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
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.
|Effective start/end date||6/1/03 → 5/31/07|
- National Science Foundation: $196,781.00