Mathematical Sciences: Higher Index for Coverings of Manifolds with Boundary

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Abstract WU This project studies the higher index problem for Dirac operators on coverings of manifolds with boundary, with global boundary condition of Atiyah-Patodi-Singer type. The main steps in solving this problem are: (1) Study a canonical boundary condition defined by the Calderon projector constructed from the bounding manifold. (2) Reduce the K-theoretic index of the boundary value problem to the noncommutative spectral flow from the original bmundary condition to the Calderon projector. These steps provide an identification of the boundary value index at the K-theory level. To get more explicit formulae for the index, one then needs to calculate the Chern character of the index in cyclic homology. Using the formula for the cyclic character for the spectral flows, this amounts to identifying the higher (cyclic) eta-invariant of the boundary Dirac and the Calderon projector in terms of the geometry of the bounding manifold. A successful solution of the higher index problem would lead to many interesting applications in differential geometry and topology. It also provides valuable theoretical understandings to the study of boundary value problems of elliptic partial differential equations arising from various applied scientific fields and engineering.