New Frontiers in Symplectic Topology

辛拓扑的新领域

• 批准号: EP/W015749/1
• 负责人: Jonathan Evans
• 金额: $ 83.63万
• 依托单位: Lancaster University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 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
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW015749%2F1
• 关键词: New; Frontiers; Symplectic; Topology

Symplectic geometry originated in the study of classical mechanics as a general setting for studying conservative dynamics. Surprisingly, in the last few decades, mathematicians have found symplectic structures are relevant in many areas very far from mechanics, including gauge theory, algebraic geometry, representation theory and string theory. The proposed research uses symplectic geometry as a bridge between some of these disparate areas of mathematics.The proposal has three strands, which are quite distinct in nature, but are tied together by ideas from symplectic geometry.In the first strand of the research, we examine a very curious conjecture at the interface between Hamiltonian dynamics and birational geometry. In birational geometry, there is an important class of singular spaces called "compound Du Val (cDV) singularities" which arise in Mori's famous minimal model program for classifying 3-dimensional algebraic varieties. If you look very close to these cDV singularities, you find a natural class of dynamical systems (Reeb flows on the link) and it seems in examples that the dynamics of the Reeb flow tells you about whether you can resolve the singularity by introducing only 1-dimensional curves. We aim to prove a strong version of this conjecture, first in a simple case (compound A_n) and then in general.In the second strand of the research, we study a class of 4-dimensional spaces from algebraic geometry (algebraic surfaces of general type). Surfaces of general type have very complicated topology, and provide a wonderful testing ground for our understanding of 4-dimensional space. There has been a lot of progress recently in understanding how such spaces can degenerate, and we want to use this to answer some long-open topological questions about these 4-dimensional spaces.In the third strand of the research, our goal is to give a construction of topological invariants of low-dimensional manifolds using algebraic geometry. Our approach is informed by the homological mirror symmetry conjecture which relates symplectic geometry to algebraic geometry.

辛几何起源于经典力学的研究，作为研究保守动力学的一般背景。令人惊讶的是，在过去的几十年里，数学家们发现辛结构在许多远离力学的领域都是相关的，包括规范理论，代数几何，表示论和弦理论。辛几何是数学中的一个重要分支，它是数学中的一个重要分支。辛几何是数学中的一个重要分支，它是数学中的一个重要分支。辛几何是数学中的一个重要分支，它是数学中的一个重要分支。在辛几何的基础上，我们提出了一个新的概念，即在哈密顿动力学和双有理几何之间的一个非常奇怪的猜想。在双有理几何中，有一类重要的奇异空间称为“复合杜瓦尔（cDV）奇异点”，它出现在Mori著名的最小模型程序中，用于分类三维代数簇。如果你仔细观察这些cDV奇点，你会发现一类自然的动力系统（链接上的Reeb流），在例子中，Reeb流的动力学似乎告诉你是否可以通过只引入一维曲线来解决奇点。我们的目标是证明这个猜想的一个强版本，首先在一个简单的情况下（复合A_n），然后在一般情况下.在研究的第二链，我们研究了一类四维空间的代数几何（代数曲面的一般类型）.一般类型的曲面具有非常复杂的拓扑结构，为我们理解四维空间提供了一个极好的实验场。近年来，我们对四维空间如何退化的认识有了很大的进展，我们希望利用这一点来回答一些关于四维空间的长期开放的拓扑问题。在研究的第三个方面，我们的目标是利用代数几何来构造低维流形的拓扑不变量。我们的方法是通知同调镜像对称猜想，涉及辛几何代数几何。