Geometric Structures in Poisson Geometry and Applications

DMS-0306665

Ping Xu

This project involves the study of Poisson geometry, with the goals of

understanding various geometric structures in connection with Poisson

manifolds, and studying their applications in analysis, integrable

systems, quantization and other related areas in mathematical physics.

One of the main tools is the theory of Lie groupoids and Lie algebroids.

In particular, the investigator will apply his previously developed

theory of Morita equivalence to investigate a geometric model of unified

momentum map theory. He will also study stacks and gerbes from the

viewpoint of differentiable geometry, and investigate their relationship

to Lie groupoids. He plans to continue his study of twisted Poisson

structures, and also the universal lifting conjecture. The latter

implies many non-trivial results in Poisson geometry including the

Karasev-Weinstein symplectic realization theorem and the integration

theorem for Lie bialgebroids of Mackenzie and the investigator. This

project also involves the study of deformation quantization. The

investigator will continue to study quantization of classical dynamical

r-matrices using his previously developed deformation quantization

techniques. Also, he will study quantization of Dubrovin Poisson

structures.

Poisson geometry is largely motivated by physics, and is in fact a

mathematical tool used to give a theoretical framework encompassing

large parts of classical mechanics. Lie groupoids are useful tools in

studying the symmetry of various geometric problems in Poisson geometry.

Quantization is developed in order to gain a better understanding of the

relationship between classical mechanics and quantum mechanics. At

present, there are various applications of Poisson geometry including

control theory, machining automation, and robotic manipulation.