(At least sketches of) proofs of the results stated will be given. This implies that theorems will not always appear in their most general form.
The analytical core of the course is the theory of classical pseudo-differential operators and its generalizations due to R. Melrose.
Differential operators on sections of vector bundles over manifolds.
Basic example: the de Rham differential. Relationship to the Euler characteristic
and the signature. The Hirzebruch signature theorem. Elliptic operators and the index. Elliptic regularity, Sobolev embedding.
Lecture 2 - Thursday, Feb. 14th : 10.30-12
Inverting elliptic operators: Pseudo-differential operators. Conormal
distributions, wave-front set, pull-back and push-forward.
Complex powers of pseudo-differential operators. Zeta and eta functions, residue trace, determinants.Comparison of the zeta function to the heat kernel trace.
Lecture 3 - Friday, Feb. 15th : 10.30-12
Lefschetz fixed point formula via complex powers. The Atiyah-Singer
Index problems on manifolds with boundary: the APS theorem. Cusp and b-operators following Melrose.
Lecture 4 - Monday, Feb. 18th : 10.30 - 12
The determinant line bundle of a family. The Quillen metric and the
Relationship with adiabatic limits. Spectral flow and the index.
lectures in room P3.10, Instituto Superior Técnico