Jacques Tits, one of the most influential mathematicians of the second half of the 20th century, passed away on December 5. His works revolutionized the way of understanding groups, central objects in algebra, which encapsulate the idea of symmetry. This new point of view allowed establishing new bridges between geometry and algebra, giving rise to the beginning of the geometrization of group theory. Tits’ mathematical contributions were recognized in 2008 with the Abel Award, the highest award in mathematics along with the Fields Medal.
Tits was born in Uccle, Belgium in 1930. The son of a mathematician, he developed an unusual passion and ability for mathematics from a very young age. At age 14 he enrolled as a Mathematics student at the Free University of Brussels, where he also completed his doctoral studies at the age of 20. Your thesis supervisor, Paul Libois, was a recognized Marxist intellectual, politician and great defender of geometric intuition as a vehicle for mathematical learning.
After holding university professor positions in Brussels and Bonn, in 1974 he obtained the Chair of Group Theory at the prestigious Collège de France (Paris). During these years, he acquired French nationality and was elected a member of the French Academy of Sciences.
Tits’ research is framed within group theory. One group is the translation of the concept of symmetry into mathematical language. For example, a square has eight different symmetries: four rotations based on the center of the square and four reflections along lines that pass through the center of the square. This set of symmetries has three fundamental properties: there is an identity symmetry (the rotation of zero angles), which leaves the square as it was; all symmetry can be undone (it has an inverse); and, finally, the result of applying one symmetry and then another (multiplying them) is again a symmetry.
By abstracting these three properties, in mathematics a group is defined as a set whose elements can be multiplied together, where each element has an inverse and where there is an identity element. Some groups have a finite number of elements, like the group of symmetries of the square; on the contrary, other groups are infinite, like the one formed by the real numbers different from zero.
Until the end of the 19th century, groups were considered primarily as abstract algebraic objects that appeared in the context of solving polynomial equations. In 1872 a great paradigm shift took place: the German mathematician Felix Klein (1849 – 1925) announced his Erlangen Program, where he proposed, for the first time, using group theory to understand problems in geometry.
The vision of Tits, supported in turn on the shoulders of giants like HSM Coxeter (1907 – 2003), can be considered as an “investment” of the Erlangen Program, since it proposes using geometric tools to study the groups. In this sense, one of the great mathematical contributions of Tits, developed with the French mathematician François Bruhat (1929 – 2007), is the concept of Bruhat-Tits buildings associated with certain very general families of groups (so-called algebraic groups).
These buildings are geometric objects built from the algebraic structure of the group, and whose group of symmetries is precisely the original group. As so often happens in mathematics, the terminology is quite suggestive: a building is made up of apartments, in turn made up of multiple rooms, glued together according to a pattern that reflects the algebraic structure of the group in question.
The buildings of Bruhat-Tits have been decisive in several of the great milestones of modern mathematics, such as the study of the Coxeter groups (generalizations of groups of reflections) or the great theorems of Margulis, Prasad, etc., on the rigidity of certain subgroups (called lattices) in algebraic groups.
As is logical, since he is a mathematician of such influence, we leave the vast majority of Tits’s contributions in the pipeline, many of enormous depth, such as the paradoxical body with an element, the well-known Tits alternative on the structure of matrix groups, or his works on extended Coxeter groups, now known as Artin-Tits groups. In addition to the Abel Prize, Tits’ work was recognized with other important awards, such as the Wolf Prize (1993) and the National Order of the French Legion of Honor (2008). But above all, the legacy of Jacques Tits’s mathematical work lives on through its influence on the thinking of a large number of researchers around the world.
Yago Antolin He is a tenured professor at the Complutense University of Madrid and a member of the Institute of Mathematical Sciences (ICMAT).
Javier Aramayona He is a tenured scientist at the Higher Council for Scientific Research, a member of ICMAT and co-director of the Mathematical Culture Unit of ICMAT.
Coffee and theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which the researchers and members of the center describe the latest advances in this discipline, share meeting points between the mathematics and other social and cultural expressions and remember those who marked its development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”
Editing and coordination: Ágata A. Timón G Longoria (ICMAT).
You can follow MATTER in Facebook, Twitter and Instagram, or sign up here to receive our weekly newsletter.
#Jacques #Tits #life #devoted #edifice #mathematics