Contributions to infinite dimensional geometry and analysis

TL;DR

This thesis explores advanced topics in infinite-dimensional geometry and analysis, including Lie groups, principal bundles, integration theory, pseudo-differential operators, and integrable systems, contributing new theoretical insights and results.

Contribution

It provides new results on the exponential map in infinite-dimensional Lie groups, holonomy in principal bundles, and developments in pseudo-differential operators and integrable hierarchies.

Findings

Existence of exponential map on infinite-dimensional Lie groups

Holonomy and Ambrose-Singer theorem for infinite-dimensional bundles

Advancements in pseudo-differential operators and KP hierarchy

Abstract

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), $\zeta-$renormalized traces, manifolds of maps and related topics. Works on the Kadomtsev-Petsviashvili (KP) hierarchy.