Contact Geometry of Links of Singularities and Affine Varieties

这个项目的目的是启动一个计划，在几何的两个不同区域之间建立联系。第一个区域称为辛几何，第二个区域称为代数几何。辛几何涉及对经典物理系统的研究，例如钟摆、运动粒子，以及可能有许多运动部件的其他装置。它提供了一种很好的方式来统一描述这些系统的位置和速度。代数几何是研究某些称为多项式方程的基本方程的解的学科。这些方程被用于数学的许多领域，甚至在数学之外，如物理和计算机科学。这些方程的解有一个形状，在某些地方是一个很好的光滑物体，但也可以有不光滑的区域称为奇点。在这个项目中，我们将探索这些奇点与某些经典物理系统之间的深层联系。奇点的许多性质在这些物理系统中有相应的纯动力学解释，这将在目前知之甚少的情况下进一步探索。由于最小模型程序的最新进展，奇点的研究在目前的代数几何中特别重要。这个项目的一部分还包括教授研究生如何使用辛几何中的各种基本工具。这类工具是本节目的特色。本项目的主要目的是了解辛几何/接触几何和仿射代数几何之间的关系。这种关系将使用伪全纯方法来研究，这些方法最初是由Gromov引入的，现在已经发展成为重要的工具，如Gromov-Witten不变量、Floer同调和辛场理论。芒福德给了我们一个判据来告诉我们，一个复杂的曲面在特定的点上是否光滑，从它的链接来看。PI证明了Mumford结果的更高维度版本，其中链接现在具有自然的接触结构。这项提议的部分目的是启动一个程序，将高维奇点的代数性质与它们作为接触流形的联系联系起来，就像芒福德的论文对曲面奇点所做的那样。这项工作的大部分也与最小模型程序的最新进展直接相关。作为一个起点，我们希望通过Neumann给我们一个来自链路的分解的拓扑，来推广至少部分结果。在一个相关的项目中，我们希望利用Floer理论的思想来理解商奇点与初解的McKay对应。这可能导致我们将这种对应关系扩展到没有确定解的商奇点，甚至到更一般的奇点。PI将给出许多不与奇点链环接触的接触流形的例子。PI还将从辛的角度来观察仿射簇，试图理解诸如对数Kodaira维和有理连通性之类的性质如何与辛结构相关。这个项目还包括教研究生如何在不同的情况下应用伪全纯方法。

The aim of this project is to start a program establishing a connection between two different areas of geometry. The first area is called symplectic geometry and the second area is called algebraic geometry. Symplectic geometry involves the study of classical physical systems, for instance pendulums, moving particles, other devices with possibly many moving parts. It gives a nice way of describing the position and velocity of these systems in a unified way. Algebraic geometry is the study of solutions of certain fundamental equations called polynomial equations. These equations are used in many areas of mathematics, and even outside of mathematics, such as physics and computer science. The solutions to these equations have a shape, which in some places is a nice smooth object, but also can have regions which are not smooth called singular points. In this project, we explore a deep relationship between these singular points and certain families of classical physical systems. Many properties of singularities have corresponding purely dynamical interpretations in these physical systems which will be explored further as very little is currently known. The study of singularities is especially important in algebraic geometry at the moment due to recent advances in an program called the minimal model program. Part of this project also involves teaching graduate students how to use various fundamental tools from symplectic geometry. Such tools are featured in this program. The primary aim of this project is to understand the relationship between symplectic/contact geometry and affine algebraic geometry. This relationship will be studied using pseudo-holomorphic methods that were originally introduced by Gromov and have now developed into important tools such as Gromov-Witten invariants, Floer homology, and Symplectic Field Theory. Mumford gave us a criterion to tell us when a complex surface is smooth at a particular point or not from its link. The PI proved a higher dimensional version of Mumford's result where the link now has a natural contact structure. Part of the aim of this proposal is to start a program relating the algebraic properties of higher dimensional singularities with their links as contact manifolds in the same way that Mumford's paper did so for surface singularities. Much of this work is also directly related to recent advances in the minimal model program. As a starting point we hope to generalize at least parts of a result by Neumann giving us the topology of a resolution from the link. In a related project, we hope to understand the McKay correspondence for quotient singularities with crepant resolutions using Floer theoretic ideas. This might lead us to extend this correspondence to quotient singularities that do not have crepant resolutions, and even to more general singularities. The PI will give many examples of contact manifolds not contactomorphic to links of singularities. The PI will also look at affine varieties from a symplectic perspective by trying to understand how properties such as log Kodaira dimension and rational connectedness relate to the symplectic structure. This project also involves teaching graduate students how to apply pseudo-holomorphic methods in various situations.