Bismut's early work was related to
stochastic differential equations, stochastic control, and
Malliavin calculus, to which he made fundamental contributions. Bismut received in 1973 his
Docteur d'État in Mathematics, from the Université Paris-VI, a thesis entitled
Analyse convexe et probabilités. In his thesis, Bismut established a stochastic version of Pontryagin's maximum principle in
control theory by introducing and studying the backward stochastic differential equations which have been the starting point of an intensive research in stochastic analysis and it stands now as a major tool in Mathematical Finance. Using the quasi-invariance of the Brownian measure, Bismut gave a new approach to the
Malliavin calculus and a probabilistic proof of Hörmander's theorem. He established his celebrated integration by parts for the Brownian motion on
manifolds. Since 1984, Bismut works on differential geometry. He found a
heat equation proof for the
Atiyah–Singer index theorem. And he established a local version of the Atiyah-Singer families index theorem for Dirac operators, by introducing the
Bismut superconnection which plays a central role in modern aspects of index theory. Bismut-Freed developed the theory of
Quillen metrics on the smooth determinant line bundle associated with a family of Dirac operators. Bismut-Gillet-Soulé gave a curvature theorem for the Quillen metric on the holomorphic determinant of a direct image by a holomorphic proper submersion. This and Bismut—Lebeau's embedding formula for analytic torsions play a crucial role in the proof of the arithmetic Riemann-Roch theorem in
Arakelov theory, in which analytic torsion is an essential analytic ingredient in the definition of the direct image. Bismut gave a natural construction of a Hodge theory whose corresponding Laplacian is a hypoelliptic operator acting on the total space of the
cotangent bundle of a
Riemannian manifold. This operator interpolates formally between the classical elliptic Laplacian on the base and the generator of the geodesic flow. One striking application is Bismut's explicit formulas for all
orbital integrals at semi-simple elements of any
reductive Lie group. He was a visiting scholar at the
Institute for Advanced Study in the summer of 1984. In 1990, he was awarded the Prix
Ampere of the Academy of Sciences. He was elected as a member of the
French Academy of Sciences in 1991. In 2021 he received the
Shaw Prize in Mathematics (jointly with
Jeff Cheeger). In 1986, he was an invited speaker in the Geometry section at the
ICM in Berkeley, and in 1998 he was a plenary speaker at the ICM in Berlin. He was a member of the Fields Medal Committee for ICM 1990. From 1999 until 2006, a member of the executive committee (from 2003 until 2006 as vice-president), International Mathematical Union (IMU). He was an editor of
Inventiones Mathematicae from 1989 until 1996 and managing editor from 1996 until 2008. ==Selected bibliography==