Function Theory on Symplectic Manifolds

University Of Chicago, Chicago IL

Investigators

Abstract

Abstract Award: DMS-1006610 Principal Investigator: Leonid Polterovich The proposed research belongs to symplectic geometry and topology, a rapidly developing field of mathematics which originally appeared as a geometric tool for problems of classical mechanics. The "symplectic revolution" of the 1980s gave rise to the discovery of surprising rigidity phenomena involving symplectic manifolds, their subsets and diffeomorphisms. A number of recent advances show that there is yet another manifestation of symplectic rigidity, taking place in function spaces associated to a symplectic manifold. These spaces exhibit unexpected properties and interesting structures, giving rise to an alternative intuition and new tools in symplectic topology, and providing a motivation to study the function theory on symplectic manifolds. Development of this new theory and its applications is the main objective of the proposed research. We focus on the following topics. First, we study robustness of the Poisson bracket. The Poisson bracket is a basic operation which involves a pair of functions and is defined by their derivatives. Certain characteristics of the Poisson bracket exhibit surprising robustness properties with respect to small perturbations in the uniform norm, even though such perturbations can dramatically change the derivatives. This phenomenon appears to be closely related to Hofer's geometry on the group of symplectic diffeomorphisms. Second, we deal with various aspects of the theory of symplectic quasi-states. Consider the space of functions on a symplectic manifold. A symplectic quasi-state is a monotone functional on this space which is linear on every Poisson-commutative subalgebra, but not necessarily on the whole space. The origins of this notion go back to foundations of quantum mechanics. Non-linear quasi-states on higher-dimensional manifolds are provided by Floer theory, the cornerstone of modern symplectic topology. Quasi-states serve as a useful tool for a number of problems in symplectic topology such as symplectic intersections and Lagrangian knots. Finally, we unify both topics and explore interrelations between symplectic quasi-states and Poisson brackets. Symplectic topology fruitfully interacts with several areas of science, which have a significant impact on society through their applications to technology. One of these areas is Hamiltonian dynamics, a mathematical discipline providing efficient tools for modeling a variety of fundamental physical and technological processes such as orbital motion of satellites, propagation of light in optical fibers and motion of charged particles through accelerators. Another one is quantum theory, a branch of physics which studies behavior of matter on microscopic scales, and whose potential applications reach as far as cryptography and computer technology. Development of function theory on symplectic manifolds that is put forward in the present proposal leads to a new insight on robust measurements in Hamiltonian dynamics and reveals a new facet of the quantum-classical correspondence, a fundamental principle of quantum theory.