Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

and

incidence geometry In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''incide ...

. He introduced

Tits building In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. B ...

s, the

Tits alternative In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups. Statement The theorem, proven by Tits, is stated as follows. Consequences A linear group is not a ...

, the

Tits group In group theory, the Tits group 2''F''4(2)′, named for Jacques Tits (), is a finite simple group of order : 211 · 33 · 52 · 13 = 17,971,200. It is sometimes considered a 27th sporadic group ...

, and the

Tits metric In mathematics, the Tits metric is a metric defined on the ideal boundary of an Hadamard space (also called a complete CAT(0) space). It is named after Jacques Tits. Ideal boundary of an Hadamard space Let (''X'', ''d'') be an Hadamard space. T ...

Life and career

Tits was born in

Uccle Uccle () or Ukkel () is one of the 19 municipalities of the Brussels-Capital Region, Belgium. In common with all of Brussels' municipalities, it is legally bilingual (French–Dutch). It is generally considered an affluent area of the city an ...

to Léon Tits, a professor, and Lousia André. Jacques attended the Athénée of Uccle and the

Free University of Brussels University of Brussels may refer to several institutions in Brussels, Belgium: Current institutions * Université libre de Bruxelles (ULB), a French-speaking university established as a separate entity in 1970 *Vrije Universiteit Brussel (VUB), a D ...

. 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 The Vrije Universiteit Brussel (VUB) () is a Dutch and English-speaking research university located in Brussels, Belgium.The Vrije Universiteit Brussel is one of the five universities officially recognised by the Flemish government. listof all ...

) (1962–1964), the

University of Bonn The Rhenish Friedrich Wilhelm University of Bonn (german: Rheinische Friedrich-Wilhelms-Universität Bonn) is a public research university located in Bonn, North Rhine-Westphalia, Germany. It was founded in its present form as the ( en, Rhine U ...

(1964–1974) and the

Collège de France The Collège de France (), formerly known as the ''Collège Royal'' or as the ''Collège impérial'' founded in 1530 by François I, is a higher education and research establishment ('' grand établissement'') in France. It is located in Paris n ...

Paris Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. S ...

, until becoming

emeritus ''Emeritus'' (; female: ''emerita'') is an adjective used to designate a retired chair, professor, pastor, bishop, pope, director, president, prime minister, rabbi, emperor, or other person who has been "permitted to retain as an honorary title ...

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 Belgian nationality law provides for the conditions in which a person holds Belgian nationality and is based on a mixture of the principles of ''jus sanguinis'' and ''jus soli''. Belgian nationality is the status of being a citizen of Belgium. N ...

did not allow

dual nationality Multiple/dual citizenship (or multiple/dual nationality) is a legal status in which a person is concurrently regarded as a national or citizen of more than one country under the laws of those countries. Conceptually, citizenship is focused on ...

at the time, he renounced his Belgian citizenship. He has been a member of the

French Academy of Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV of France, Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific me ...

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 In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there ...

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 reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...

, and

Coxeter graph In the mathematical field of graph theory, the Coxeter graph is a 3-regular graph with 28 vertices and 42 edges. It is one of the 13 known cubic distance-regular graphs. It is named after Harold Scott MacDonald Coxeter. Properties The Coxeter ...

Siobhan Roberts Siobhan Roberts is a Canadians, Canadian science journalist, biographer, and history of mathematics, historian of mathematics. Education Roberts was born in Belleville, Ontario. She earned a degree in history at Queen's University at Kingston ...

"Donald Coxeter: The man who saved geometry"
''

Toronto Life ''Toronto Life'' is a monthly magazine about entertainment, politics and life in Toronto, Ontario, Canada. ''Toronto Life'' also publishes a number of annual special interest guides about the city, including ''Real Estate'', ''Stylebook'', ''Eatin ...

'', January 2003

Honors

Tits received the Wolf Prize in Mathematics in 1993, the

Cantor Medal The Cantor medal of the Deutsche Mathematiker-Vereinigung is named in honor of Georg Cantor, the first president of the society. It is awarded at most every second year during the yearly meetings of the society. The prize winners are mathematician ...

from the

Deutsche Mathematiker-Vereinigung The German Mathematical Society (german: Deutsche Mathematiker-Vereinigung, DMV) is the main professional society of German mathematicians and represents German mathematics within the European Mathematical Society (EMS) and the International Mathe ...

(German Mathematical Society) in 1996, and the German distinction "

Pour le Mérite The ' (; , ) is an order of merit (german: Verdienstorden) established in 1740 by Frederick the Great, King Frederick II of Prussia. The was awarded as both a military and civil honour and ranked, along with the Order of the Black Eagle, the Or ...

". In 2008 he was awarded the

Abel Prize The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Prizes. ...

, along with

John Griggs Thompson John Griggs Thompson (born October 13, 1932) is an American mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992, and the Abel Prize in 2008. ...

, "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 The Norwegian Academy of Science and Letters ( no, Det Norske Videnskaps-Akademi, DNVA) is a learned society based in Oslo, Norway. Its purpose is to support the advancement of science and scholarship in Norway. History The Royal Frederick Unive ...

. He became a foreign member of the

Royal Netherlands Academy of Arts and Sciences The Royal Netherlands Academy of Arts and Sciences ( nl, Koninklijke Nederlandse Akademie van Wetenschappen, abbreviated: KNAW) is an organization dedicated to the advancement of science and literature in the Netherlands. The academy is housed ...

in 1988.

Death

Tits died on 5 December 2021, at the age of 91 in the 13th arrondissement, Paris.

Contributions

He introduced the theory of ''buildings'' (sometimes known as '' Tits buildings''), which are combinatorial structures on which groups act, particularly in

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...

theory (including

finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

s, and groups defined over the

p-adic number In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensi ...

s). The related theory of (B, N) pairs is a basic tool in the theory of

groups of Lie type In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phras ...

. Of particular importance is his classification of all irreducible buildings of spherical type and rank at least three, which involved classifying all

polar space In mathematics, in the field of geometry, a polar space of rank ''n'' (), or ''projective index'' , consists of a set ''P'', conventionally called the set of points, together with certain subsets of ''P'', called ''subspaces'', that satisfy these ax ...

s of rank at least three. The existence of these buildings initially depended on the existence of a

group of Lie type In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phra ...

in each case, but in joint work with

Mark Ronan Mark Andrew Ronan (born 1947) is Emeritus Professor of Mathematics at the University of Illinois at Chicago and Honorary Professor of Mathematics at University College London. He has lived and taught in: Germany (at the University of Braunschwei ...

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 polygon In mathematics, Moufang polygons are a generalization by Jacques Tits of the Moufang planes studied by Ruth Moufang, and are irreducible buildings of rank two that admit the action of root groups. In a book on the topic, Tits and Richard Weiss c ...

s). In collaboration with

François Bruhat François Georges René Bruhat (; 8 April 1929 – 17 July 2007) 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 "

": if ''G'' is a finitely generated

subgroup In group theory, a branch of mathematics, 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 a

linear group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a f ...

, then either ''G'' has a solvable subgroup of

finite index In mathematics, specifically group theory, the index of a subgroup ''H'' in a group ''G'' is the number of left cosets of ''H'' in ''G'', or equivalently, the number of right cosets of ''H'' in ''G''. The index is denoted , G:H, or :H/math> or (G ...

or it has a free subgroup of rank 2. The

and the

Kantor–Koecher–Tits construction In algebra, the Kantor–Koecher–Tits construction is a method of constructing a Lie algebra from a Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axiom ...

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

Publications

* * * *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.

References

External links

* *
Biography at the Abel Prize site
(pdf)
List of publications
at the Université libre de Bruxelles {{DEFAULTSORT:Tits, Jacques 1930 births 2021 deaths People from Uccle 20th-century French mathematicians 21st-century French mathematicians Abel Prize laureates Belgian expatriates in France Belgian mathematicians Collège de France faculty Free University of Brussels (1834–1969) alumni Group theorists Members of the French Academy of Sciences Members of the Norwegian Academy of Science and Letters Members of the Royal Netherlands Academy of Arts and Sciences Foreign associates of the National Academy of Sciences Recipients of the Pour le Mérite (civil class) Nicolas Bourbaki Officers of the Ordre national du Mérite University of Bonn faculty Wolf Prize in Mathematics laureates Free University of Brussels (1834–1969) faculty