Parametrized Pseudodifferential Operators and Geometric Invariants

This is based on joint work with R. T. Seeley. The introduction presents the problem of parameter-dependent calculi for do's and the question of trace asymptotics for Atiyah-Patodi-Singer operators. Chapter 2 establishes relations between the three operator functions: resolvent, heat operator and power operator (zeta function). Chapter 3 explains our parameter-dependent do calculus with weak polyhomogeneity, showing how logarithmic terms appear in trace formulas. In Chapter 4, the APS problem is treated in the case with a product structure near the boundary, where functional calculus on the cylinder leads to precise formulas for heat trace expansions and zeta function pole structure. Finally, Chapter 5 treats the APS problem in the non-product case where the weakly polyhomoge-neous do calculus is used to get asymptotic trace expansions generalizing those in the product case.