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Contributions to infinite dimensional geometry and analysis

The works prsented in this habilitation thesis can be gathered in six themes. Works on the implicit function theorem and the geometry of numerical schemes. On the existence of an exponential map on an infinite dimensioal Lie group. Holonomy and Ambrose-Singer theorem for connections on infinite dimensional principal bundles. Results on integration theory and discretized Yang-Mills theory. Works on non-formal pseudo-differential operators (PDOs), $ζ-$renormalized traces, manifolds of maps and related topics. Works on the Kadomtsev-Petsviashvili (KP) hierarchy.