Singularities, symplectic topology and mirror symmetry

Symplectic geometry is a rapidly developing field, with tools drawn from many different areas of mathematics. Modern geometry studies manifolds, smooth objects that at small enough scale look like the standard space of a fixed dimension. For instance, the surface of a ball is a 2D-manifold, standard space-time is a 4D-manifold, and the parameter space for a biological experiment might be an 18D-manifold. Symplectic manifolds are equipped with an extra structure that generalises conservation laws from classical mechanics. This makes them the natural formal framework for studying orbits of satellites or space probes. Also, some models in string theory, a branch of physics, allow any symplectic manifold in lieu of space-time. Duality ideas in physics have led to mirror-symmetry, a booming field that relates symplectic geometry with a very different looking part of mathematics: algebraic geometry, which studies solutions of polynomial equations in several variables.

This project is guided by the major open question: `What are the transformations (that is, global symmetries) of a symplectic manifold?' By transformation, we mean a rule for taking each point to another, which is smooth (no breaks), invertible (you can go backwards), and preserves the additional symmetries.

We don't understand symplectic transformations well: for a lot of spaces, the one real source is something called Dehn twists. Let me describe these for 2D surfaces. (2D surfaces are symplectic if they have orientations: the surface of a ball or of an inner tube does, a Mobius strip does not.) Start with a closed curve without self-intersections - for instance, a circle around the thin part of an inner tube. Cut the surface open along it: the inner tube is now a long annulus, with two boundary components, each a circle. Twist each of the boundaries to the right by 180 degrees and glue the edges together again. You have got the same surface back! This transformation is a Dehn twist. Circles on surfaces are 1D-spheres, and in general, we can define Dehn twists analogously in higher dimensions, by using higher dimensional spheres inside symplectic manifolds - for instance, copies of the usual sphere (the surface of a ball) in four-dimensional symplectic manifolds.

In 2D, all transformations can be decomposed into sequences of twists. A major goal of the project is to show that the higher-dimensional situation can be radically different, by constructing large families of new examples of transformations, inspired by mirror symmetry. These translate to a different sort of transformation in the world of algebraic geometry, where we propose to settle questions of independent interest.

A long-term goal is to compare dynamical properties of transformations of surfaces with the ones in higher dimensions. For instance, Dehn twists on surfaces have linear dynamics: the number of fixed points grows linearly with iteration. However, a generic surface transformation, called a pseudo-Anosov map, has exponential dynamics. For large families of examples, we will study the possible growth-rates of fixed points of transformations, and whether there is a generic behaviour.

Many of the objects that will be studied in the project arise naturally in singularity theory, a field tied to the parts of mathematics that explain discontinuities and abrupt changes - for instance, the cuspy caustic curve that appears when light shines through water. We also propose to use ideas from symplectic geometry to study classical structural questions about spaces of deformations of generalised caustics.

Lots of other geometric structures enter the project too: for instance, braid groups, which are mathematical formalisations of the braids you can make with hair or ribbons; and Coxeter groups, which are transformations of space generalising the ones you can obtain from reflections in configurations of (physical, light-reflecting) mirrors.