Cohomology of classes of symbols and classi cation of traces on corresponding classes of operators with non positive order

This thesis is devoted to the classification issue of traces on classical pseudo-differential operators with fixed non positive order on closed manifolds of dimension $n>1$. We describe the space of homogeneous functions on a symplectic cone in terms of Poisson brackets of appropriate homogeneous functions, and we use it to find a representation of a pseudo-differential operator as a sum of commutators. We compute the cohomology groups of certain spaces of classical symbols on the $n$--dimensional Euclidean space with constant coefficients, and we show that any closed linear form on the space of symbols of fixed order can be written either in terms of a leading symbol linear form and the noncommutative residue, or in terms of a leading symbol linear form and the cut-off regularized integral. On the operator level, we infer that any trace on the algebra of classical pseudo-differential operators of order $a\in\Z$ can be written either as a linear combination of a generalized leading symbol trace and the residual trace when $-n+1\leq2a\leq0$, or as a linear combination of a generalized leading symbol trace and any linear map that extends the $L^2$--trace when $2a\leq-n\leq a$. In contrast, for odd class pseudo-differential operators in odd dimensions, any trace can be written as a linear combination of a generalized leading symbol trace and the canonical trace. We derive from these results the classification of determinants on the Fr\'echet Lie group associated to the algebras of classical pseudo-differential operators with non positive integer order