Floer Cohomology and Birational Geometry

从广义上讲，这个项目是关于两个学科之间的关系：代数几何和辛几何。代数几何学研究被称为变种的几何对象，由加法和乘法建立的方程创建。辛几何涉及与经典力学相关的几何对象。我们感兴趣的“拓扑”属性的一些品种，如有多少“洞”，他们有和“枚举”的属性，如数量的曲线，他们包含。PI感兴趣的是成对的品种，它们彼此是双理性的，这意味着它们在去除较小的品种后变得相同。我们希望知道它们有什么共同的拓扑或枚举性质。这个项目的主要目的之一是看看如何从辛几何工具可以用来调查这些问题。PI认为这些想法是新的，可以用来探索其他领域的代数和辛几何使用这个不同的角度。PI将帮助一个名为“海狼数学”的高中生项目，还将帮助斯托尼布鲁克的高中生举办数学夏令营。PI还将帮助为研究生设计的研讨会学习密切相关的研究领域。这个项目的主要目的是了解双有理几何和某些Floer理论不变量之间的关系。 这些不变量包括辛/接触上同调，辛同构的Floer上同调和福谷范畴。PI将使用这些Floer不变量来证明与双有理几何相关的某些陈述。 这个NSF资助的项目的主要目标之一是使用扩展版本的辛上同调给出一个全新的方法来解决crepant分辨率猜想。PI将使用这些方法证明这个猜想的弱版本。PI认为这些方法指向这个猜想的某些概括。PI将使用Floer理论研究终端3重奇点和牛顿非退化奇点。研究这种奇点可能会导致新的见解，就像研究商奇点导致PI研究crepant分辨率猜想一样。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Broadly speaking, this project is about the relationship between two subjects: algebraic geometry and symplectic geometry. Algebraic geometry studies geometric objects called varieties, created by equations that are built from addition and multiplication. Symplectic geometry involves geometric objects related to classical mechanics. We are interested in `topological' properties of some of these varieties such as how many `holes' they have and `enumerative' properties such as the number of curves they contain. The PI is interested in pairs of varieties which are birational to each other, which means that they become identical after removing smaller varieties. We wish to know what topological or enumerative properties they have in common. One of the main aims of this project is to see how tools from symplectic geometry can be used to investigate such issues. The PI believes these ideas are new and can be used to explore other areas of algebraic and symplectic geometry using this different perspective. The PI will help out with a program involving high school students called Seawolf math, and will also help out with a math summer camp for high school students at Stony Brook. The PI will also help out in workshops designed for graduate students learning closely related fields of study.The primary aim of this project is to understand the relationship between birational geometry and certain Floer theoretic invariants. These invariants include symplectic/contact cohomology, Floer cohomology of a symplectomorphism and the Fukaya category. The PI will use these Floer invariants to prove certain statements related to birational geometry. One of the main goals of this NSF funded project is to give a completely new approach to the crepant resolution conjecture using an extended version of symplectic cohomology. The PI will prove a weak version of this conjecture using these methods. The PI believes these methods point to certain generalizations of this conjecture. The PI will study terminal 3-fold singularities and also Newton non-degenerate singularities using Floer theory. Studying such singularities could lead to new insights just as studying quotient singularities led the PI to study the crepant resolution conjecture.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.