Jean-Michel Bismut | |
|---|---|
 Bismut in 2004 (photo from MFO) | |
| Born | (1948-02-26)26 February 1948 (age 77) Lisbon, Portugal |
| Nationality | French |
| Alma mater | Ecole Polytechnique |
| Known for | Backward stochastic differential equations, Probabilistic proof of Atiyah–Singer index theorem, Bismut connection, Bismut superconnection, Geometric hypoelliptic Laplacian, Explicit formulas for orbital integrals |
| Awards | Prix Ampère (French Academy of Sciences), 1990 Shaw Prize, 2021 |
| Scientific career | |
| Fields | Mathematics |
| Institutions | Université Paris-Sud |
| Doctoral advisor | Jacques-Louis Lions Jacques Neveu |
Jean-Michel Bismut (born 26 February 1948) is a French mathematician who has been a professor at theUniversité Paris-Sud since 1981.[1]His mathematical career covers two apparently different branches ofmathematics: probability theory and differential geometry. Ideas from probability play an important role in his works on geometry.
Bismut's early work was related tostochastic differential equations, stochastic control, andMalliavin calculus, to which he made fundamental contributions.
Bismut received in 1973 hisDocteur 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 incontrol theory by introducing and studying the backwardstochastic 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.[2][3]
Using the quasi-invariance of the Brownian measure, Bismut gave a new approach to theMalliavin 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 aheat equation proof for theAtiyah–Singer index theorem.And he established a local version of the Atiyah-Singer families index theorem for Dirac operators, by introducing theBismut superconnection which plays a central role in modern aspects of index theory.
Bismut-Freed developed the theory ofQuillen metrics on the smooth determinant line bundle associated with a family of Dirac operators. Bismut-Gillet-Soulé gave a curvature theorem for theQuillen 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 inArakelov 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 aRiemannian 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 allorbital integrals at semi-simple elements of any reductive Lie group.
He was a visiting scholar at theInstitute for Advanced Study in the summer of 1984.[4] In 1990, he was awarded the PrixAmpere of the Academy of Sciences. He was elected as a member of theFrench Academy of Sciences in 1991. In 2021 he received theShaw Prize in Mathematics (jointly withJeff Cheeger).[5]
In 1986, he was an invited speaker in the Geometry section at theICM in Berkeley,[6] and in 1998 he was a plenary speaker at theICM in Berlin.[7][8]
He was a member of the Fields Medal Committee forICM 1990.[9]From 1999 until 2006, a member of the executive committee(from 2003 until 2006 as vice-president), International Mathematical Union (IMU).[10]He was an editor ofInventiones Mathematicae from 1989 until 1996 and managing editor from 1996 until 2008.[11]
