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New Advances through the boundaries of Poisson Geometry

Final Report Summary - NEW-POETRY (New Advances through the boundaries of Poisson Geometry)

Poisson Geometry lies on the cross road of Mathematical Physics and Geometry; it arises as the geometric structure that governs Hamiltonian mechanics and as the semi-classical limit of quantum systems. The PI’s own philosophy has been that Poisson Geometry has the power to exchange fundamental ideas with other areas in geometry such as Foliation Theory, Symplectic Geometry, Lie Theory, Equivariant Geometry, thereby providing new insights in all these fields. It is this global view which provided the PI with the new tools and the new ideas needed to solve concrete open problems. Poisson Geometry saw spectacular developments over the last 20 years- e.g. Kontsevich's solution to the “quantization problem” or "Lie's third theorem" for Lie algebroids by the PI and R.L. Fernandes.

In this project, building on his philosophy, the PI discovered and investigated completely new interactions of Poisson Geometry- e.g. with the theory of Exterior Differential Systems (the geometric theory of PDE), Symplectic Topology, Integral Affine Geometry, Contact Geometry and with geometric deformation theories. Here are some of the main outcomes of the project:
- the theory of PMCTs (Poisson manifolds of compact types): Poisson Geometry is very closely related to Lie Theory- the mathematical theory that studies symmetries of various objects (spaces, PDEs, etc). Lie theory pervades modern Mathematics and Mathematical Physics, and the basics are part of the standard education of any geometer. In particular, compact Lie groups are very well understood/classified. This project initiated the theory of PMCTs- which is the analogue in Poisson Geometry of the compactness in Lie Theory. We have discovered in this way completely new and exciting interactions with various other classical fields such as Symplectic Topology, Integral Affine Geometry, Algebraic Topology, etc. Another remarkable finding is that a large part of the Lie Theory itself is entirely Poisson geometric (namely a particular case of our theory of PMCTs). It is clear that the theory of PMCTs is still at the beginning- in particular it is now a longer term project (joint with R.L. Fernandes and D.M. Torres),
- deformations of geometric structures: one of the most known principles in Geometry (going back to Klein's Erlangen program) is that geometric structures are best understood via their symmetries. As above, symmetries are encoded in Lie groups or, more generally, Lie groupoids. In this project we have discovered that there is a good deformation theory of Lie groupoids: they can be controlled linearly (by certain cohomologies) and that allows one to achieve rigidity results. In particular, this gives an unified approach to deformations of a large class of geometric structures. As an application, we discovered a geometric approach to "Weinstein's linearization conjecture": under compactness assumptions, Lie groupoids are locally rather simple ("linearizable").
- Cartan's work (from around 1900) on symmetries of PDEs (Lie pseudogroups), and the resulting techniques, had a tremendous influence on shaping modern Differential Geometry. Despite that, his original writings on pseudogroups remained at the level of 1900:
occasionally un-precise, with hidden assumptions, always very local/computational (and notoriously hard to penetrate). In this project we finally achieved a modern formulation of Cartan's work. In particular, we have discovered the mathematical structure that governs symmetries of PDEs: Pfaffian groupoids.

This project supported a large team: next to the PI we had six PostDocs (each for at least one year), two PhD students with full support and three with partial support. That result was a very exciting research environment, with 21 papers published in refereed journals (and a few more under revision) and five PhD theses. Looking ahead, it gave rise to exciting new research projects and a strong international network (e.g. all the six PostDocs, and even one of the former PhD students, now hold permanent positions elsewhere). Moreover, one of the PostDocs (I. Marcut) received in 2014 the Lichnerowicz prize for contributions to Poisson Geometry, while the PI received the N.G. De Bruijn prize for the best publication by a mathematician hosted in The Netherlands during the period 2011-2014.