## by

(Universität Hamburg)

- These lectures will give an overview of certain problems
in Analysis on manifolds which share two common features:
- They are formulated in terms of spectral properties of differential operators.
- They are best attacked via pseudo-differential algebras methods.

(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.

Contents

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 theorem.

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
Bismut-Freed connection.

Relationship with adiabatic limits. Spectral flow and the index.

All
lectures in room P3.10, Instituto Superior Técnico

Sponsorship: Centro de Análise Matemática, Geometria e Sistemas Dinâmicos

Organizers: Jorge Silva
, Gustavo Granja