Periodic Reporting for period 1 - SingSymp (Singularities and symplectic mapping class groups)
Okres sprawozdawczy: 2023-07-01 do 2025-12-31
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.
The overarching question for the project is: `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 will focus on classes of examples which we expect to be “paradigmatic” -- in particular, capturing key phenomena. These examples belong to the field of singularity theory. This is the part of mathematics that explain discontinuities and abrupt changes - for instance, the cuspy caustic curve that appears when light shines through water.
We don't understand symplectic transformations well: 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.
Here are the main objectives of the project:
Project 1: In 2D, all transformations can be decomposed into sequences of twists. We plan 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 are in 4D. We expect them to translate to interesting transformations in the world of algebraic geometry, and propose to study applications to questions there. We will also consider generalisations to 6D and above.
Project 2: We plan to use ideas from symplectic geometry to study classical structural questions in singularity theory – for instance, about topological properties of spaces of deformations of generalised caustics.
Project 3: This is the most ambitious component (and hardest to explain in lay-terms). Loosely, we study “universal” objects which naturally appear when thinking about generalisations of Project 2. These would give spaces which classify all symplectic transformations of a large family of symplectic manifolds, at the cost of the classification being done “stably” (loosely, up to some asymptote-like error term).
Both Projects 2 and 3 involve lots of other geometric structures: for instance, generalisations of braid groups; these 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.
Finally, in Project 4, the 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. One subproject is to study the possible growth-rates of fixed points of transformations for large families of examples. Other subprojects consider possible other dynamical questions, notably about “generic” behaviours, including again for examples from singularity theory.
The overarching question for the project is: `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 will focus on classes of examples which we expect to be “paradigmatic” -- in particular, capturing key phenomena. These examples belong to the field of singularity theory. This is the part of mathematics that explain discontinuities and abrupt changes - for instance, the cuspy caustic curve that appears when light shines through water.
We don't understand symplectic transformations well: 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.
Here are the main objectives of the project:
Project 1: In 2D, all transformations can be decomposed into sequences of twists. We plan 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 are in 4D. We expect them to translate to interesting transformations in the world of algebraic geometry, and propose to study applications to questions there. We will also consider generalisations to 6D and above.
Project 2: We plan to use ideas from symplectic geometry to study classical structural questions in singularity theory – for instance, about topological properties of spaces of deformations of generalised caustics.
Project 3: This is the most ambitious component (and hardest to explain in lay-terms). Loosely, we study “universal” objects which naturally appear when thinking about generalisations of Project 2. These would give spaces which classify all symplectic transformations of a large family of symplectic manifolds, at the cost of the classification being done “stably” (loosely, up to some asymptote-like error term).
Both Projects 2 and 3 involve lots of other geometric structures: for instance, generalisations of braid groups; these 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.
Finally, in Project 4, the 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. One subproject is to study the possible growth-rates of fixed points of transformations for large families of examples. Other subprojects consider possible other dynamical questions, notably about “generic” behaviours, including again for examples from singularity theory.
Due to unforeseen circumstances, the project, which was originally supposed to run for 60 months, was instead terminated after 12 months. As a result, only a subset of the proposed work was performed. This included:
- The construction of new 4D transformations from Project 1, jointly with Paul Hacking, along with establishing mirror symmetric properties. Work is ongoing on later parts of the projects, including higher-dimensional aspects.
- For Projects 2 and 3, only preliminary work was performed (including in-depth exchanges with experts on the other geometric structures involved).
- For Project 4, one objective was largely resolved in a joint paper with Ward (which in fact went beyond what was proposed, drawing on universality ideas in project 3).
- The construction of new 4D transformations from Project 1, jointly with Paul Hacking, along with establishing mirror symmetric properties. Work is ongoing on later parts of the projects, including higher-dimensional aspects.
- For Projects 2 and 3, only preliminary work was performed (including in-depth exchanges with experts on the other geometric structures involved).
- For Project 4, one objective was largely resolved in a joint paper with Ward (which in fact went beyond what was proposed, drawing on universality ideas in project 3).
This is a project in pure mathematics. The impact of pure maths research on areas outside of mathematics is generally acknowledged to be on timescales of a least a decade; this project has no immediate applications outside of maths. In the shorter term, the project has generated significant interest within mathematics, especially the geometry & topology communities. In particular, the project has already provided a range of ways in which symplectic transformations in 4D and above can behave significantly differently to 2D; this contrasts with what we knew previously, and some project results are already being built upon by other research teams.