New Frontiers in Symplectic Topology

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与力学之外的许多领域都有关系，包括规范理论、代数几何、表示理论和弦理论。提议的研究使用辛几何作为这些不同数学领域之间的桥梁。这个提议有三个方面，它们在本质上是截然不同的，但却被辛几何的思想联系在一起。在研究的第一部分中，我们研究了哈密顿动力学和两族几何之间的一个非常奇怪的猜想。在二元几何中，有一类重要的奇异空间称为“复合Du - Val （cDV）奇点”，它出现在Mori著名的用于分类三维代数变量的最小模型规划中。如果你仔细观察这些cDV奇点，你会发现一个自然的动力系统（连杆上的里布流）在例子中，里布流的动力学告诉你是否可以通过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。在研究的第二部分，我们从代数几何中研究了一类四维空间（一般类型的代数曲面）。一般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很好的试验场。最近在理解这些空间如何退化方面有了很大的进展，我们想用它来回答一些关于这些四维空间的长开拓扑问题。在研究的第三部分，我们的目标是用代数几何给出低维流形的拓扑不变量的构造。我们的方法是由同调镜像对称猜想，它将辛几何与代数几何联系起来。

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.