Categorical Symplectic Topology

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This is an intra-disciplinary proposal to study classical (and novel) questions in topology and dynamical systems by using sophisticated new ideas from algebra, which were in turn developed following insight from ``mirror symmetry" in quantum field theory. A fundamental question in dynamics is to understand periodic orbits of systems (asteroids, satellites, fluid flows, motions of rigid jointed bodies). Remarkably, some of our most powerful methods for detecting such periodic orbits make essential use of complex analysis, and partial differential equations for ``holomorphic curves", which are closely related to area-minimising surfaces like soap films. In the last twenty years, it has been understood that counts of these special surfaces give numbers which are not independent of one another, but which should be bundled together into complicated algebraic structures, and which satisfy remarkable identities. Aspects of this insight arose first in theoretical physics of quantum field theory, via a duality in string theory called mirror symmetry, which can be viewed as a far-reaching generalisation of Maxwell's classical electric-magnetic duality. Mirror symmetry relates different physical theories which are models for a single structure in nature but which are superficially described by very different kinds of mathematics. This enables insights and structures which seem natural in one area to be carried ``to the other side of the mirror" where they yield powerful new methods, and intriguing predictions, which are just beginning to be understood.This Fellowship will study the algebraic structures in topology that have been developed following mirror symmetry, and apply them to new questions in topology and dynamics. These questions relate to the structure of the set of symmetries of a mechanical system, to the entropy and mixing properties of such symmetries, to the complexity of ``random" knots in space, and to periodic orbit problems for games of billiards on a polygonal table.