I was born in Lisbon on February 26th 1948. My father had been assigned to teach in the French high school in Lisbon. Lisbon was a pleasant city, preserved from the war. In Lycée Charles-Lepierre, I received an extraordinary education, at the contact of several cultures, with mathematics, physics, biology and humanities playing an important role. My father did not leave me much choice on the field on which I should concentrate: mathematics was as important as literature, Latin, ancient Greek and history. One uncle who was an engineer, and an aunt and her husband, both theoretical physicists, opened my eyes at an early age to what physics was accomplishing at the time.
At fifteen, I moved to Greece with my parents, stayed one year there, and spent my last year of high school at Lycée Louis-le-Grand, as well as the next two years to prepare for Grandes Écoles. In the fall of 1967, I was accepted at École Polytechnique. There came a real shock: the mathematics introductory course was given by Laurent Schwartz, a Fields medalist and the inventor of distributions. He gave a deep, witty, extraordinary dense course. Twice a week, he organized an evening seminar that started at 8 pm and lasted for two hours, whose scope was an introduction to mathematics.
In May 1968, while universities were closed, École Polytechnique was still functioning. With no teaching duties, university professors came to teach us. In mathematics, three times the usual number of courses were being given. No wonder so many mathematicians came out of École Polytechnique under these circumstances.
Under J-L Lions and J Neveu, I completed my PhD thesis at Université Paris VI in 1973. It contained a mixture of probability and the calculus of variations. Since I had been ranked first in a competitive environment, I had many choices. As I was concerned by the atmosphere in universities at the time, I decided to discover the real world, while teaching at École Polytechnique, and still practicing mathematics at night. Ultimately, I realized that mathematics was infinitely more interesting to me.
In my work, I have been concerned with connections between probability, the calculus of variations, and geometry. Into that study, I imported ideas from classical mechanics, and stochastic differential equations. This way, I obtained Hamiltonian equations, unforeseen at the time, and which are still in use, in particular in mathematical economics. Links with geometry appeared progressively, first under the influence of Paul Malliavin, later because of the index theory of Dirac operators, and unforeseen connections with equivariant localization on loop spaces proposed by Michael Atiyah and Edward Witten. After working on local versions of the families index theorem for Dirac operators, in joint work with Henri Gillet, Christophe Soulé, and Gilles Lebeau, I turned to analytic questions motivated by Arakelov geometry, that led us to a detailed study of Quillen metrics. I discovered that probabilistic ideas combined with algebra could be relevant in questions not connected with probability. With collaborators that included Jeff Cheeger, I turned to refined version of index theory, that included eta invariants, real torsion, and their families version.
Later, I discovered an object whose potential is still not fully explored, the hypoelliptic Laplacian. Such an operator was known before in statistical physics as the Fokker-Planck operator. My own contribution was to show that it is part of a geometric deformation scheme, with unsuspected preserved quantities. With Gilles Lebeau, we showed that real torsion is one of those. Later, I showed that holomorphic torsion is also included. This interpolation process has two related aspects: an analytic aspect, connected with differential operators, and a probabilistic aspect connected with dynamics. Geometry and classical physics unify these two points of view. This construction creates nonclassical and even nonphysical links between classical objects, the origin of which lies in path integrals and index theory.
On symmetric spaces of non-compact type, more is true: no spectral information is lost in the deformation. This led to new geometric perspectives on classical objects in the harmonic analysis of real reductive groups: the orbital integrals. Using the hypoelliptic deformation in complex geometry also opened up new possibilities for getting rid of the restrictions of Kähler geometry.
In my mathematical life, I have been fortunate to work with many collaborators, who gave me considerable courage and the benefit of their friendship. I was also fortunate to have had many excellent students, who became extraordinary collaborators. I also owe my family so much kind support in difficult times