Geometric Structures in Poisson Geometry and Quantization

DMS-0072171

Ping Xu

This project involves the study of Poisson structures

using the theory of Lie groupoids and Lie algebroids,

in particular, Poisson groupoids and Lie bialgebroids.

The theory of Poisson groupoids was developed as a

unification of both Drinfel'd's Poisson group theory

and the theory of symplectic groupoids of Karasev-Weinstein.

The investigator aims to apply this theory to study

integrable systems such as Calogero-Moser systems. He will

also continue his study on Courant algebroids and Dirac

structures from the viewpoint of Dirac generating operators,

as applied to objects in Poisson geometry such as moment

maps and equivariant cohomology. This project also involves

the study of deformation quantization, in particular on quantum groupoids. More specifically, it includes the study of universal enveloping algebras of Courant algebroids, Kontsevich's formality

type conjecture for Lie algebroids, and cohomology theory of deformation of Hopf algebroids, all of which are components in quantization of Lie bialgebroids. An important application is to study quantization of classical dynamical r-matrices.

Poisson geometry is largely motivated by physics, which is

in fact a mathematical tool used to give a theoretical framework

encompassing large parts of classical mechanics. Quantization

is developed in order to gain a better understanding between

classical mechanics and quantum mechanics. At present, Poisson

geometry finds various applications including control theory,

machining automation and robotic manipulation.