Categorical Symplectic Topology

这是一个跨学科的建议，通过使用来自代数的复杂的新思想来研究拓扑和动力系统中的经典（和新颖）问题，这些新思想是继量子场论中的“镜像对称”的见解之后发展起来的。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、刚体运动）的周期性轨道。值得注意的是，我们探测这种周期轨道的一些最强大的方法，基本上是利用复分析和“全纯曲线”的偏微分方程，这与肥皂膜等面积最小化表面密切相关。在过去的二十年里，人们已经了解到，对这些特殊曲面的计数给出的数字不是彼此独立的，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。这种见解的各个方面首先出现在量子场论的理论物理学中，通过弦理论中被称为镜像对称的对偶，它可以被视为对麦克斯韦经典电磁对偶的深远推广。镜像对称涉及不同的物理理论，这些理论是自然界中单一结构的模型，但它们是由非常不同的数学类型表面上描述的。这使得在一个领域看起来很自然的见解和结构能够被“带到镜子的另一边”，在那里它们产生强大的新方法和有趣的预测，而这些只是刚刚开始被理解。本奖学金将研究自镜像对称发展而来的拓扑学代数结构，并将其应用于拓扑学和动力学的新问题。这些问题涉及到一个机械系统的对称集的结构，涉及到这种对称的熵和混合特性，涉及到空间中“随机”结的复杂性，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.