# Background on 2002 Fields and Nevanlinna Awardees

Laurent Lafforgue

Laurent Lafforgue has made an enormous advance in the so-called Langlands Program by proving the global Langlands correspondence for function fields. His work is characterized by formidable technical power, deep insight, and a tenacious, systematic approach.

The Langlands Program, formulated by Robert P. Langlands for the first time in a famous letter to Andre Weil in 1967, is a set of far-reaching conjectures that make precise predictions about how certain disparate areas of mathematics might be connected. The influence of the Langlands Program has grown over the years, with each new advance hailed as an important achievement.

One of the most spectacular confirmations of the Langlands Program came in the 1990s, when Andrew Wiles's proof of Fermat's Last Theorem, together with work by others, led to the solution of the Taniyama-Shimura-Weil Conjecture. This conjecture states that elliptic curves, which are geometric objects with deep arithmetic properties, have a close relationship to modular forms, which are highly periodic functions that originally emerged in a completely different context in mathematical analysis. The Langlands Program proposes a web of such relationships connecting Galois representations, which arise in number theory, and automorphic forms, which arise in analysis.

The roots of the Langlands program are found in one of the deepest results in number theory, the Law of Quadratic Reciprocity, which goes back to the time of Fermat in the 17th century and was first proved by Carl Friedrich Gauss in 1801. An important question that often arises in number theory is whether, upon dividing two prime numbers, the remainder is a perfect square. The Law of Quadratic Reciprocity reveals a remarkable connection between two seemingly unrelated questions involving prime numbers p and q: "Is the remainder of p divided by q a perfect square?" and "Is the remainder of q divided by p a perfect square?" Despite many proofs of this law (Gauss himself produced six different proofs), it remains one of the most mysterious facts in number theory. Other reciprocity laws that apply in more general situations were discovered by Teiji Takagi and by Emil Artin in the 1920s. One of the original motivations behind the Langlands Program was to provide a complete understanding of reciprocity laws that apply in even more general situations.

The global Langlands correspondence proved by Lafforgue provides this complete understanding in the setting not of the ordinary numbers but of more abstract objects called function fields. One can think of a function field as consisting of quotients of polynomials; these quotients can be added, subtracted, multiplied, and divided just like the rational numbers. Lafforgue established, for any given function field, a precise link between the representations of its Galois groups and the automorphic forms associated with the field. He built on work of 1990 Fields Medalist Vladimir Drinfeld, who proved a special case of the Langlands correspondence in the 1970s. Lafforgue was the first to see how Drinfeld's work could be expanded to provide a complete picture of the Langlands correspondence in the function field case.

In the course of this work Lafforgue invented a new geometric construction that may prove to be important in the future. The influence of these developments is being felt across all of mathematics.

Laurent Lafforgue was born on 6 November 1966 in Antony, France. He graduated from the Ecole Normale Superieure in Paris (1986). He became an attache de recherche of the Centre National de la Recherche Scientifique (1990) and worked in the Arithmetic and Algebraic Geometry team at the Universite de Paris-Sud, where he received his doctorate (1994). In 2000 he was made a permanent professor of mathematics at the Institut des Hautes Etudes Scientifiques in Bures-sur-Yvette, France.

The following is an article about the work of Lafforgue: "Fermat's Last Theorem's First Cousin," by Dana Mackenzie. Science, Volume 287, Number 5454, 4 February 2000, pages 792-793.

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Vladimir Voevodsky made one of the most outstanding advances in algebraic geometry in the past few decades by developing new cohomology theories for algebraic varieties. His work is characterized by an ability to handle highly abstract ideas with ease and flexibility and to deploy those ideas in solving quite concrete mathematical problems.

Voevodsky's achievement has its roots in the work of 1966 Fields Medalist Alexandre Grothendieck, a profound and original mathematician who could perceive the deep abstract structures that unite mathematics. Grothendieck realized that there should be objects, which he called "motives," that are at the root of the unity between two branches of mathematics, number theory and geometry. Grothendieck's ideas have had widespread influence in mathematics and provided inspiration for Voevodsky's work.

The notion of cohomology first arose in topology, which can be loosely described as "the science of shapes." Examples of shapes studied are the sphere, the surface of a doughnut, and their higher-dimensional analogues. Topology investigates fundamental properties that do not change when such objects are deformed (but not torn). On a very basic level, cohomology theory provides a way to cut a topological object into easier-to-understand pieces. Cohomology groups encode how the pieces fit together to form the object. There are various ways of making this precise, one of which is called singular cohomology. Generalized cohomology theories extract data about properties of topological objects and encode that information in the language of groups. One of the most important of the generalized cohomology theories, topological K-theory, was developed chiefly by another 1966 Fields Medalist, Michael Atiyah. One remarkable result revealed a strong connection between singular cohomology and topological K-theory.

In algebraic geometry, the main objects of study are algebraic varieties, which are the common solution sets of polynomial equations. Algebraic varieties can be represented by geometric objects like curves or surfaces, but they are far more "rigid" than the malleable objects of topology, so the cohomology theories developed in the topological setting do not apply here. For about forty years, mathematicians worked hard to develop good cohomology theories for algebraic varieties; the best understood of these was the algebraic version of K-theory. A major advance came when Voevodsky, building on a little-understood idea proposed by Andrei Suslin, created a theory of "motivic cohomology." In analogy with the topological setting, there is a strong connection between motivic cohomology and algebraic K-theory. In addition, Voevodsky provided a framework for describing many new cohomology theories for algebraic varieties. His work constitutes a major step toward fulfilling Grothendieck's vision of the unity of mathematics.

One consequence of Voevodsky's work, and one of his most celebrated achievements, is the solution of the Milnor Conjecture, which for three decades was the main outstanding problem in algebraic K-theory. This result has striking consequences in several areas, including Galois cohomology, quadratic forms, and the cohomology of complex algebraic varieties. Voevodsky's work may have a large impact on mathematics in the future by allowing powerful machinery developed in topology to be used for investigating algebraic varieties.

Vladmir Voevodsky was born on 4 June 1966 in Russia. He received his B.S. in mathematics from Moscow State University (1989) and his Ph.D. in mathematics from Harvard University (1992). He held visiting positions at the Institute for Advanced Study, Harvard University, and the Max-Planck-Institut fuer Mathematik before joining the faculty of Northwestern University in 1996. In 2002 he was named a permanent professor in the School of Mathematics at the Institute for Advanced Study in Princeton, New Jersey.

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Madhu Sudan has made important contributions to several areas of theoretical computer science, including probabilistically checkable proofs, non-approximability of optimization problems, and error-correcting codes. His work is characterized by brilliant insights and wide-ranging interests.

Sudan has been a main contributor to the development of the theory of probabilistically checkable proofs. Given a proof of a mathematical statement, the theory provides a way to recast the proof in a form where its fundamental logic is encoded as a sequence of bits that can be stored in a computer. A "verifier" can, by checking only some of the bits, determine with high probability whether the proof is correct. What is extremely surprising, and quite counterintuitive, is that the number of bits the verifier needs to examine can be made extremely small. The theory was developed in papers by Sudan, S. Arora, U. Feige, S. Goldwasser, C. Lund, L. Lovasz, R. Motwani, S. Safra, and M. Szegedy. For this work, these authors jointly received the 2001 Goedel Prize of the Association for Computing Machinery.

Also together with other researchers, Sudan has made fundamental contributions to understanding the non-approximability of solutions to certain problems. This work connects to the fundamental outstanding question in theoretical computer science: Does P equal NP? Roughly, P consists of problems that are "easy" to solve with current computing methods, while NP is thought to contain problems that are fundamentally harder. The term "easy" has a technical meaning related to the efficiency of computer algorithms for solving problems. A fundamentally hard problem in NP has the property that a proposed solution is easily checked but that no algorithm is known that will easily produce a solution from scratch. Some NP hard problems require finding an optimal solution to a combinatorial problem such as the following: Given a finite collection of finite sets, what is the largest size of a subcollection such that every two sets in the subcollection are disjoint? What Sudan and others showed is that, for many such problems, approximating an optimal solution is just as hard as finding an optimal solution. This result is closely related to the work on probabilistically checkable proofs. Because the problems in question are closely related to many everyday problems in science and technology, this result is of immense practical as well as theoretical significance.

The third area in which Sudan made important contributions is error-correcting codes. These codes play an enormous role in securing the reliability and quality of all kinds of information transmission, from music recorded on CDs to communications over the Internet to satellite transmissions. In any communication channel, there is a certain amount of noise that can introduce errors into the messages being sent. Redundancy is used to eliminate errors due to noise by encoding the message into a larger message. Provided the coded message does not suffer too many errors in transmission, the recipient can recover the original message. Redundancy adds to the cost of transmitting messages, and the art and science of error-correcting codes is to balance redundancy with efficiency. A class of widely used codes is the Reed-Solomon codes (and their variants), which were invented in the 1960s. For 40 years it was assumed that the codes could correct only a certain number of errors. By creating a new decoding algorithm, Sudan demonstrated that the Reed-Solomon codes could correct many more errors than previously thought possible.

Madhu Sudan was born on 12 September 1966, in Madras (now Chennai), India. He received his B. Tech. degree in computer science from the Indian Institute of Technology in New Delhi (1987) and his Ph.D. in computer science at the University of California at Berkeley (1992). He was a research staff member at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York (1992-1997). He is currently an associate professor in the Department of Electrical Engineering and Computer Science at the Massachusetts Institute of Technology.

Here are two articles about the work of Sudan: "The easy way to check hard maths," by Arturo Sangalli. New Scientist, 8 May 1993, pages 24-28; and "Coding theory meets theoretical computer science," by Sara Robinson. SIAM News, 34(10):216--217, December 2001 (also available online).