Nonlocal Boundary Value Problems: Index Theory and Semiclassical Asymptotics

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Nonlocal problems arise in many areas of mathematics and its applications to science and technology. In this project, we are interested in nonlocal boundary value problems associated with group actions on manifolds. In these problems, both the main operator and the boundary operator belong to operator algebras generated by (pseudo)differential operators on the manifold and so-called shift operators associated with diffeomorphisms, which define a group action. Nonlocal problems of this type arise in plasma physics, in the theory of multi-layer plates and shells used in aviation and astronautics, in optical systems with two-dimensional feedback, etc. They are also interesting from the point of view of noncommutative geometry, because symbols of nonlocal operators form essentially noncommutative algebras. We intend to study elliptic and hyperbolic nonlocal boundary value problems associated with such group actions. Our aim is (i) to investigate the analytical aspects of the theory (i.e. to introduce a notion of ellipticity and to prove the Fredholm property of elliptic elements), (ii) to apply methods of topology and noncommutative geometry in order to establish index formulas and (iii) to use semiclassical methods to obtain asymptotics in hyperbolic problems.