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Categorical Symplectic Topology

分类辛拓扑

基本信息

批准号：
EP/N01815X/1
负责人：
Ivan Smith
金额：
$ 128.19万
依托单位：
University of Cambridge
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2016
资助国家：
英国
起止时间：
2016 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN01815X%2F1
关键词：
Categorical Symplectic Topology

项目摘要

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.

这是一个跨学科的提案，旨在通过使用代数中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，这些新思想又是根据量子场论中“镜像对称”的见解而发展起来的。动力学的一个基本问题是理解系统的周期轨道（小行星、卫星、流体流动、刚性关节体的运动）。值得注意的是，我们的一些最强大的方法为了检测这种周期性轨道，必须使用复杂分析和“全纯曲线”的偏微分方程，它们与肥皂膜等面积最小化表面密切相关。在过去的二十年里，人们已经认识到，这些特殊曲面的计数给出的数字并不是彼此独立的，而是应该捆绑在一起形成复杂的代数结构，并且满足显着的恒等式。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，通过弦理论中称为镜像对称的对偶性，这可以被视为麦克斯韦经典电磁对偶性的深远概括。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，但表面上是由非常不同类型的数学描述的。这使得在一个领域中看似自然的见解和结构能够被带到“镜子的另一边”，在那里它们产生了强大的新方法和有趣的预测，而这些刚刚开始被理解。该奖学金将研究在镜像对称之后发展起来的拓扑中的代数结构，并将它们应用于拓扑和动力学中的新问题。这些问题与机械系统对称集的结构有关，这种对称性的熵和混合特性，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。

项目成果

期刊论文数量（10）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Fukaya categories of surfaces, spherical objects and mapping class groups

表面、球形物体和映射类组的 Fukaya 类别

DOI：
10.1017/fms.2021.21
发表时间：
2021
期刊：
Sigma
影响因子：
0
作者：
Auroux, Denis;Smith, Ivan
通讯作者：
Smith, Ivan

Khovanov homology from Floer cohomology

Floer 上同调的 Khovanov 同调

DOI：
10.1090/jams/902
发表时间：
2018
期刊：
Journal of the American Mathematical Society
影响因子：
3.9
作者：
Abouzaid M
通讯作者：
Abouzaid M

Bounds on Wahl singularities from symplectic topology

DOI：
10.14231/ag-2020-003
发表时间：
2017-08
期刊：
Algebraic Geometry
影响因子：
1.5
作者：
J. Evans;I. Smith
通讯作者：
J. Evans;I. Smith

Quantitative Heegaard Floer cohomology and the Calabi invariant

DOI：
10.1017/fmp.2022.18
发表时间：
2021-05
期刊：
Forum of Mathematics, Pi
影响因子：
0
作者：
Daniel Cristofaro-Gardiner;Vincent Humilière;C. Mak;Sobhan Seyfaddini;I. Smith
通讯作者：
Daniel Cristofaro-Gardiner;Vincent Humilière;C. Mak;Sobhan Seyfaddini;I. Smith

Symplectomorphisms of exotic discs