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. Born on August 12, 1930 in Uccle (Belgium), this child prodigy defended 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 are the Wolf Prize, in 1993, and the Abel Prize, in 2008.
The concept of groups is central to contemporary mathematics. Didn’t Henri Poincaré say 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 to the Academy of Sciences: “The theory of groups can be summarily defined as a theory of symmetry, indistinguishability and homogeneity; the link between these notions is clear: an object possesses a certain symmetry if different angles of view give indistinguishable images of it, a medium is homogeneous if
its points are indistinguishable. The idea already appeared in Greek mathematics, where figures with a high degree of symmetry played an essential role.
He geometrized algebra
Tits has in fact devoted his scientific life to a long reflection on symmetries in a very general sense. Groups appeared in science at the beginning of the 19th century thanks to the imagination of Evariste Galois. At that time, they were purely algebraic ideas: equations were manipulated and symmetries were sought. Towards the end of the century, Felix Klein published his “Erlangen program”, which asserted that the study of geometry was the same as the study of groups. Geometry was thus subordinated to algebra. Jacques Tits worked in the other direction: he geometrized algebra.
In order to realize 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 that have very little to do with the meaning given to them in everyday language, which often contributes to the fact that we do not understand them. These Tits buildings have apartments, rooms and walls, but the analogy ends there, as a room can be located in two different apartments at the same time.
To tell the truth, the terminology initially proposed by Tits was in very bad taste: there were cemeteries, ossuaries and skeletons! And yet his buildings are concrete, made of segments, triangles or tetrahedrons assembled together, in the manner of Plato’s polyhedra. At a conference in his honor in 2000, he explained that he preferred “palpable” mathematics, which might surprise a neophyte who dares to read one of his articles. To caricature in the extreme, one 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 geometer. He was always joking and good-humored, and he gave me a black look one day when I dared to suggest that he was also an algebraist.
Tits attached great importance to his role as editor of the Mathematical Publications of the Institut des hautes études scientifiques, which he held for twenty years. While proofreading an article of mine that was to appear in this journal, I noticed to my surprise that a letter “i” inside a word had been printed upside down, with the dot underneath rather than above. When I asked Tits about this, he told me with a broad smile that this magazine was still printed in the old fashioned way, on a Linotype, with lead type, and sometimes a typeface would turn over. He then showed me that by closing your eyes and stroking the paper, you could feel the mathematical content. The mathematics of Tits was really palpable.
Etienne Ghys (perpetual secretary of the Academy of Sciences, director of research (CNRS) at the ENS Lyon)