Geometric Structures in Poisson Geometry and Applications

Project: Research project

Project Details

Description

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.
StatusFinished
Effective start/end date6/1/035/31/07

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