Geometric Structures in Poisson Geometry and Quantization

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DMS-0072171Ping XuThis project involves the study of Poisson structuresusing 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 formalitytype 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 isin fact a mathematical tool used to give a theoretical framework encompassing large parts of classical mechanics. Quantizationis developed in order to gain a better understanding betweenclassical mechanics and quantum mechanics. At present, Poisson geometry finds various applications including control theory, machining automation and robotic manipulation.