Jacques Tits
Encyclopedia
Jacques Tits is a Belgian and French
mathematician
who works on group theory
and geometry and who introduced Tits buildings, the Tits alternative
, and the Tits group
.
in mathematics at the age of 20. 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 Belgium does not allow dual nationalities, 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
.
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 is a member of several Academies of Sciences.
He is a member of the Norwegian Academy of Science and Letters
.
theory (including finite group
s, and groups defined over the p-adic number
s). The related theory of (B, N) pair
s 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 space
s of rank at least three. 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.
French people
The French are a nation that share a common French culture and speak the French language as a mother tongue. Historically, the French population are descended from peoples of Celtic, Latin and Germanic origin, and are today a mixture of several ethnic groups...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who works on group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
and geometry and who introduced Tits buildings, the Tits alternative
Tits alternative
In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups. It states that every such group is either virtually solvable In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about...
, and the Tits group
Tits group
In mathematics, the Tits group 2F4′ is a finite simple group of order 17971200 = 211 · 33 · 52 · 13 found by ....
.
Career
Tits received his doctorateDoctorate
A doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder to teach in a specific field, A doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder...
in mathematics at the age of 20. His academic career includes professorships at the Free University of Brussels
Free University of Brussels
The Free University of Brussels was a university in Brussels, Belgium. In 1969, it split into the Université Libre de Bruxelles and the Dutch-speaking Vrije Universiteit Brussel....
(now split into the Université Libre de Bruxelles
Université Libre de Bruxelles
The Université libre de Bruxelles is a French-speaking university in Brussels, Belgium. It has 21,000 students, 29% of whom come from abroad, and an equally cosmopolitan staff.-Name:...
and the Vrije Universiteit Brussel
Vrije Universiteit Brussel
The Vrije Universiteit Brussel is a Flemish university located in Brussels, Belgium. It has two campuses referred to as Etterbeek and Jette.The university's name is sometimes abbreviated by "VUB" or translated to "Free University of Brussels"...
) (1962-1964), the University of Bonn
University of Bonn
The University of Bonn is a public research university located in Bonn, Germany. Founded in its present form in 1818, as the linear successor of earlier academic institutions, the University of Bonn is today one of the leading universities in Germany. The University of Bonn offers a large number...
(1964-1974) and the Collège de France
Collège de France
The Collège de France is a higher education and research establishment located in Paris, France, in the 5th arrondissement, or Latin Quarter, across the street from the historical campus of La Sorbonne at the intersection of Rue Saint-Jacques and Rue des Écoles...
in Paris, until becoming emeritus
Emeritus
Emeritus is a post-positive adjective that is used to designate a retired professor, bishop, or other professional or as a title. The female equivalent emerita is also sometimes used.-History:...
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 Belgium does not allow dual nationalities, he renounced his Belgian citizenship. He has been a member of the French Academy of Sciences
French Academy of Sciences
The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...
since then.
Tits was an "honorary" member of the Nicolas Bourbaki
Nicolas Bourbaki
Nicolas Bourbaki is the collective pseudonym under which a group of 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality...
group; as such, he helped popularize Harold Scott MacDonald Coxeter
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, was a British-born Canadian geometer. Coxeter is regarded as one of the great geometers of the 20th century. He was born in London but spent most of his life in Canada....
's work, introducing terms such as Coxeter number
Coxeter number
In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group, hence also of a root system or its Weyl group. It is named after H.S.M. Coxeter.-Definitions:...
, Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...
, and Coxeter graph
Coxeter graph
In the mathematical field of graph theory, the Coxeter graph is a 3-regular graph with 28 vertices and 42 edges. All the cubic distance-regular graphs are known. The Coxeter graph is one of the 13 such graphs....
.
Honors
Tits received the Wolf Prize in MathematicsWolf Prize in Mathematics
The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts...
in 1993, the Cantor Medal
Cantor medal
The Cantor medal of the Deutsche Mathematiker-Vereinigung is named in honor of Georg Cantor. It is awarded at most every second year during the yearly meetings of the society...
from the Deutsche Mathematiker-Vereinigung (German Mathematical Society) in 1996, and the German distinction "Pour le Mérite
Pour le Mérite
The Pour le Mérite, known informally as the Blue Max , was the Kingdom of Prussia's highest military order for German soldiers until the end of World War I....
". In 2008 he was awarded the Abel Prize
Abel Prize
The Abel Prize is an international prize presented annually by the King of Norway to one or more outstanding mathematicians. The prize is named after Norwegian mathematician Niels Henrik Abel . It has often been described as the "mathematician's Nobel prize" and is among the most prestigious...
, along with John Griggs Thompson, “for their profound achievements in algebra and in particular for shaping modern group theory.” He is a member of several Academies of Sciences.
He is a member of the Norwegian Academy of Science and Letters
Norwegian Academy of Science and Letters
The Norwegian Academy of Science and Letters is a learned society based in Oslo, Norway.-History:The University of Oslo was established in 1811. The idea of a learned society in Christiania surfaced for the first time in 1841. The city of Throndhjem had no university, but had a learned...
.
Contributions
He introduced the theory of buildings (sometimes known as Tits buildings), which are combinatorial structures on which groups act, particularly in algebraic groupAlgebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
theory (including finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
s, and groups defined over the p-adic number
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
s). The related theory of (B, N) pair
(B, N) pair
In mathematics, a pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs. Roughly speaking, it shows that all such groups are similar to the general linear group over a field...
s 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 space
Polar space
In mathematics, in the field of combinatorics, a polar space of rank n , or projective index n−1, consists of a set P, conventionally the set of points, together with certain subsets of P, called subspaces, that satisfy these axioms :* Every subspace, together with its own subspaces, is isomorphic...
s of rank at least three. 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
François Bruhat
François Georges René Bruhat was a French mathematician who worked on algebraic groups. The Bruhat order of a Weyl group, the Bruhat decomposition, and the Schwartz–Bruhat functions are named after him....
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
Tits alternative
In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups. It states that every such group is either virtually solvable In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about...
": if G is a finitely generated subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
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
Tits group
In mathematics, the Tits group 2F4′ is a finite simple group of order 17971200 = 211 · 33 · 52 · 13 found by ....
and the Tits–Koecher construction are named after him.