Floer Cohomology and Birational Geometry

弗洛尔上同调和双有理几何

• 批准号: 1811861
• 负责人: Mark McLean
• 金额: $ 23.92万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-06-01 至 2022-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811861&HistoricalAwards=false
• 关键词: Floer; Cohomology; Birational; Geometry

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.

从广义上讲，这个项目是关于两个主题之间的关系：代数几何和符号几何形状。代数几何研究几何对象称为品种，由添加和乘法构建的方程式创建。符号几何形状涉及与经典力学有关的几何对象。我们对其中一些品种的“拓扑”特性感兴趣，例如它们具有多少个“孔”以及“枚举”特性，例如它们所包含的曲线数量。 PI对彼此不同的品种兴趣感兴趣，这意味着它们在去除较小品种后变得相同。我们希望知道它们具有什么共同点。该项目的主要目的之一是查看如何使用来自符号几何形状的工具来调查此类问题。 PI认为这些想法是新的，可以用来使用这种不同的观点来探索代数和象征性几何的其他领域。 PI将为一项名为Seawolf Math的高中生的计划提供帮助，还将为Stony Brook的高中学生提供数学夏令营。 PI还将在为研究生学习密切相关的研究领域的研讨会上提供帮助。该项目的主要目的是了解Birational几何形状与某些浮标理论不变的关系之间的关系。 这些不变的人包括符号/接触共同体，符号切除型的浮子共同体和福卡亚类别。 PI将使用这些浮标不变式来证明与Birational几何相关的某些陈述。 该NSF资助的项目的主要目标之一是使用Symblectic Cromology的扩展版本为毛茸茸的解决方案提供一种全新的方法。 PI将使用这些方法证明该猜想的弱版本。 PI认为这些方法指出了这种猜想的某些概括。 PI将使用浮点理论研究终端3倍的奇异点，并研究牛顿非分类奇异性。研究这种奇异性可能会导致新的见解，就像研究商的奇异性导致PI研究了Crepant解决方案的猜想一样。该奖项反映了NSF的法定任务，并且被认为是值得通过基金会的智力优点和更广泛影响的审查标准通过评估来获得支持的。