Singularities, symplectic topology and mirror symmetry

奇点、辛拓扑和镜像对称

基本信息

  • 批准号:
    EP/W001780/1
  • 负责人:
  • 金额:
    $ 92.03万
  • 依托单位:
  • 依托单位国家:
    英国
  • 项目类别:
    Fellowship
  • 财政年份:
    2022
  • 资助国家:
    英国
  • 起止时间:
    2022 至 无数据
  • 项目状态:
    未结题

项目摘要

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.
辛几何是一个迅速发展的领域,其工具来自许多不同的数学领域。现代几何学研究流形,在足够小的尺度上看起来像固定维度的标准空间的平滑对象。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,该结构概括了经典力学中的守恒定律。这使得它们成为研究卫星或空间探测器轨道的自然正式框架。此外,作为物理学的一个分支,弦理论中的一些模型允许任何辛流形来代替时空。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,它研究多项式方程的几个变量的解。这个项目是由一个重大的悬而未决的问题指导的: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移到另一个点的规则,该规则是平滑的(没有中断)、可逆的(可以倒退),并且保留了额外的对称性。我们不能很好地理解辛变换:对于很多空间来说,一个真正的来源是一种叫做德恩扭曲的东西。让我描述一下2D曲面的这些特性。(如果2D曲面具有方向,则它们是辛的:球或内管的曲面是辛的,而Mobius带的曲面不是。)从没有自交点的闭合曲线开始-例如,围绕内管较薄部分的圆。沿着它切割曲面:内管现在是一个很长的环形,有两个边界组件,每个组件都是一个圆。将每个边界向右旋转180度,然后再次将边缘粘合在一起。你得到了相同的表面回来了!这种转变是德恩的转折。曲面上的圆是一维球体,一般来说,我们可以通过在辛流形中使用更高维的球体来类似地定义高维Dehn扭曲-例如,四维辛流形中通常的球体(球的表面)的副本。在2D中,所有变换都可以分解为扭曲序列。该项目的一个主要目标是通过构建受镜像对称启发的新变换大家族,来证明更高维的情况可以是完全不同的。这些转化成了代数几何世界中的一种不同类型的变换,我们建议解决独立感兴趣的问题。一个长期目标是比较曲面变换的动力学性质与高维的变换。例如,曲面上的Dehn扭曲具有线性动力学:固定点的数量随着迭代而线性增长。然而,一种被称为伪Anosov映射的通用曲面变换具有指数动力学。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。该项目中将研究的许多物体都是在奇点理论中自然产生的,奇点理论是一个与数学中解释不连续性和突变的部分有关的领域--例如,当光线穿过水时出现的尖锐焦散曲线。我们还建议使用辛几何的思想来研究关于广义焦散的变形空间的经典结构问题。许多其他几何结构也参与了该项目:例如,辫子群,它是你可以用头发或丝带编成的辫子的数学形式化;以及科克塞特群,它是空间的变换,你可以从(物理的,光反射的)镜子的配置中反射得到的空间变换。

项目成果

期刊论文数量(1)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Symplectomorphisms and spherical objects in the conifold smoothing
圆锥形平滑中的辛同胚和球形物体
  • DOI:
  • 发表时间:
    2023
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Keating AM
  • 通讯作者:
    Keating AM
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Ailsa Keating其他文献

On symplectic stabilisations and mapping classes
关于辛稳定性和映射类
Symplectic properties of Milnor fibres
Milnor 纤维的辛性质
  • DOI:
  • 发表时间:
    2014
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Ailsa Keating
  • 通讯作者:
    Ailsa Keating
Lagrangian tori in four-dimensional Milnor fibres
四维 Milnor 纤维中的拉格朗日环面
On the order of Dehn twists
按照 Dehn 曲折的顺序
  • DOI:
  • 发表时间:
    2021
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Ailsa Keating;O. Randal
  • 通讯作者:
    O. Randal
Dehn twists and free subgroups of symplectic mapping class groups
Dehn 扭曲和辛映射类群的自由子群
  • DOI:
  • 发表时间:
    2012
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Ailsa Keating
  • 通讯作者:
    Ailsa Keating

Ailsa Keating的其他文献

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