Alexander Beilinson &

David Kazhdan

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 prizewinners 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

21 May 2020 Hong Kong