Bismut received in 1973 his Docteur d'État in Mathematics, from the Université Paris-VI, a thesis entitled Analyse convexe et probabilités.

[2] [3] 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.

Bismut-Freed developed the theory of Quillen metrics on the smooth determinant line bundle associated with a family of Dirac operators.

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.

One striking application is Bismut's explicit formulas for all orbital integrals at semi-simple elements of any reductive Lie group.