# Essay

Alexander Beilinson and David Kazhdan have made profound contributions to the branch of mathematics known as representation theory, and 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 would otherwise have been possible.

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 reduce many group-theoretic 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. In loose terms, representation theory is the study of the basic symmetries of mathematics and physics. Many different kinds of groups occur naturally as symmetry groups. An obvious example is finite groups, but there are also more “continuous” groups known as Lie groups that are hugely important in physics, as well as algebraic groups, which are groups defined using polynomial equations, and several other classes of groups. This partly explains 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, he introduced an important family of polynomials, as well as formulating a very influential pair of (equivalent) conjectures concerning them. One of Alexander Beilinson’s achievements was to prove these conjectures with Joseph Bernstein. (These conjectures 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 is bringing extraordinary mathematical insight into this circle of ideas. Indeed in a first attempt to prove the fundamental lemma, Goresky, Kottwitz and MacPherson were missing a way to organize certain algebraic objects in families. Kazhdan recently came up with the insight that this can be achieved by switching to formal algebraic geometry in infinite dimensions. This is the source of promising current work. 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.

20 May 2021 Hong Kong