The mathematician Jacques Tits is dead

The immense work of the mathematician Jacques Tits, who died on December 5, in Paris, at the age of 91, profoundly transformed geometry in the 20th century.e century. Born August 12, 1930 in Uccle (Belgium), this child prodigy supported a doctorate in Brussels at the age of 20. After a stint in Germany, he spent most of his career at the Collège de France, where he held the Group Theory chair from 1973 to 1999. Among the many prizes he has received, we can mention the Wolf Prize. , in 1993, and the Abel Prize, in 2008.

The concept of groups is central in contemporary mathematics. Didn’t Henri Poincaré assert that “Mathematics is only a story of groups” ? Here is how Jacques Tits described his research theme in the introduction to the notice presenting his work at the Academy of Sciences: “Group theory can be roughly defined as a theory of symmetry, indistinguishability and homogeneity; the link between these notions is clear: an object has a certain symmetry if different viewing angles give indistinguishable images, a medium is homogeneous if its points are indistinguishable. The idea already appears in Greek mathematics where figures with a high degree of symmetry play an essential role. “

He geometrized algebra

Tits has indeed devoted his scientific life to a long reflection around symmetries in a very general sense. Groups appeared in science at the beginning of the XIXe century thanks to the imagination of Evariste Galois. These were purely algebraic ideas: we manipulated equations and looked for symmetries. Towards the end of the century, Felix Klein published his “Erlangen Program”, which asserted that the study of geometry came back to that of groups. Geometry thus found itself subservient to algebra. Jacques Tits works the other way around: he geometrized algebra.

To carry out his program, he invented what we now call “Tits buildings”, which are geometric objects that embody algebraic groups. It must be said that mathematicians often use words which have very little connection with the meaning given to them in everyday language, which often contributes to the fact that they are not understood. These Tits buildings have apartments, bedrooms, and walls, but the analogy ends there, as a bedroom can be located in two different apartments at the same time.

Read also (archive from 2009): From Poincaré to Gromov, a French tradition

To tell the truth, the terminology initially proposed by Tits was in very bad taste: there were cemeteries, ossuaries and skeletons! And yet its buildings are concrete, made up of segments, triangles or tetrahedra assembled together, like Plato’s polyhedra. At a conference in his honor in 2000, he explained that he preferred mathematics “Palpable”, which might surprise a neophyte who would risk reading one of his articles. By caricaturing to the extreme, we can indeed say that algebra is the domain of abstraction while geometry deals with more manipulable objects. Geometers and algebraists have very different approaches to mathematical activity. Jacques Tits was above all a surveyor. He who was always joking and in a good mood had given me a black look one day when I had dared to imply that he was also an algebraist.

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