Properties and Applications of the Microlocal Category

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The geometric structure that lies behind classical Newtonian mechanics is called symplectic geometry, where the coordinates of position and momentum, along with the conservation laws of mechanics, are modeled in even-dimensional geometries with distinctive structures. While the familiar geometry of Euclidean spaces takes length and angle as fundamental notions, the structure of symplectic geometry begins from the measurement of two-dimensional area. Much of the modern study of symplectic geometry is devoted to understanding distinguished structures on symplectic manifolds, a study that can reveal useful invariants for those manifolds. The research to be pursued in this project is focused on questions in this area arising out of string theory.This research project focuses on developing the theory of microlocal categories, such as the full sub-category of objects supported on a Lagrangian submanifold, and on developing an appropriate ground category for the microlocal category, which may be topological in origin. The project also aims to find a description of the microlocal categories associated to flag varieties. Further investigations are motivated by ideas of quantization, which is here understood in geometric terms abstracted from concepts in physics.