Jacques Tits

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 then.

Tits was an "honorary" member of the Nicolas Bourbaki group; as such, he helped popularize Harold Scott MacDonald Coxeter's work, introducing terms such as Coxeter number, Coxeter group, and Coxeter graph.[1]

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 Tits–Koecher construction are named after him. He introduced the Kneser–Tits conjecture.

• Tits, Jacques (1964). "Algebraic and abstract simple groups". Annals of Mathematics. Second Series. 80 (2): 313–329. doi:10.2307/1970394. ISSN 0003-486X. JSTOR 1970394. MR 0164968.
• Tits, Jacques (1974). Buildings of spherical type and finite BN-pairs. Lecture Notes in Mathematics, Vol. 386. 386. Berlin, New York: Springer-Verlag. doi:10.1007/978-3-540-38349-9. ISBN 978-3-540-06757-3. MR 0470099.[5]
• Tits, Jacques; Weiss, Richard M. (2002). Moufang polygons. Springer Monographs in Mathematics. Berlin, New York: Springer-Verlag. ISBN 978-3-540-43714-7. MR 1938841.
• J. Tits, Oeuvres - Collected Works, 4 vol., Europ. Math. Soc., 2013. J. Tits, Résumés des cours au Collège de France, S.M.F., Doc.Math. 12, 2013.

• Jacques Tits at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "Jacques Tits", MacTutor History of Mathematics archive, University of St Andrews.
• Biography at the Abel Prize site (pdf)
• List of publications at the Université libre de Bruxelles