Symplectic Topology, Symmetries, and Singularities

这个项目属于纯数学领域称为辛几何。现代几何学的中心是流形的研究，流形是光滑的物体，在足够小的尺度下看起来像一个固定维度的标准空间;例如，球的表面是一个二维流形。辛流形配备了额外的结构，推广了经典力学的守恒定律。另外，物理学的一个分支弦理论中的一些模型允许用任何辛流形来代替时空。这个项目研究的问题：什么是辛流形的变换（即，整体对称性）？特别是，研究者将使用弦论预测的对偶来进一步理解这些变换。本项目旨在使用Floer理论，奇点理论和镜像对称的工具来研究辛流形的对称性。最朴素的这种对称是辛同胚，即保持辛形式的双同胚。这些形成一个拓扑群，其连通分支给出辛映射类群。这推广了黎曼曲面的“通常”映射类群。本计画研究辛映射类群的结构性质，重点是来自奇点理论的四维例子。研究者还将致力于理解代数几何中的镜像现象。辛同构诱导福谷范畴的自同构，这是拉格朗日弗洛尔理论的代数强化。该项目还将考虑辛流形的“隐藏”对称性：不是由辛同构诱导的福谷范畴的自同构，例如C*-作用。

This project pertains to an area of pure mathematics called symplectic geometry. Modern geometry is centered around the study of 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 two-dimensional manifold. Symplectic manifolds are equipped with extra structure that generalizes conservation laws from classical mechanics. Also, some models in string theory, a branch of physics, allow any symplectic manifold in lieu of space-time. This project studies the question: What are the transformations (that is, global symmetries) of a symplectic manifold? In particular, the investigator will use dualities predicted by string theory to further our understanding of these transformations.This project aims to investigate symmetries of symplectic manifolds, using tools from Floer theory, singularity theory, and mirror symmetry. The most naive such symmetries are symplectomorphisms, i.e. diffeomorphisms which preserve the symplectic form. These form a topological group, the connected components of which give the symplectic mapping class group. This generalizes the "usual" mapping class group for Riemann surfaces. This project studies structural properties of symplectic mapping class groups, with a focus on four-dimensional examples coming from singularity theory. The investigator will also aim to understand the mirror phenomena in algebraic geometry. Symplectomorphisms induce automorphisms of the Fukaya category, an algebraic strengthening of Lagrangian Floer theory. The project will also consider "hidden" symmetries of symplectic manifolds: automorphisms of the Fukaya category that are not induced by symplectomorphisms, such as C*-actions.