Jean-Michel Bismut worked first in probability theory and on the control of stochastic processes. His interests in probability theory led him to study refinements of the index theorem of Atiyah-Singer. He constructed eta forms and analytic torsion forms, which are local extensions of well-known spectral invariants, and established their functorial properties. He participated in the proof of a Riemann-Roch theorem in arithmetic geometry. He devoted part of his work to certain aspects of symplectic geometry, in connection with the Verlinde formulas. He has also constructed an exotic version of Hodge theory, for real and complex manifolds, whose corresponding Laplacian is a hypoelliptic operator acting on the total space of the tangent or cotangent bundle of a manifold, which interpolates between the standard elliptic Laplacian and the geodesic flow.
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