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

Alexander Beilinson, Professor in Mathematics, shares the 1999 Ostrowski Prize with Helmut Hofer of New York University's Courant Institute of Mathematical Sciences. The 1999 prize was awarded this past summer at the University of Basel in Switzerland.

Beilinson received the prize for achievements in representation theory, arithmetic geometry and modern mathematical physics. The prize carries an award of 150,000 Swiss francs (approximately $87,000), and two fellowships each of 30,000 Swiss francs.

The Ostrowski Prize is awarded every other year by the Ostrowski Foundation, which was created by Alexander Ostrowski, a longtime professor at the University of Basel. Ostroswki left his estate to the foundation in order to establish a prize for outstanding achievements in pure mathematics and the foundations of numerical mathematics.

The prize jury consists of representatives from the universities of Basel, Jerusalem and Waterloo, and from the academies of Denmark and the Netherlands.

1.2. Beilinson and Hofer Share Ostrowski Prize, Notices of the American Mathematical Society 47 (8) (2000), 885.

Alexander Beilinson of the University of Chicago and Helmut Hofer of the Courant Institute shared the 1999 Ostrowski Prize. The prize carries a monetary award of 150,000 Swiss francs (approximately $87,000) and two fellowships each of 30,000 Swiss francs.

Beilinson received the Ostrowski Prize for achievements in the areas of representation theory, arithmetic geometry, and modern mathematical physics. His proof with J Bernstein of the Jantzen conjectures for reductive Lie groups involved the earlier development of D-modules and perverse sheaves in which he also played a key role. His conjectures and computations in $K$-theory continue to be very influential, as for example in his motivic treatment of D Zagier's polylogarithm conjectures in joint work with P Deligne. And Beilinson's total rebuilding of the theory of vertex operator algebras with V Drinfeld contributes to the understanding of two-dimensional conformal field theory and string theory and has furthermore led to progress in the geometric Langlands program.

The Ostrowski Foundation was created by Alexander Ostrowski, for many years a professor at the University of Basel. He left his entire estate to the foundation and stipulated that the income should provide a prize for outstanding recent achievements in pure mathematics and the foundations of numerical mathematics. The prize is awarded every other year. Previous recipients of the Ostrowski Prize are Louis de Branges, Jean Bourgain, Miklós Laczkovich, Marina Ratner, Andrew Wiles, Yuri Nesterenko, and Gilles Pisier. The prize jury consists of representatives from the universities of Basel, Jerusalem, and Waterloo, and from the academies of Denmark and the Netherlands.

The 1999 prize was awarded on 9 June 2000, at the University of Basel.

2.1. L Lerner, Two UChicago mathematicians awarded one of field's top prizes, UChicago News (13 February 2013).

Profs Alexander Beilinson and Vladimir Drinfeld win Wolf Prize.

University of Chicago mathematicians Alexander Beilinson and Vladimir Drinfeld have been awarded the prestigious Wolf Prize for Mathematics "for their ground-breaking work in algebraic geometry, representation theory and mathematical physics."

Awarded by the Israeli Wolf Foundation, the prize honours the greatest achievements every year in the fields of agriculture, chemistry, mathematics, physics, medicine and the arts. The award for each subject area carries a $100,000 prize.

"It is a great pleasure to see such deserving people recognized with this prestigious prize," said Prof Edward W "Rocky" Kolb, dean of the Division of the Physical Sciences. "Their work in algebraic geometry is truly remarkable."

Beilinson, the David and Mary Winton Green University Professor, and Drinfeld, the Harry Pratt Judson Distinguished Service Professor, specialise in algebraic geometry, which uses abstract algebra to solve questions of geometry. Frequent collaborators, their association dates back to 1975, when they were both students of Yuri Manin at Moscow State University.

Several mathematical techniques and conjectures bear their names, including the Beilinson Conjectures, cited as a guiding influence in number theory and algebraic geometry; and the Drinfeld module, which Drinfeld used in 1974 to prove parts of the Langlands program.

"The Geometric Langlands Program is a far-reaching network of conjectures, and sometimes theorems, connecting number theory, algebraic geometry, representation theory and mathematical physics in unexpected and illuminating ways," said Prof Kevin Corlette, who chairs the Department of Mathematics. "It is wonderful to see Profs Beilinson and Drinfeld recognised for their work, which has been fundamental to the development of this subject."

Beilinson specialises in geometric representation theory and mathematical physics. His honours include the Ostrowski Prize and the Moscow Mathematical Society Prize.

Both Beilinson and Drinfeld joined the University of Chicago in 1998. They frequently work together - they co-authored a 2004 textbook called Chiral Algebras, one of the most prominent texts on the subject - and they jointly run a seminar called the "Geometric Langlands Seminar," which runs Mondays from 4:30 p.m. "until both the speaker and the participants are regularly exhausted," according to a 2006 collection of mathematics articles titled Algebraic Geometry and Number Theory.

Drinfeld called the Wolf Prize "a great honour." "We're in good company," Beilinson added. "To receive a prize together with Paul McCartney - who would think it would happen?" (McCartney received the Wolf Prize in Music this year.)

The Wolf Foundation was established by the German-born inventor, diplomat and philanthropist Ricardo Wolf; he later served as Fidel Castro's ambassador to Israel, where he lived until his death in 1981. The prizes will be awarded by Israeli president Reuven Rivlin at a May ceremony in Jerusalem.

2.2. Wolf Prizes 2018, International Centre for Theoretical Physics (15 February 2018).

Two recipients of the Wolf Foundation's 2018 Prizes in Mathematics and Physics, which were announced on 12 February, have strong ties to ICTP (International Centre for Theoretical Physics). The Prize for Mathematics will be awarded to Professors Alexander Beilinson and Vladimir Drinfeld, both of the University of Chicago, for their ground-breaking work in algebraic geometry (a field that integrates abstract algebra with geometry), in mathematical physics and in presentation theory, a field which helps to understand complex algebraic structures.

Beilinson has lectured frequently at ICTP; most recently, he delivered a seminar on algebraic cycles as part of the ICTP Mathematics section's Basic Notions Seminars, as well as an introductory course on Hodge theory. The Wolf Prize is one of the highest prizes in mathematics, along with the Abel Prize and the Fields Medal, with the latter restricted to mathematicians under 40 years of age.

Alexander Beilinson's outstanding achievements include proofs of the Kashdan-Lustig 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.

2.3. E Kehoe, Beilinson and Drinfeld awarded 2018 Wolf Prize in mathematics, Notices of the American Mathematical Society 65 (6) (2018), 697-698.

Alexander Beilinson and Vladimir Drinfeld of the University of Chicago have been awarded the Wolf Foundation Prize for Mathematics for 2018 by the Wolf Foundation.

The prize citation reads in part: "The Wolf Foundation Prize for Mathematics in 2018 will be awarded to Professors Alexander Beilinson and Vladimir Drinfeld, both of the University of Chicago, for their ground-breaking work in algebraic geometry (a field that integrates abstract algebra with geometry), in mathematical physics and in representation theory, a field which helps to understand complex algebraic structures. An 'algebraic structure' is a set of objects, including the actions that can be performed on those objects, that obey certain axioms. One of the roles of modern algebra is to research, in the most general and abstract way possible, the properties of various algebraic structures (including their objects), many of which are amazingly complicated."

Beilinson's "outstanding achievements include proofs of the Kashdan-Lustig 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.

Drinfeld and Beilinson together created a geometric model of algebraic theory that plays a key role in both field theory and physical string theory, thereby further strengthening the connections between abstract modern mathematics and physics. In 2004 they jointly published their work in a book [Chiral Algebras, AMS], that describes important algebraic structures used in quantum field theory, which is the theoretical basis for the particle physics of today. This publication has since become the basic reference book on this complex subject.

Biographical Sketch

Alexander Beilinson was born in 1957 in Moscow and received his PhD in 1988 from the Landau Institute of Theoretical Physics. He was a researcher at the Landau Institute from 1987 to 1993 and professor of mathematics at the Massachusetts Institute of Technology from 1988 to 1998, when he moved to his present position at the University of Chicago. He was awarded the Moscow Mathematical Society Prize in 1985 and the Ostrowski Prize in 1999. He was elected to the National Academy of Sciences of the USA in 2017.

About the Prize

The Wolf Prize carries a cash award of US$100,000. The science prizes are given annually in the areas of agriculture, chemistry, mathematics, medicine, and physics. Laureates receive their awards from the President of the State of Israel in a special ceremony at the Knesset Building (Israel's Parliament) in Jerusalem.

3.1. Contribution of Alexander Beilinson & David Kazhdan, The Shaw Prize (21 May 2020).

The Shaw Prize in Mathematical Sciences 2020 is awarded in equal shares to Alexander Beilinson, David and Mary Winton Green University Professor at the University of Chicago, USA and David Kazhdan, Professor of Mathematics at the Hebrew University of Jerusalem, Israel, for their huge influence on and profound contributions to representation theory, as well as many other areas of mathematics.

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.

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.

One of Kazhdan's most influential ideas has been the introduction of a property of groups that is known as Kazhdan's property (T). Among the representations of a group there is always the not very interesting "trivial representation" where we associate with each group element the "transformation" that does nothing at all to the object. While the trivial representation is not interesting on its own, much more interesting is the question of how close another representation can be to the trivial one. Property (T) gives a precise quantitative meaning to this question. Kazhdan used Property (T) to solve two outstanding questions about discrete subgroups of Lie groups. Since then it has had important applications to group representation theory, lattices in algebraic groups over local fields, ergodic theory, geometric group theory, expanders, operator algebras and the theory of networks, and has been recognised as a truly fundamental concept in representation theory.

After this first breakthrough Kazhdan solved several other outstanding problems about lattices in Lie groups and representation theory such as the Selberg conjecture about non-uniform lattices, and the Springer conjecture on the classification of affine Hecke algebras.

While working with George Lusztig on this last problem, Kazhdan introduced an important family of polynomials, as well as formulating a very influential pair of (equivalent) conjectures. One of Alexander Beilinson's achievements was to prove these conjectures with Joseph Bernstein. (They were also proved independently by Jean-Luc Brylinski and Masaki Kashiwara.) The methods introduced in this proof led to the area known as geometric representation theory, an area that Kazhdan also played an important part in developing, which aims to understand the deeper geometric and categorical structures that often underlie group representations. The resulting insights have been used to solve several open problems.

Another famous concept, this one established by Beilinson, Bernstein and Pierre Deligne, is that of a perverse sheaf. It is not feasible to give a non-technical explanation of what a perverse sheaf is - one well-known account begins by helpfully stating that it is neither perverse nor a sheaf - but it is another concept that can be described as a true discovery, in that it has a far from obvious definition, but it is now seen to be "one of the most natural and fundamental objects in topology" (to quote from the same account). One of the central goals of mathematics, the Langlands programme, has been deeply influenced by this concept. For example, the whole work of Ngô on the "fundamental lemma" and the contributions of Laurent and Vincent Lafforgue (all three of them major prize winners for this work) would have been unthinkable without it. Kazhdan too has brought extraordinary mathematical insight into this circle of ideas. By pointing out that orbital integrals could be interpreted as counting points on certain algebraic varieties over finite fields, he and Lusztig opened the way to the proof of the fundamental lemma, and since then Kazhdan has had and continues to have an enormous influence on the subject. Beilinson is also famous for formulating deep conjectures relating $L$-functions and motivic theory, which have completely changed the understanding of both topics and led to an explosion of related work.

Beilinson and Kazhdan are at the heart of many of the most exciting developments in mathematics over the last few decades, developments that continue to this day. It is for this that they are awarded the 2020 Shaw Prize in Mathematical Sciences

Mathematical Sciences Selection Committee
The Shaw Prize
Hong Kong
21 May 2020

3.2. Autobiography of Alexander Beilinson, The Shaw Prize (21 May 2020).

I was born in 1957 in Moscow. 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. I got to know A Parshin, and in 1972 he took me to I Gelfand's seminar; this was the start of mathematical life. After school I entered the Pedagogical Institute, and in 1977 moved to the University to be Y Manin's student. Attendance was not enforced; skipping all classes of no interest (an officer informed me I was the champion of playing truant) gave much time for walking in the woods and for doing maths. I graduated in 1980, and V Alexeev took me to his computer lab at the Cardiological Centre. Sadly, he died soon after. By the word of Gelfand, I became an engineer with no responsibilities at the biological division of the Centre and gained the same freedom as in my student years.

To those who wished to see things with one's own eyes, and valued free time to think - not focusing on their career - the Moscow of my time was a very nice place to be. Since Khrushchev announced that his predecessor had been a criminal, much of the public relegated all things related to the powers that be to the domain of the ridiculous. People were connected by the flow of books; poetry was learned by heart. A unique art, light and free, came to life: to get a taste, one can watch Yu Norstein's animated movies "Hedgehog in the Fog" and "Tale of Tales", and read Yury Koval (it's a pity the best of Koval's books - the finest Russian prose of the time - are not translated into English). Mathematics was largely a part of that culture.

Doing maths is akin to unfolding a melody; its first sounds are usually a gift from someone else. My first paper was written in the footsteps of the ADHM (Atiyah-Drinfeld-Hitchin-Manin) classification of instantons. I caught the idea of higher regulators while preparing a talk on S Bloch's work; this led to conjectures on the values of $L$-functions (still widely open) and to speculations about mixed motives (largely realised by V Voevodsky and A Suslin). Conversations with R MacPherson and P Deligne brought forth our work with J Bernstein on the Kazhdan-Lusztig conjecture; we played happily with D-modules and perverse sheaves until Bernstein left Russia at the beginning of 1981. In the mid-80s, A Belavin taught me the basics of string theory and conformal field theory; his work with A Polyakov and A Zamolodchikov came to be a source of the idea of factorisation geometry developed later together with V Drinfeld.

At the end of the 80s "perestroika" brought into Moscow streets immense crowds calling for changes. These arrived: the country was split and pillaged by the robber barons, the life losses on par with those in the Civil War 74 years earlier. In 1989 I went to the Landau Institute and, for two months in the Fall, I was at the Massachusetts Institute of Technology. Around 1993 I began to work with Drinfeld on his approach to geometric Langlands theory via the quantization of Hitchin's fibration, and on factorisation geometry. In 1998 we moved to Chicago. Drinfeld joined us within a year. Since then, we have run the "geometric Langlands" seminar at the University of Chicago, which resembles the Gelfand seminar of yore.

In America there are still some woods; the trees are magnificent, the animals full of grace and wisdom. Life is very kind to me. But one can't help seeing everywhere the madman's effort to build in his own image a fake world by destroying the real one, together with its live magic we are all part of. In his Dachau diaries E Kupfer-Koberwitz wrote that the worst of what humans do to themselves is a direct consequence of what they do to animals. Perhaps the death spiral cannot be stopped unless a phase transition in our attitude to ourselves and to Nature happens, and we realise that animal lives matter no less than human ones.

Hong Kong
20 May 2021

3.3. Alexander Beilinson: The Shaw Prize Speech.

Ladies and Gentlemen, I am deeply grateful to the Shaw Prize Foundation for the great honour of receiving the 2020 prize in Mathematical Sciences. I was born in Moscow into a life that was not poisoned by wars and strife, nor by search for money and power. People were doing mathematics for the sole reason that it was extremely interesting. Among them were my teachers - Belavin, Bernstein, Drinfeld, Feigin, Gelfand, Manin, Parshin - to whom I owe my gratitude for becoming a mathematician.

Two features of mathematics are prominent: 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. It is dear to me that this prize comes from the country of Tao Te Ching. 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. And I am so happy to share this prize with Kazhdan, a dear friend whose works were at the base of a large part of my own research. Thank you.

3.4. UChicago mathematician Alexander Beilinson wins prestigious Shaw Prize, UChicago News (22 May 2020).

Prof Alexander Beilinson, the David and Mary Winton Green University Professor of Mathematics at the University of Chicago, is one of two recipients of the prestigious Shaw Prize in Mathematical Sciences.

He shares this year's honour jointly with David Kazhdan of the Hebrew University of Jerusalem for their "huge influence on and profound contributions to representation theory, as well as many other areas of mathematics."

The Shaw Prize honours individuals who have recently achieved distinguished and significant advances in the fields of astronomy, life science and medicine, and mathematical sciences. Each category carries a monetary award of $1.2 million.

Beilinson has done pioneering work in algebraic geometry, geometric representation theory and mathematical physics. His "Beilinson Conjectures" serve as a guiding influence in the field of arithmetic geometry, and he has made substantial contributions to geometric representation theory. He introduced the concept of motivitic sheaves with cohomological properties, sometimes referred the "Beilinson dream" and is regarded as a founder in the field of derived noncommutative algebraic geometry.

Beilinson's work with UChicago colleague Vladimir Drinfeld, the Harry Pratt Judson Distinguished Service Professor in Mathematics, is critical to geometric Langlands theory. They jointly received the 2018 Wolf Prize in Mathematics.

Established in 2002 under the auspices of the late Run Run Shaw, the Shaw Prize honours individuals who have achieved significant breakthroughs in academic and scientific research or applications and whose works have resulted in positive and profound impacts on mankind.