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.
|Effective start/end date||6/15/00 → 5/31/03|
- National Science Foundation: $113,903.00