Index theory for Fourier integral operators

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The development of index theory, culminating with the proof of the Atiyah-Singer index theorem, was one of the great achievements of mathematics in the last century. Since then, index theory has become a central research area in pure mathematics with applications in fields as diverse as topology, geometry, analysis, number theory, and theoretical physics.While initially, only local (or pseudo-local) operators were considered, it has meanwhile become possible - due to the work of Savin and Sternin and of Perrot - to apply methods of noncommutative geometry of Connes and treat also operators which combine the action of local operators with that of group actions on the underlying space. In this project we shall extend their work to the case, where the group acts by quantized canonical transformations, so that we obtain index theory for an algebra of Fourier integral operators. The task of determining the index of a single quantized canonical transformation is known as the Atiyah-Weinstein index problem; it was solved by Epstein and Melrose and, in a more general setting, by Leichtnam, Nest and Tsygan. For Dirac operators on Lorentzian spacetimes, Bär and Strohmaier set up and solved an index problem. For time-independent metrics, the index is given as the index of a Toeplitz variant of a quantized canonical transformation.We will study the index theory for these operator algebras in three geometric situations: for closed manifolds, in the context of Sobolev problems, and for manifolds with singularities.In all these settings, we will establish criteria for the Fredholm property and derive index formulae. Of particular interest is the case of Sobolev problems (or relative elliptic theory), where the operator algebras are constructed for pairs consisting of a smooth manifold and a smooth submanifold. Via composition with boundary and coboundary operators, an operator on the full manifold induces a 'trace' on the submanifold. One of the exciting and unexpected features of the theory is the fact that these traces are operators of a completely new nature; in particular we obtain new important classes of operators.Manifolds with singularities present an additional challenge. Up to now, there is no satisfactory index theory even for the standard differential operators on these spaces. We will derive Fredholm criteria for the full algebra and obtain index formulae only for special classes of operators satisfying certain symmetry properties.