Sasha himself put it on multiple occasions describing other people, a person's ability to produce beautiful mathematics stems from this person's inner freedom. This is exactly the quality that Sasha is amply endowed with, and his unique vision and perception of mathematics is a manifestation of this freedom.The paper [16], written in 2017, ends:-

Sasha continues to surprise and delight us by his amazing insight. For example, a couple of years ago he solved the long-standing problem of defining the notion of singular support for étale sheaves. In a sense, his solution is completely elementary; it uses Radon transform as its main tool. Yet, this is something that has eluded experts for decades.Our final quote is from Beilinson himself [13]:-

Mathematics is simple, but to perceive this your vision must not be rigid and fixed in one direction. Also appreciation of mathematics is virtually impossible without actively doing it - like Nature, it is reality you have to be part of in order to see it. Remarkably, mathematics shares these traits with our common life. ... To me, understanding ourselves as tiny parts no more important than other living beings, of the great flow of Unknown is of key importance both for everyone's normal life and for our common survival.

### References (show)

- Accolades,
*The University of Chicago Chronicle***20**(6) (2000).

http://chronicle.uchicago.edu/001130/accolades.shtml - Alexander A Beilinson,
*American Academy of Arts & Sciences*.

https://www.amacad.org/person/alexander-beilinson - Alexander Beilinson,
*Institute for Advanced Study*.

https://www.ias.edu/scholars/alexander-beilinson - Alexander Beilinson,
*Academia Europaea*.

https://www.ae-info.org/ae/Member/Beilinson_Alexander - Alexander Beilinson, Douglas Diamond elected to National Academy of Sciences,
*UChicago News*(3 May 2017).

https://news.uchicago.edu/story/alexander-beilinson-douglas-diamond-elected-national-academy-sciences - Autobiography of Alexander Beilinson,
*The Shaw Prize*(21 May 2020).

https://www.shawprize.org/autobiography/alexander-beilinson/ - An Essay on the Prize,
*The Shaw Prize*(21 May 2020).

https://www.shawprize.org/laureates/2020-mathematical-sciences/?type=Essay&laureate=1 - Beilinson and Hofer Share Ostrowski Prize,
*Notices of the American Mathematical Society***47**(8) (2000), 885. - Beilinson and Kazhdan awarded 2020 Shaw Prize,
*Notices of the American Mathematical Society***67**(8) (2020), 1252-1253. - Beilinson, Alexander A, MIT Museum.

https://webmuseum.mit.edu/detail.php?module=people&type=browse&id=1&term=Beilinson&page=1&kv=12700&record=0&module=people - A Beilinson, AMS Should Sever Ties to NSA,
*Notices of the American Mathematical Society***60**(11) (2013), 1432. - A Beilinson, I M Gelfand and his seminar - A presence,
*Notices of the American Mathematical Society***63**(3) (2016), 295-298. - Alexander Beilinson: The Shaw Prize Speech,
*youtube.com*.

https://www.youtube.com/watch?app=desktop&v=NvktjngsxdM - Contribution of Alexander Beilinson & David Kazhdan,
*The Shaw Prize*(21 May 2020).

https://www.shawprize.org/laureates/2020-mathematical-sciences/?type=Contribution - J Egenhoff, Math as a tool of anti-Semitism,
*The Mathematics Enthusiast***11**(3) (2014).

https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=1320&context=tme - M Finkelberg, D Gaitsgory, A Goncharov and A Polishchuk, A tribute to Sasha Beilinson,
*Selecta Mathematica***24**(2018), 1-5. - C Glass, Beilinson and Zakiniaeiz lead world's only student-run scientific journal, Yale News (31 October 2017).

https://yaledailynews.com/blog/2017/10/31/beilinson-and-zakiniaeiz-lead-worlds-only-student-run-scientific-journal/ - Helen Beilinson,
*Instagram*(14 May 2019).

https://www.instagram.com/p/BxcW5_9hkT1/?img_index=1 - A Katok, Moscow dynamics seminars of the Nineteen seventies and the early career of Yasha Pesin,
*Discrete and Continuous Dynamical Systems***22**(1-2) (2008), 1-22. - E Kehoe, Beilinson and Drinfeld awarded 2018 Wolf Prize in mathematics,
*Notices of the American Mathematical Society***65**(6) (2018), 697-698. - S Koppes, Math department welcomes latest addition to its stellar team of recruits,
*The University of Chicago Chronical***18**(8) (1999).

http://chronicle.uchicago.edu/990121/drinfeld.shtml - L Lerner, Two UChicago mathematicians awarded one of field's top prizes,
*UChicago News*(13 February 2013).

https://news.uchicago.edu/story/two-uchicago-mathematicians-awarded-one-fields-top-prizes - News from the National Academy of Sciences,
*National Academy of Sciences*(2 May 2017).

https://www.nasonline.org/news-and-multimedia/news/may-2-2017-NAS-Election.html - The Shaw Prize in Mathematical Sciences 2020,
*The Shaw Prize*(2020).

https://www.shawprize.org/laureates/2020-mathematical-sciences/ - UChicago mathematician Alexander Beilinson wins prestigious Shaw Prize,
*UChicago News*(22 May 2020).

https://news.uchicago.edu/story/uchicago-mathematician-alexander-beilinson-wins-prestigious-shaw-prize - F L Williams, Review: Localisation de g-modules, by A Beilinson and J Bernstein, Mathematical Reviews
*MR0610137***(82k:14015)**. - Wolf Prizes 2018,
*International Centre for Theoretical Physics*(15 February 2018).

https://www.ictp.it/news/2018/2/wolf-prizes-2018

### Additional Resources (show)

Other pages about Alexander Beilinson:

Other websites about Alexander Beilinson:

### Honours (show)

Honours awarded to Alexander Beilinson

### Cross-references (show)

Written by J J O'Connor and E F Robertson

Last Update March 2024

Last Update March 2024