Complex geometry and Toeplitz quantization

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Symplectic geometry provides a mathematical framework for classical mechanics. It enables a description of behavior of certain physical systems (for example, motion of a particle) in an elegant way. On a very small scale, in which equations of classical mechanics do not provide good results, laws of quantum mechanics serve as an appropriate replacement. Trying to understand the transition from classical mechanics to quantum mechanics leads to interesting mathematical problems. In this proposal some of these problems and possible approaches are described. On the classical side, I will study geometry of the phase space (for motion of a particle the phase space can be viewed as the space that parametrizes values of its position and velocity) and functions on this space (for example, energy of the particle). On the quantum side functions produce operators. In my research these operators are most often matrices. It is intuitively clear that everything is interconnected: geometry of the space, properties of functions and properties of matrices. My work will be concerned with establishing precise mathematical statements that describe these connections and how one affects the other. I will also develop approaches to creating these relationships (matrices from functions, or functions from geometry) for systems where existing techniques are not applicable and it is not immediately clear how to adapt them. Proposed research will provide contribution that will touch a number of areas of mathematics including differential geometry, semiclassical analysis, automorphic forms. Many of the projects have an underlying physical interpretation or motivation.