Categorical Symplectic Topology

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.

The proposed Fellowship would have a significant academic impact in symplectic topology, algebraic geometry and low-dimensional topology. There is the potential to have a transformative effect on geometric group theory, by opening up symplectic mapping class groups as a different natural generalisation of classical mapping class groups to the usually studied hyperbolic groups and lattices in Lie groups. There would also be significant impact for people working in the parts of classical dynamics concerned with Hamiltonian systems, and the intended applications of theory would be relevant for the geometric Langlands program.

Very recently ideas from big data analysis (barcodes and persistent homology) have been brought into symplectic topology and the geometry of displacement energy. We hope to take advantage of the newly created Alan Turing Institute to extend and deepen that connection, which would have impact on the inter-relationship of complexity theory and data processing to the mainstream of more traditional pure mathematics.

The role of entropy in two-dimensional Hamiltonian systems can be easily visualised by mixing fluids, and ``topological rod stirring" optimises the mixing efficiency of practical mechanical devices (used in making toffee, for instance). On the other hand, basic questions in four-dimensional topology concerning the structure of space-time can be encoded in questions of the topology of surfaces bounding knots, which can in turn be visualised through soap films on knotted wires. This makes the basic instances of the questions which the Fellowship goals hope to address and generalise readily visualisable, and opens the opportunity for impact through outreach. The PI has given informal talks to school-children (Knots in art and mathematics), trainee teachers (Knots, dynamics and space-time) and undergraduate audiences (Billiards and beyond) drawn from across the Arts and Sciences, on the broad theme of why geometry matters.

The proposed research would both raise the international research profile of the UK in an active area, helping it maintain its world-leading status in geometry and in particular in the interactions of geometry with quantum field theory, and also identify it as the (almost unique?) centre in Europe for research in the emerging discipline of categorical symplectic topology. The PI will train two postdoctoral Research Associates to independence, who would then be well-poised to continue their own research projects as group leaders themselves. A mid-term Workshop will bring international experts from all corners of the field and all corners of the world to the UK, to cement the visibility of this vital research presence and to help initiate collaborations which should go beyond the remits and members of the PI's group itself.