Jacques Tits
Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric. Life and career. Tits was born in Uccle to Léon Tits, a professor, and Lousia André. Jacques attended the Athénée of Uccle and the Free University of Brussels. His thesis advisor was Paul Libois, and Tits graduated with his doctorate in 1950 with the dissertation "Généralisation des groupes projectifs basés sur la notion de transitivité". His academic career includes professorships at the  Free University of Brussels (now split into the Université Libre de Bruxelles and the Vrije Universiteit Brussel) (1962–1964), the University of Bonn (1964–1974) and the Collège de France in Paris, until becoming emeritus in 2000. He changed his citizenship to French in 1974 in order to teach at the Collège de France, which at that point required French citizenship. Because Belgian nationality law did not allow dual nationality at the time, he renounced his Belgian citizenship. He has been a member of the French Academy of Sciences since 1979. Tits was an "honorary" member of the Nicolas Bourbaki group; as  such, he helped popularize H.S.M. Coxeter's work, introducing terms such as Coxeter number, Coxeter group, and Coxeter graph. Honors. Tits received the Wolf Prize in Mathematics in 1993, the Cantor Medal from the Deutsche Mathematiker-Vereinigung (German Mathematical Society) in 1996, and the German distinction "Pour le Mérite". In 2008 he was awarded the Abel Prize, along with John Griggs Thompson, "for their profound achievements in algebra and in particular for shaping modern group theory". He was a member of several Academies of Sciences. He was a member of the Norwegian Academy of Science and Letters. He became a  foreign member of the Royal Netherlands Academy of Arts and Sciences in 1988. Death. Tits died on 5 December 2021, at the age of 91. Contributions. He introduced the theory of "buildings" (sometimes known as "Tits buildings"), which are combinatorial structures on which groups act, particularly in algebraic group theory (including finite groups, and groups defined over the p-adic numbers). The related theory of (B, N) pairs is a basic tool in the theory of groups of Lie type. Of particular importance is his classification of all irreducible buildings of spherical type and rank at least three, which involved  classifying all polar spaces of rank at least three. The existence of these buildings initially depended on the existence of a group of Lie type in each case, but in joint work with Mark Ronan he constructed those of rank at least four independently, yielding the groups directly. In the rank-2 case spherical building are generalized n-gons, and in joint work with Richard Weiss he classified these when they admit a suitable group of symmetries (the so-called Moufang polygons). In collaboration with François Bruhat he developed the theory of affine buildings, and later he classified all irreducible buildings  of affine type and rank at least four. Another of his well-known theorems is the "Tits alternative": if "G" is a finitely generated subgroup of a linear group, then either "G" has a solvable subgroup of finite index or it has a free subgroup of rank 2. The Tits group and the Kantor–Koecher–Tits construction are named after him. He introduced the Kneser–Tits conjecture.