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。代数几何研究的几何对象被称为变化，由加法和乘法建立的方程产生。辛几何涉及与经典力学有关的几何对象。我们感兴趣的是其中一些品种的“拓扑”性质，比如它们有多少“洞”，以及“枚举”性质，比如它们包含的曲线数量。PI感兴趣的是一对彼此有亲缘关系的品种，这意味着它们在去掉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了解如何使用辛几何中的工具来研究这些问题。PI相信这些想法是新的，可以用这种不同的视角来探索代数和辛几何的其他领域。PI将帮助一个涉及高中生的名为海狼数学的项目，还将帮助石溪高中学生的数学夏令营。PI还将在为研究生学习密切相关的研究领域而设计的研讨会上提供帮助。本计画的主要目的是了解两族几何与某些花理论不变量之间的关系。这些不变量包括辛/接触上同调、辛同态的花上同调和Fukaya范畴。PI将使用这些花不变量来证明与两族几何有关的某些陈述。这个由国家科学基金会资助的项目的主要目标之一是使用辛上同的扩展版本给出一种全新的方法来解决蠕变分辨猜想。PI将使用这些方法证明这个猜想的弱版本。PI相信这些方法指向了这个猜想的某些推广。PI将使用Floer理论研究终端3倍奇点和牛顿非退化奇点。研究这样的奇点可能会带来新的见解，就像研究商奇点导致PI研究蠕变分辨率猜想一样。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.