Application Abstract: Since the Atiyah-Singer index theory was established,enormous efforts have been taken to generalize it.Index theory on non-compact manifolds with group actions has become an important topic in geometry.This project mainly focuses on the non-compact manifolds with boundary,admitting proper cocompact group actions,and deals with the following problems:1Introduce an invariant index for G-equivariant Dirac operators under some Lopatinski-Schapiro boundary conditions.2Generalize the Hodge-Morrey-Friedrichs decomposition and study the relationships between Hopf cyclic cohomology and index theory.3Investigate the relationships of indexes under different boundary conditions and discover the relationships of the two Hopf cyclic cohomology for non-compact manifolds with boundary.
