Singularities, symplectic topology and mirror symmetry

辛几何是一个快速发展的领域，其工具来自许多不同的数学领域。现代几何研究流形，在足够小的尺度上看起来像一个固定维度的标准空间的光滑物体。例如，球的表面是2d流形，标准时空是4d流形，生物实验的参数空间可能是18d流形。辛流形配备了一个额外的结构，从经典力学推广守恒定律。这使它们成为研究卫星或太空探测器轨道的自然形式框架。此外，弦理论（物理学的一个分支）中的一些模型允许用任何辛流形代替时空。物理学中的对偶思想导致了镜像对称，这是一个蓬勃发展的领域，它将辛几何与数学中一个非常不同的部分联系起来：代数几何，它研究多个变量多项式方程的解。这个项目是由一个主要的开放问题引导的：“辛流形的变换（即全局对称）是什么？”通过变换，我们指的是将每个点转换为另一个点的规则，该规则是平滑的（没有中断），可逆的（可以向后走），并保留额外的对称性。我们不太了解辛变换：对于很多空间来说，一个真正的来源是一种叫做Dehn扭曲的东西。我来描述一下二维曲面。（二维表面如果有方向就是辛的：球或内管的表面有方向，而莫比乌斯带没有。）从一条没有自交的封闭曲线开始——例如，一个围绕内管薄部分的圆。沿着它切开表面：内管现在是一个长环，有两个边界组件，每个都是一个圆。将每个边界向右扭转180度，并再次将边缘粘合在一起。你得到了同样的表面！这种转变是Dehn的转折。表面上的圆是一维球体，一般来说，我们可以在高维中类似地定义Dehn扭曲，通过在辛流形中使用高维球体-例如，在四维辛流形中复制通常的球体（球的表面）。在2D中，所有的转换都可以分解成一系列的扭曲。该项目的一个主要目标是通过构建大量新的变换实例，在镜像对称的启发下，展示高维情况可以完全不同。这些转化为代数几何世界中另一种不同的变换，在代数几何中，我们建议解决独立感兴趣的问题。一个长期的目标是比较曲面变换与高维曲面变换的动力学性质。例如，表面上的Dehn扭曲具有线性动力学：固定点的数量随着迭代线性增长。然而，一般的曲面变换，称为伪anosov映射，具有指数动态。对于大族的例子，我们将研究变换不动点的可能增长率，以及是否存在一般行为。该项目将研究的许多对象都是在奇点理论中自然产生的，奇点理论是一个与解释不连续和突变的数学部分相关的领域——例如，当光穿过水时出现的尖角苛性曲线。我们还提出利用辛几何的思想来研究广义焦散变形空间的经典结构问题。许多其他几何结构也进入了这个项目：例如，辫子组，这是你可以用头发或丝带编成的辫子的数学形式化；Coxeter群，它们是空间变换的推广，你可以从（物理的，反射光的）镜子的构型反射中获得。

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.