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Home , Abel Prize, Jacques Tits, Lennart Carleson, Michael Atiyah, Richard Borcherds

Thompson and Tits Receive 2008

On March 27, 2008, the Norwegian Academy of Science and Letters announced it would award the Abel Prize for 2008 to John Griggs Thompson, , and , Collège de France. Thompson and Tits receive the Abel Prize “for their profound achievements in and in particular for shaping modern theory”.

Citation of the Abel Prize Committee Modern algebra grew out of two ancient traditions in , the art of solving equations, and the use of as for example in the pat- John Thompson Jacques Tits terns of the tiles of the Alhambra. The two came together in late eighteenth century, when it was belong to certain standard families, except for first conceived that the key to understanding even 26 sporadic groups. Thompson and his students the simplest equations lies in the of played a major role in understanding the fascinat- their solutions. This vision was brilliantly realized ing properties of these sporadic groups, including by two young , the largest, the so-called Monster. and Evariste Galois, in the early nineteenth century. Tits created a new and highly influential vision Eventually it led to the notion of a group, the most of groups as geometric objects. He introduced powerful way to capture the idea of symmetry. what is now known as a Tits , which en- In the twentieth century, the group theoretical codes in geometric terms the algebraic structure approach was a crucial ingredient in the develop- of linear groups. The theory of buildings is a cen- ment of modern physics, from the understanding tral unifying principle with an amazing range of of crystalline symmetries to the formulation of applications, for example to the classification of models for fundamental particles and forces. algebraic and Lie groups as well as finite simple In mathematics, the idea of a group proved groups, to Kac-Moody groups (used by theoreti- enormously fertile. Groups have striking proper- cal physicists), to combinatorial (used ties that unite many phenomena in different areas. The most important groups are finite groups, aris- in computer science), and to the study of rigidity ing for example in the study of permutations, and phenomena in negatively curved spaces. Tits’s linear groups, which are made up of symmetries geometric approach was essential in the study and that preserve an underlying geometry. The work realization of the sporadic groups, including the of the two laureates has been complementary: Monster. He also established the celebrated “Tits John Thompson concentrated on finite groups, alternative”: every finitely generated while Jacques Tits worked predominantly with is either virtually solvable or contains a copy of linear groups. the on two generators. This result has Thompson revolutionized the theory of finite inspired numerous variations and applications. groups by proving extraordinarily deep theorems The achievements of John Thompson and of that laid the foundation for the complete classifi- Jacques Tits are of extraordinary depth and influ- cation of finite simple groups, one of the greatest ence. They complement each other and together achievements of twentieth century mathematics. form the backbone of modern . Simple groups are atoms from which all finite Biographical Sketch: John G. Thompson groups are built. In a major breakthrough, Feit and Thompson proved that every non-elementary John Griggs Thompson was born in Ottawa, simple group has an even number of elements. Kansas, on October 13, 1932, and now holds a Later Thompson extended this result to establish position as Graduate Research at the a classification of an important kind of finite Department of Mathematics, University of Florida simple group called an N​-group. At this , the in Gainesville. classification project came within reach and was Thompson received his bachelor’s degree from carried to completion by others. Its almost incred- in 1955 and his Ph.D. from the Uni- ible conclusion is that all finite simple groups versity of Chicago in 1959. His advisor was

June/July 2008 Notices of the AMS 707 tutoring students four years older to contribute to More on Buildings and the Monster the household expenses. He passed the entrance Two installments of the Notices “What is…?” column relate to exam at the Free University of Brussels at the age the work of this year’s Abel Prizewinners: “What is the Mon- of fourteen and received his in 1950 at ster?” by (October 2002), and “What is a twenty years of age. Building?” by Kenneth S. Brown (November 2002). Find these Tits was promoted to professor at the Free Uni- articles online at http://www.ams.org/notices. versity of Brussels in 1962 and remained in this posi- tion for two years before accepting a professorship Saunders Mac Lane, one of the founding fathers of at the in 1964. In 1973 he moved category theory. to Paris, taking up a position as Chair of Group Thompson was an assistant professor at Har- Theory in the Collège de France. Shortly after, in vard University until he was appointed as a profes- 1974, he became a naturalized French citizen. Tits sor at the University of Chicago in 1962. In 1970 held this chair until he retired in 2000. he moved to Britain to take up the Rouse Ball Jacques Tits has been a member of the French Professorship of Mathematics at the University of Académie des Sciences since 1974. In 1992 he was Cambridge. After 23 years in Cambridge he moved elected a foreign member of the U.S. National Acad- back to the United States to his present position emy of Sciences and the American Academy of Arts at the University of Florida. He is currently Profes- and Sciences. In addition he holds memberships sor Emeritus of Mathematics at the University of in science academies in Holland and Belgium. He Cambridge. has been awarded honorary from the John Griggs Thompson has been awarded hon- Universities of Utrecht, Ghent, Bonn, and Leuven. orary doctorates at the Universities of Illinois, Yale, Tits has received many awards, such as the , and Ohio State. He was elected a member Wolf Prize, the Cantor Medal, the Grand Prix des of the U.S. of Sciences in 1967 Sciences Mathématiques et Physiques, and the and became a of the American Academy Wettrems Prize. He was appointed Chevalier de la of Arts and Sciences in 1998. He is a foreign Légion d’Honneur in 1995 and Officier de l’Ordre member of the Royal Society of and of National du Mérite in 2001. the Accademia Nazionale dei Lincei. Thompson In addition to his mathematical research Tits has received many international awards for his has played a major role in international mathemati- outstanding contribution to mathematics. Among cal life. He was editor-in-chief for mathematical them are the , the Senior Berwick publications at the Institut des Hautes Études Prize, the , the Wolf Prize, and the Scientifiques from 1980 to 1999. He served on the Poincaré Medal. In 1965 he and Walter Feit received committee awarding the Fields Medal in 1978 and the AMS Cole Prize in Algebra for their paper “Solv- again in 1994. He has also served since 1985 on ability of groups of odd order”, Pacific Journal of the committee awarding the Balzan Prize. Mathematics, volume 13 (1963), pp. 775–1029. In The publications of Jacques Tits contain a 2000 he was awarded the National Medal of Science remarkable number of fundamental and path- by U.S. President Bill Clinton. breaking mathematical ideas, making him one of The late Walter Feit, with whom Thompson the most influential and original mathematicians proved one of the key theorems of group theory, of our time. said of him: “He is a who studies About the Abel Prize important problems and does not let difficulties The Niels Henrik Abel Memorial Fund was es- discourage him. He frequently overcomes such dif- tablished in 2002, to award the Abel Prize for ficulties by introducing new ideas which then have outstanding scientific work in the of math- an enormous impact on future developments.” ematics. The prize amount is 6 million NOK (ap- This impact is all the greater for the generosity proximately US$1.2 million), and it was awarded with which he shares the ideas with his many re- for the first time in 2003. The prize is awarded by search students and other mathematicians. the Norwegian Academy of Science and Letters, and Biographical Sketch: Jacques Tits its choice of Abel Laureate is based on a recom- Jacques Tits was born in Uccle, in the southern mendation by the Abel Prize Committee consisting outskirts of Brussels, Belgium, on August 12, 1930. of five internationally recognized mathematicians. He retired from his professorship at the Collège The 2008 prize will be presented by His Majesty de France in Paris in 2000 and has since then been King Harald at the Abel Prize Award Ceremony in professor emeritus. Oslo, May 20, 2008. His father a mathematician, Jacques had mathe- Previous recipients of the Abel Prize are Jean- matical talent that showed early. At the age of three Pierre Serre (2003), and I. M. Singer he was able to do all the operations of arithmetic. (2004), (2005), (2006), He skipped several years at school. His father died and S. R. S. Varadhan (2007). when Jacques was only thirteen years old. Since the —From Norwegian Academy of Science and family had very little to live on, Jacques started Letters news releases

708 Notices of the AMS Volume 55, Number 6