Jacques Tits

Jaques Tits, 2008

Jacques Leon Tits (born August 12, 1930 in Uccle / Ukkel ; † December 5, 2021 ) was a Belgian-French mathematician who mainly worked in the field of group theory and algebraic geometry . He was an honorary professor at the Collège de France in Paris .

Life

Tits was born in Belgium and went to school in Uccle / Ukkel near Brussels. He studied at the French-speaking Free University in Brussels ( Université Libre de Bruxelles ), where he received his doctorate under Paul Libois in 1950 (Généralisations des groupes projectifs basées sur la notion de transitivité) . From 1956 to 1962 he was an assistant and from 1962 to 1964 professor at the Université Libre de Bruxelles, then until 1974 at the University of Bonn before becoming a professor at the Collège de France (retired 2000). After his retirement he was the first Vallée-Poussin visiting professor at the Université catholique de Louvain in Louvain-la-Neuve .

Tits married the historian Marie-Jeanne Dieuaide in 1956. He was a French citizen.

1980 to 1999 he was editor of the Publications Mathematiques de l ' IHES .

Works

In the 1960s Tits worked a lot with Armand Borel on algebraic groups . He is the inventor of the theory of buildings , combinatorial structures on which groups operate, which Tits originally investigated in order to investigate generalizations of simple Lie groups over arbitrary bodies . Buildings can not only be defined using real and complex bodies, but also using p-adic numbers or finite bodies and thus have applications in algebraic geometry and number theory . In developing the theory of buildings, he worked with François Bruhat ( Bruhat Tits Building ). Tits classified irreducible spherical buildings with a rank greater than or equal to 3. Connected to the theory of buildings is the theory of the pairs of a group, where (Borel subgroup) and are subgroups of that produce. In addition, the intersection of , (the Cartan subgroup ) is a normal subgroup in , and the factor group (the Weyl group ) is generated by elements of order 2 (ie “reflections”). The whole thing is generalized from the theory of Lie algebras and allows simplified and more generally valid proofs. With -pairs, buildings can also be constructed that are neither affine nor spherical, but result from infinite-dimensional Lie groups ( Kac-Moody algebras ). ${\displaystyle (B,N)}$ ${\displaystyle B}$ ${\displaystyle N}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle B}$ ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle W=N/H}$ ${\displaystyle (B,N)}$

Tits alternative : Every finitely generated subgroup of the linear group has either a resolvable subgroup of finite index or a free subgroup of rank 2.

Tits 1967

He also worked on finite simple groups , for example on the " monster ". A finite simple group, the Tits group , is named after him. The Tits group is the commutator group of the twisted Chevalley group of the type over the finite field GF (2) . ${\displaystyle {}^{2}F_{4}}$

Tits classified the polar spaces of finite rank greater than or equal to three and, more generally, the so-called spherical buildings of rank greater than or equal to three. Further work is devoted to geometries defined by groups, for example Moufang octagons, projective planes with the Suzuki groups as automorphisms, generalized -gons, automorphism groups of trees , square shapes , Buekenhout-Tits geometries and Clifford algebras . ${\displaystyle n}$

Observations by Tits in an essay from 1956 are regarded as the beginning of dealing with the body with an element . He considered the cardinality of an algebraic group of the Lie type over the finite field of elements in the limit value and found that this is given by the cardinality of the Weyl group of (except for a prefactor that depends on the rank, which disappears at the limit ) . He then suggested looking at the Weyl group as a group of points from above the body with an element. ${\displaystyle F_{1}}$ ${\displaystyle G}$ ${\displaystyle F_{q}}$ ${\displaystyle q}$ ${\displaystyle q\to 1}$ ${\displaystyle G}$ ${\displaystyle G}$

honors and awards

In 1976 he received the Grand Prize of the French Academy of Sciences , of which he has been a corresponding member since 1977 and of which he has been a member since 1979. In 1962 in Stockholm (Groupes simples et geometries associees) and 1974 in Vancouver (On Buildings and their Application) he gave a plenary lecture at the ICM . In 1970 he was invited speaker at the ICM in Nice (Homomorphismes et automorphismes "abstraits" de groupes algébriques et arithmétiques) . He received the Wolf Prize in 1993 and from 1995 was a member of the Pour le Mérite order for science and the arts . In 1996 he received the Georg Cantor Medal of the German Mathematicians Association and in March 2008, together with John Griggs Thompson, the Abel Prize for fundamental contributions to algebra.

He was a member of the German Academy of Sciences Leopoldina (since 1977), the Academia Europaea , the Belgian and Dutch Academy of Sciences , the Norwegian Academy of Sciences , the American Academy of Arts and Sciences (since 1992), the National Academy of Sciences and of the London Mathematical Society (honorary member). He was a Knight of the Legion of Honor (1995) and an honorary doctorate from the Universities of Utrecht, Ghent, Bonn and Leuven. In 2009 he received the Great Cross of Merit of the Federal Republic of Germany .

Fonts

• Read groups and algebras. Springer Hochschultext 1983, ISBN 978-3-540-12547-1 , doi: 10.1007 / 978-3-642-69189-8 .
• Tables for the simple Lie groups and their representations. Springer 1967.
• Lectures on algebraic groups. Yale, 1967.
• Formes quadratiques, groupes orthogonaux et algèbres de Clifford. In: Inventiones Mathematicae , 5 (1968), pp. 19-41.
• with Armand Borel : Groupes réductifs. In: Publications Mathématiques de l'IHÉS , No. 27, 1965, pp. 55-150.
• Free subgroups in linear groups. In: Journal of Algebra, 20 (1972), pp. 250-270.
• with A. Borel: Homomorphismes “abstraits” de groupes algébriques simples. In: Annals of Mathematics , (2) 97 (1973), pp. 499-571.
• Buildings of spherical type and finite pairs ${\displaystyle (B,N)}$ (= Lecture Notes in Mathematics ; 386). Springer-Verlag, Berlin et al., 1974, ISSN  0075-8434 , ISBN 978-3-540-06757-3 .
• with François Bruhat : Groupes reductifs sur un corps local. In: Publications Mathématiques de l'IHÉS, No. 41 (1972), pp. 5-251. II: Publications Mathématiques de l'IHÉS, No. 60 (1984), pp. 197-376. III: J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987) No. 3, pp. 671-698.
• A local approach to buildings. In: Chandler Davis, Branko Grünbaum, FA Sherk (Ed.): The geometric vein: the Coxeter festschrift. Springer-Verlag, Berlin et al., 1981, ISBN 0-387-90587-1 .
• On R. Griess' “Friendly giant”. In: Inventiones Mathematicae 1984 (on the group also Seminar Bourbaki Expose 620, 1983/4), online .
• Uniqueness and presentation of Kac-Moody groups over fields. In: Journal of Algebra, 105/2 (1987), pp. 542-573.
• Symmetry. In: Peter Hilton, Friedrich Hirzebruch, Reinhold Remmert (eds.): Miscellanea Mathematica. Springer, Berlin 1991. Reprint: 2011, ISBN 978-3-642-76711-1 .
• Twin buildings and groups of Kac-Moody type. Groups, combinatorics & geometry (Durham, 1990), pp. 249-286. London Math. Soc. Lecture Note Ser., 165, Cambridge Univ. Press, Cambridge, 1992.
• with R. Weiss: Moufang polygons. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2002, ISBN 3-540-43714-2 .
• Collected Works , 4 volumes. Edited by Francis Buekenhout, Bernhard Matthias Mühlherr, Jean-Pierre Tignol, Hendrik Van Maldeghem. European Mathematical Society 2013.

literature

• Article on the occasion of the awarding of the Cantor Medal of the DMV in the annual report of the DMV 2001

Web links

• John J. O'Connor, Edmund F. Robertson :  Jacques Tits. In: MacTutor History of Mathematics archive .
• Homepage Collège de France
• Jacques Tits receives the Federal Cross of Merit
• Literature by and about Jacques Tits in the catalog of the German National Library
• Literature by and about Jacques Tits in the SUDOC catalog (Association of French University Libraries)

Individual evidence

1. ↑ Nous apprenons avec tristesse le décès de Jacques Tits le 5 décembre 2021. In: smf.emath.fr. December 5, 2021, accessed December 6, 2021 (French). 
2. ↑ Jacques Tits in the Mathematics Genealogy Project (English)
3. ↑ Historique de la Chaire ( Memento from September 23, 2009 in the Internet Archive )
4. ↑ Mark Ronan: Buildings. Accessed January 4, 2021 . 
5. ↑ Tits, Sur les analogues algèbriques des groups semi-simples complexes, Colloque de'l algèbre supérieure, Brussels 1956, Gauthier Villars 1957, pp. 261–289
6. ^ Francis Buekenhout, A Biography of Jacques Tits , in: Helge Holden, Ragne Piene (Ed.), The Abel Prize 2008-2012 , Berlin 2014, pp. 35–53, ISBN 978-3-642-39448-5 , doi : 10.1007 / 978-3-642-39449-2_3
7. ↑ Member entry by Prof. Dr. Jacques Tits at the German Academy of Sciences Leopoldina , accessed on July 22, 2016.