Geometric Structures on Lie Groupoids and their Applications

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Groups typically arise as the symmetries of some given object. The concept of a groupoid allows for more general symmetries, acting on a collection of objects rather than just a single one. The concept of a Lie groupoid, as well as its infinitesimal counterpart, the notion of a Lie algebroid, have played an increasing important role in various branches of mathematics, as well as its applications. For example, one can now find them in the formulation of dynamics in field theory in High Energy Physics, in various models for populations dynamics in Biology, or in evolutionary dynamics in Game Theory. Over the last two decades, the theory of Lie groupoids and Lie algebroids has undergone several exciting developments and this project aims to develop further both foundational aspects and significant new applications of the theory. This project includes collaborations with various researchers in the field working in Europe and South America and the training of 3 PhD students. The PI will be involved in the organization of several international meetings, workshops and summer courses in the field, as well as a weekly seminar at UIUC.The project is organized into 4 main tasks. The first task continues the PI study of Poisson manifolds of compact type, which are central objects in Poisson Geometry, playing a role analogous to compact Lie algebras in Lie Theory. The second task is dedicated to multiplicity free fibrations, a.k.a. non-commutative integrable systems, and initiates the study of its generic singularities. The third task concerns the study of strict (non-formal) deformation quantization and aims to establish the first known obstructions to the existence of strict deformation quantizations of a Poisson manifold. The fourth task concerns Cartan’s realization problem underlying many classification problems in geometry, namely the ones that can be formulated in terms of (finite dimensional) families of G-structures. It aims to establish a theory of such realization problems, based on Lie groupoids and Lie algebroids, as well as to give a complete solution to the classification problem. All these tasks rely heavily on Lie groupoid techniques, many of which have yet to be developed. Some of these ideas extend and expand results and techniques previously developed by the PI and collaborators.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.