Categorical Symplectic Topology

这是一个学科内部的提议，利用代数中复杂的新概念来研究拓扑学和动力系统中的经典(和新的)问题，这些新概念是根据量子场论中的“镜像对称性”而发展起来的。动力学中的一个基本问题是理解系统(小行星、卫星、流体流动、刚性连接体的运动)的周期轨道。值得注意的是，我们一些最强大的检测周期轨道的方法基本使用了复数分析，以及`全纯曲线‘的偏微分方程式，这与肥皂片等面积最小的表面密切相关。在过去的二十年里，人们已经认识到，这些特殊曲面的计数给出的数不是彼此独立的，而是应该捆绑在一起形成复杂的代数结构，并且满足显著的恒等式。这种洞见的某些方面首先出现在量子场论的理论物理中，通过弦理论中的对偶性被称为镜像对称性，这可以被视为麦克斯韦经典电磁对偶性的深远推广。镜像对称性将不同的物理理论联系在一起，这些理论是自然界中单一结构的模型，但表面上是由非常不同的数学类型描述的。这使得在一个领域看似自然的洞见和结构能够被带到镜子的另一边，在那里它们产生了强大的新方法，以及刚刚开始被理解的有趣的预测。本奖学金将研究在镜像对称之后发展起来的拓扑学中的代数结构，并将它们应用于拓扑学和动力学中的新问题。这些问题涉及机械系统的对称组的结构，这种对称性的熵和混合性质，空间中`随机‘结点的复杂性，以及多边形桌上台球游戏的周期轨道问题。

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.