On the B-pseudodifferential Calculus on Manifolds with Corners on the B-pseudodifferential Calculus on Manifolds with Corners

Structure theorems for both the resolvent and the heat kernel of b-pseudodifferential operators on a compact manifold with corners (of arbitrary codimension) are presented. In both cases, the kernels are realized as classical conormal functions on appropriate manifolds with corners. To prove these results, a space of operators with complex parameter (or tempered operators) is introduced. These tempered operators are shown to be classical conormal functions on a manifold with corners called the Tempered space. The resolvent of a b-pseudodifferential operator is shown to be a tempered operator (for large values of the parameter) and so it follows that the resolvent is a classical conormal function. The Laplace transforms of holomorphic tempered operators are shown to be operators of order -oo for positive times and are also shown to be classical conormal functions on a manifold with corners called the Heat space. Since the heat kernel of a b-pseudodifferential operator is the Laplace transform of the operators resolvent, the heat kernel is of order -oo for positive times and is also a classical conormal function. The structure result for the heat kernel is used to generalize the Index formula of Atiyah, Patodi, and Singer for Dirac operators on a manifold with boundary to Fredholm b-pseudodifferential operators on arbitrary compact manifolds with corners. The formula expresses the index of an operator as a sum of two terms, the usual 'interior term' given by the integral of the Atiyah-Patodi-Singer density associated to the operator and a second contribution given by a generalization of the eta-invariant associated to the induced operators on each of the corners of the manifold. Thesis Supervisor: Richard Melrose Title: Professor of Mathematics Acknowledgments I would like to thank M.I.T. for blessing me with the opportunity to come here. I thank Richard Melrose, my very caring advisor who always had the time to meet with me. I thank my church for all their prayers and support. I thank David Jerison who served on my thesis committee. I thank Robert Lauter who served on my committee, and who read much of this thesis and helped to correct many errors. I thank my fellow advisees': Sang Chin, Dimitri Kountourogiannis, Sergiu Moroianu, Boris Vaillant, and Jared Wunsch. There is one more person that I cannot miss. One who has always been there for me, through times of trouble and stress, through precious moments, through times of joy, one whose hand has always guided me to green pastures. One who I can always trust, not only with the events of the day, but indeed with my very life. He is Jesus Christ, my Lord and Savior, and it is he that I thank the most, for he is the one who put all these people in my life. I also thank him because even after all of mathematics and the present world passes away, in heaven, there will always be plenty of days to sing God's praise.