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Pierre Deligne was born on 3 October 1944 as a hobby for his own personal enjoyment. in Etterbeek, Brussels, . He is Profes- There, as a student of , Deligne sor Emeritus in the School of at was pleased to discover that, as he says, the Institute for Advanced Study in Princeton, “one could earn one’s living by playing, i.e. by New Jersey, USA. Deligne came to Prince- doing research in mathematics.” ton in 1984 from Institut des Hautes Études After a year at École Normal Supériure in Scientifiques (IHÉS) at Bures-sur-Yvette near as auditeur libre, Deligne was concur- Paris, France, where he was appointed its rently a junior scientist at the Belgian National youngest ever permanent member in 1970. Fund for Scientific Research and a guest at When Deligne was around 12 years of the Institut des Hautes Études Scientifiques age, he started to read his brother’s university (IHÉS). Deligne was a visiting member at math books and to demand explanations. IHÉS from 1968-70, at which time he was His interest prompted a high-school math appointed a permanent member. teacher, J. Nijs, to lend him several volumes Concurrently, he was a Member (1972– of “Elements of Mathematics” by Nicolas 73, 1977) and Visitor (1981) in the School of Bourbaki, the pseudonymous grey eminence Mathematics at the Institute for Advanced that called for a renovation of French mathe- Study. He was appointed to a faculty position matics. This was not the kind of reading mat- there in 1984. ter that one would normally dream of offering Pierre Deligne is a research mathemati- a 14-year old, but for Deligne it became a life cian who has excelled in making connections changing experience. From then on he never between various fields of mathematics. His looked back. research has led to several important discov- Although his father wanted him to be- eries. One of his most famous contributions come an engineer and to pursue a career was his proof of the Weil in 1973. that would afford him a good living, Deligne This earned him both the (1978) knew early on that he should do what he and the (1988), the latter joint- loved, and what he loved was mathematics. ly with Alexandre Grothendieck. Deligne was He studied mathematics at the Université awarded the Balzan Prize in 2004 and Wolf Libre de Bruxelles (University of Brussels) and Prize in 2008. received his Licence en mathématiques, the When Deligne was awarded the Fields equivalent of a B.A., in 1966 and his Ph.D., Medal, and , both Doctorat en mathématiques, in 1968. In at the Harvard Mathematics Department, 1972, Deligne received the doctorat d’État wrote in Science magazine that “There are ès Sciences Mathématiques from Université few mathematical subjects that Deligne’s Paris-Sud 11. questions and comments do not clarify, for Deligne went to the University of Brus- he combines powerful technique, broad sels with the ambition of becoming a high- knowledge, daring imagination, and unfailing school teacher, and of pursuing mathematics instinct for the key idea.”

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photo: Cliff Moore

Abel Laureate 2013

Pierre Deligne Institute for Advanced Study in Princeton, New Jersey, USA

“for seminal contributions to and for their transformative impact on number theory, , and related fields”

For full citation see www.abelprize.no Pierre Deligne and the

Arne B. Sletsjøe Deligne’s best known achievement is his new solutions to the equation x2-y2=3, spectacular solution of the last and deep- e.g. x=0 and y=α since 02-α2=-2=3 when est of the Weil conjectures, namely the counting modulo 5. Another solution is analogue of the for given by x=α and y=2. All together we algebraic varieties over a finite . André find 24 different solutions in the extended Weil wrote in 1949, in the paper Numbers number system. The two numbers 4 and of solutions of equations in finite fields: 24 decides the two first terms of the zeta “... and other examples which we cannot function in the example. discuss here, seem to lend some support The Weil conjectures are formulated to the following conjectural statements, in four statements. Weil proved himself the which are known to be true for curves, conjectures in the curve case. For more but which I have not so far been able to general equations, three of the four state- prove for varieties of higher dimension.” ments were proved by other mathemati- The statements Weil was not able to cians in the following 10-15 years after the prove have been named the Weil conjec- publishing of Weil´s paper in 1949. The last tures. The issue of the Weil conjectures statement, the most difficult, analogous to is so-called zeta functions. Zeta func- the Riemann hypothesis, was proved by tions are mathematical constructions that Pierre Deligne in 1974. keep track of the number of solutions of Soon after the announcement of the an equation, in different number systems. conjectures it was clear that they would be When Weil says that the conjectural state- proved to be true if one could find a certain ments are known to be true for curves, he type of cohomology, called Weil cohomol- means that they are true for equations in ogy. Cohomology are mathematical tools two unknown. Varieties in higher dimen- that were developed in 1920- and 30’s to sions, as referred to, correspond to equa- understand and systematize knowledge tions in three or more unknowns. about geometric shapes and structures. The equation x2-y2=3 describes a The more complicated the structure, the plane curve, and in the inset Counting more cohomology. Weil had no sugges- modulo 5 we have showed that the equa- tions on how to define Weil cohomology, tion has 4 solutions in the number system but he knew what qualities cohomology {0,1,2,3,4} when counting modulo 5. should have to provide a proof of the Weil We notice that none of the numbers conjectures. 0,1,2,3,4 has square equal to 2. We there- At the end of the 1940s nobody knew fore introduce a new number α, the square any cohomology which could solve the root of 2. This number is not an element of conjectural problem and thus unify the the original set 0,1,2,3,4 and is determined geometric aspect, related to the solution of by the equation α2=2. Extending the num- equations and the arithmetic aspect, rep- ber system to include α, gives us many resented by the finite fields (number sys-

4 tems). The solution came in 1960. At that fortunately he had a young student, Pierre time introduced Deligne, who succeeded in this task. the concept of étale cohomology and pro- By a complicated argument, where he posed that it should play the role of the based his arguments on several previous mysterious, unknown, but essential Weil achievements made by​​ other mathemati- cohomology. The problem however, was cians, Deligne was able to prove the Weil to prove that the étale cohomology­ satisfy conjectures in full generality. The result the requirements to be a Weil cohomology. provoked attention and brought Deligne Grothendieck was not able to do so, but into the mathematical elite.

The first one leads

In 2006 Pierre Deligne was ennobled As three hens head for the fields, by the Belgian king as viscount. In The first one leads, connection with his viscount honour, The second follows the first, Deligne designed a coat of arms inspired The third one is last. by the song “As three hens head for the As three hens head for the fields, fields”. The song is a tautology, Deligne The first one leads. explained, “and one can view mathe­ matics as being also (long) chains of tautologies.”

Counting modulo 5

Counting modulo 5 means that instead of (22 and 32) and two of square 1 (12 and 42), counting 0,1,2,3,4,5,6,7,8,..., we count thus we get all together four solutions, x=2 0,1,2,3,4,0,1,2,3,..., i.e. we start again at and y=1, x=2 and y=4, x=3 and y=1, and 0 every time we reach 5. The computation x=3 and y=4. 4+2 means counting 2 steps further from 4. An example of modulo-counting is Counting modulo 5, doesn’t bring us to 6, time, where we count modulo 12. If we but rather to 1, i.e.. 4+2=1. The compu- leave home at 10 o’clock and stay out for tation 3·4 modulo 5, means counting to 4 four hours, then we return at 2 o’clock. three times, i.e. 1,2,3,4,0,1,2,3,4,0,1,2, which gives 3·4=2. The number system 0 1 2 3 4 {0,1,2,3,4} with these computation rules is 0 0 0 0 0 0 called a finite field of 5 elements. Our aim is to find the solutions of the 1 0 1 2 3 4 equation x2-y2=3 within this number sys- 2 0 2 4 1 3 2 tem. Computing all the squares, 0 =0, 3 0 3 1 4 2 12=1, 22=4, 32=4, and 42=1, we see that the only possibility to achieve a difference 4 0 4 3 2 1 3 between two squares is when x2=4 and y2=1. There are two numbers of square 4 Multiplication table, modulo 5

5 About the

The Abel Prize is an international award for and stimulating the interest of children and outstanding scientific work in the field of young people in mathematics. The prize mathematics, including mathematical as- amounts to 6 million NOK (about 800,000 pects of computer science, mathematical Euro or about 1 million USD) and was first physics, probability, numerical analysis, sci- awarded in 2003. entific computing, statistics, and also appli- The Norwegian Academy of Science cations of mathematics in the sciences. The and Letters appoints the Abel Board which Norwegian Academy of Science and Letters is responsible for the activities related to the awards the Abel Prize based upon recom- Abel Prize, and which supports The Abel mendations from the Abel Committee. Symposia, The International Mathematical The Prize is named after the exceptional Union’s Commission for Developing Coun- Norwegian Niels Henrik Abel, tries, and The born in 1802. Upon his early death at the Memorial Prize for excellence in teaching age of 26, he left a large body of work, in- mathematics in . In addition, the cluding the proof of the impossibility of solving Abel Board funds national mathematical the general quintic equation by radicals. contests and various projects and activities The Niels Henrik Abel Memorial Fund in order to stimulate interest in mathematics was established by the Norwegian Govern- among children and youth. The Norwegian ment on January 1, 2002. According to the Academy of Science and Letters co-organ- statutes, the objective is both to award the izes the annual Abel Conference at the In- annual Abel Prize, and to contribute towards stitute for Mathematics and its Applications raising the status of mathematics in society in Minnesota.

Abel celebration in . meets enthusiastic school children in Bergen.

For more information see www.abelprize.no

6 Abel Prize Laureates

2013 2012 Pierre Deligne Endre Szemerédi “for seminal contributions to algebraic “for his fundamental contributions to dis­ geometry and for their transformative crete mathematics and theoretical computer impact on number theory, representation science, and in recognition of the profound theory, and related fields.” and lasting impact of these contributions on additive number theory and ergodic theory.”

2011 2010 2009 John Torrence Tate Mikhail Leonidovich Gromov “for pioneering discoveries in , “for his vast and lasting impact on the “for his revolutionary contributions geometry and algebra.” theory of numbers.” to geometry.”

2008 2007 2006 John Griggs Thompson Srinivasa S. R. Varadhan and Jacques Tits “for his fundamental contributions to “for his profound and seminal contributions “for their profound achievements in probability theory and in particular for to harmonic analysis and the theory of algebra and in particular for shaping creating a unified theory of large smooth dynamical systems.” modern theory.” deviations.”

2005 2004 2003 Peter D. Lax Sir Michael Francis Atiyah Jean-Pierre Serre “for his groundbreaking contributions to the and Isadore M. Singer “for playing a key role in shaping the theory and application of partial differential “for their discovery and proof of the modern form of many parts of mathemat­ equations and to the computation of their index theorem, bringing together topology, ics, including topology, algebraic geometry solutions.” geometry and analysis, and their outstand­ and number theory.” ing role in new7 bridges between mathe­matics and theoretical physics.” Programme Abel Week 2013

May 20 Wreath-laying ceremony at the Abel Monument The Palace Park May 21 Abel Prize Award Ceremony University Aula, University of Oslo May 21 Reception and interview with the Abel Laureate Theatersalen, Hotel Continental May 22 Holmboe Prize Award Ceremony Oslo May 22 The Abel Lectures Georg Sverdrups Hus, Aud. 1, University of Oslo May 22 The Abel Party The Norwegian Academy of Science and Letters

Register online at www.abelprize.no from mid-April, or contact [email protected]

Press contact: For other information: Anne-Marie Astad Trine Gerlyng [email protected] [email protected] The Norwegian Academy +47 22 12 10 92 +47 22 12 10 93 of Science and Letters +47 415 67 406