Cohomology in algebra, geometry, physicsand statistics

On the differential geometry of groups of diffeomorphisms and of non-formal pseudo-differential operators

Summary:
After reviewing a class of infinite dimensional groups based on the central extension of a group of diffeomorphisms by a group of pseudo-differential operators (PDOs), I will explain:
1) how the action of the group of diffeomorphisms generates the dressing operator of a KP hierarchy, which is shown to be well-posed in a class of NON FORMAL PDOs
2) How renormalized traces enables to define pseudo-Riemannian metrics on some of these groups of PDOs, different from the classical sobolev metrics present in the literature
3) how the geodesic equation of one of these metrics admit an infinite number of independent  integrals of the motion
Part of the results of this talk are obtained from works in collaboration with Enrique G. Reyes. Arxiv identifiers of related publications/preprints are: 2104.08159 ; 2007.00387 ; 1808.03791 and 1407.1427
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