Jean-Michel Bismut

The analysis of elliptic families

In this paper we specialize the results obtained in [BF1] to the case of a family of Dirac operators. We first calculate the curvature of the unitary connection on the determinant bundle which we introduced in [BF1].We also calculate the odd Chern forms of Quillen for a family of self-adjoint Dirac operators and give a simple proof of certain results of Atiyah-Patodi-Singer on êta invariants.We finally give a heat equation proof of the holonomy theorem, in the form suggested by Witten [W 1, 2].

An Introductory Approach to Duality in Optimal Stochastic Control

The purpose of this paper is to compare the results which have been recently obtained in optimal stochastic control. Various maximum principles are shown to derive from a general Pontryagin princip...

Analytic torsion and holomorphic determinant bundles. II. Direct images and Bott-Chern forms

On etudie les proprietes principales des fibrations de Kahler: On introduit la superconnexion associee de Levi-Civita pour construire des formes torsion analytiques pour des images directes holomorphes

The Atiyah-Singer index theorem for families of Dirac operators: Two heat equation proofs

SummaryThe purpose of this paper is to give two heat equation proofs of the Index Theorem of Atiyah-Singer for a family of Dirac operators.

The analysis of elliptic families. I. Metrics and connections on determinant bundles

In this paper, we construct the Quillen metric on the determinant bundle associated with a family of elliptic first order differential operators. We also introduce a unitary connection on λ and calculate its curvature. Our results will be applied to the case of Dirac operators in a forthcoming paper.

Linear Quadratic Optimal Stochastic Control with Random Coefficients

The purpose of this paper is to apply the methods developed in [1] and [2] to solve the problem of optimal stochastic control for a linear quadratic system.After proving some preliminary existence results on stochastic differential equations, we show the existence of an optimal control.The introduction of an ad joint variable enables us to derive extremality conditions: the control is thus obtained in random “feedback” form. By using a method close to the one used by Lions in [4] for the control of partial differential equations, a priori majorations are obtained.A formal Riccati equation is then written down, and the existence of its solution is proved under rather general assumptions.For a more detailed treatment of some examples, the reader is referred to [1].

Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion

We attach secondary invariants to any acyclic complex of holomorphic Hermitian vector bundles on a complex manifold. These were first introduced by Bott and Chern [Bot C]. Our new definition uses Quillens superconnections. We also give an axiomatic characterization of these classes. These results will be used in [BGS2] and [BGS3] to study the determinant of the cohomology of a holomorphic vector bundle.

Complex immersions and Quillen metrics

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Flat vector bundles, direct images and higher real analytic torsion

We prove a Riemann-Roch-Grothendieck-type theorem concerning the direct image of a flat vector bundle under a submersion of smooth manifolds. We refine this theorem to the level of differential forms. We construct associated secondary invariants, the analytic torsion forms, which coincide in degree 0 with the Ray-Singer real analytic torsion. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Analytic torsion and holomorphic determinant bundles. III. Quillen metrics on holomorphic determinants

In this paper, we derive the main properties of Kähler fibrations. We introduce the associated Levi-Civita superconnection to construct analytic torsion forms for holomorphic direct images. These forms generalize in any degree the analytic torsion of Ray and Singer. In the case of acyclic complexes of holomorphic Hermitian vector bundles, such forms are calculated by means of Bott-Chern classes.