### Abstract

In this talk we will present applications of the noncommutative residue in Riemannian and CR geometry. It will be divided into 3 main parts. 1. **Lower dimensional volumes in Riemannian geometry.**
Given a Riemannian
manifold (M^n,g) a natural geometric is how to define the k'th
dimensional volume of $M$ for $k=1,...,n$. Classical Riemannian
geometry provides us with an answer for $k=n$ only. We will explain
that by extending an idea of A. Connes we can make use of the
noncommutative residue for classical PsiDO's and of the
framework of
noncommutative geometry to define in a purely *differential- geometric*
fashion the k'th dimensional volumes for any k.

2. **Lower dimensional volumes in Riemannian geometry.**
CR structures
naturally arise in varous contexts. We also can define
lower dimensional volumes in CR geometry. This involves
constructing a
noncommutative residue trace for the Heisenberg calculus, which
is the
relevant pseudodifferential calculus at stake in the CR setting.

3. **New invariants for CR and contact manifolds.** We can
define new global
invariants of CR and contact structures in terms of noncommutative
residues of various geometric projections in the Heisenberg calculus.
This allows us to recover recent results of Hirachi and Boutet de
Monvel
and to answer a question of Fefferman.