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 pseudodifferential operators on the manifold and so-called 'shift operators', Fourier integral operators associated with diffeomorphisms of the base space.

Nonlocal problems of this type arise in concrete applications (plasma physics, theory of multi-layer plates and shells used in aviation and astronautics, in optical systems with two-dimensional feedback,  etc.), but they  are also interesting from the point of view of noncommutative geometry, because their symbols form 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 analytic aspects of the theory, in
  particular ellipticity and the Fredholm property
  (ii) to establish
  index formulae and
  (iii) to use semiclassical methods to obtain asymptotics for
  hyperbolic problems.

This is part of a joint German-Russian project with Anton Savin (RUDN, Moscow) and Vladimir Nazaikinkii (Russian Academy of Sciences).