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In these talks I shall give examples of some applications of elliptic operators on manifolds centering around the fixed point formula and the index theorem. In his lectures at this session, R. T. Seeley will review pseudo-differential operators on manifolds and the elementary properties of the index. This will allow me to assume much of the analytical facts concerning elliptic operators. Since the primary discipline of the majority of the participants is analysis, I will spend more time with the geometry and topology but motivate the proofs of the main theorems from the analytic viewpoint. By and large analysts find the geometrical and topological background needed here formidable. It is my hope that these expository talks will help orient the listener and allow him to read the relevant papers listed at the end more easily.