Geometric Structures in Poisson Geometry and Applications

点击翻译按钮获取中文摘要

DMS-0306665Ping XuThis project involves the study of Poisson geometry, with the goals ofunderstanding various geometric structures in connection with Poissonmanifolds, and studying their applications in analysis, integrablesystems, 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 developedtheory of Morita equivalence to investigate a geometric model of unifiedmomentum map theory. He will also study stacks and gerbes from theviewpoint of differentiable geometry, and investigate their relationshipto Lie groupoids. He plans to continue his study of twisted Poissonstructures, and also the universal lifting conjecture. The latterimplies many non-trivial results in Poisson geometry including theKarasev-Weinstein symplectic realization theorem and the integrationtheorem for Lie bialgebroids of Mackenzie and the investigator. Thisproject also involves the study of deformation quantization. Theinvestigator will continue to study quantization of classical dynamicalr-matrices using his previously developed deformation quantizationtechniques. Also, he will study quantization of Dubrovin Poissonstructures.Poisson geometry is largely motivated by physics, and is in fact amathematical tool used to give a theoretical framework encompassinglarge parts of classical mechanics. Lie groupoids are useful tools instudying the symmetry of various geometric problems in Poisson geometry.Quantization is developed in order to gain a better understanding of therelationship between classical mechanics and quantum mechanics. Atpresent, there are various applications of Poisson geometry includingcontrol theory, machining automation, and robotic manipulation.