Geometric Structures in Poisson Geometry

In this project, the investigator aims to apply methods of symplectic and Poisson geometry, and Lie groupoids and Lie algebroids, to the study of various differential structures related to Poisson manifolds and the quantization problem. More specifically, the project involves the study of bihamiltonian structures and hyperkahler structures using the theory of Lie bialgebroids. Also, this project will investigate deformation quantization of Lagrangian submanifolds, the characteristic class of star products on symplectic manifolds, quantization of Poisson manifolds, and their application in knot theory. Symplectic geometry is a mathematical tool used to lay a theoretical framework encompassing large parts of classical mechanics of Newton. Ideas from symplectic geometry and Poisson geometry can be used to explain and predict various mechanical phenomena - e.g., locomotion generation and motion control. Knot theory has profound application in biotechnology such as the understanding of different DNA structures.