权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Higher Structure in Low-Dimensional Floer Theories

低维Floor理论中的高级结构

基本信息

批准号：
1810893
负责人：
Robert Lipshitz
金额：
$ 23万
依托单位：
University of Oregon Eugene
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：
2018-09-01 至 2022-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1810893&HistoricalAwards=false
关键词：
Higher Structure Low Dimensional Floer

项目摘要

Humanity has been interested in the geometry of three-dimensional spaces since long before recorded history; since Einstein's work on general relativity, the geometry of four-dimensional spaces has also been of central importance, with applications now as ubiquitous as the global positioning system. Mathematicians have divided the study of spaces into two parts: qualitative questions (like whether a space has a hole in it), which is the field of topology, and quantitative questions (like the diameter of the hole), which is the field of geometry. Most, though not all, applications require quantitative answers, but before one can start on quantitative answers---the geometry---one needs to understand the qualitative behavior---low-dimensional topology, the focus of this National Science Foundation funded project. Indeed, in four-dimensions, even many of the most basic topological questions remain unanswered---questions about how many different simple spaces there are, how spaces can be cut up into simpler pieces, and what kinds of surfaces one can fit in given four-dimensional spaces. By using relations with other parts of mathematics and high-energy physics, this proposal seeks to advance human understanding by developing tools to answer some of these basic questions. This project, which focuses on applications of symplectic topology, algebraic topology, and representation theory to low-dimensional topology, has four interrelated goals. The first goal is to extend Lipshitz-Ozsváth-Thurston's "bordered Heegaard Floer homology" invariant of three-manifolds with boundary from the "hat" variant to the richer "minus" variant, for three-manifolds with torus boundary. The second goal is to prove a higher naturality result for Heegaard Floer homology, giving a functor from an appropriate quasi-category of decorated cobordisms to the quasi-category of chain complexes. The third goal is to prove new localization results relating the Heegaard Floer homology of a space with the Heegaard Floer homology of its branched covers. The fourth is to extend Lipshitz-Sarkar's stable homotopy refinement of Khovanov homology to cobordisms between knots and to other variants of Khovanov homology, including Lee homology. These tools are expected to have applications to concordance, homology cobordism, and other problems at the interface of three- and four-dimensional topology.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.

早在有历史记载之前，人类就对三维空间的几何学感兴趣；自爱因斯坦的广义相对论工作以来，四维空间的几何学也发挥了核心作用，现在的应用与全球定位系统一样无处不在。数学家将对空间的研究分为两部分：定性问题(如空间中是否有洞)是拓扑学领域，定量问题(如洞的直径)是几何学领域。虽然不是全部，但大多数应用需要定量的答案，但在开始定量答案之前-几何--需要理解定性的行为-低维拓扑，这是国家科学基金会资助的项目的重点。事实上，在四维空间中，甚至许多最基本的拓扑问题仍然没有得到回答-关于有多少不同的简单空间，如何将空间分割成更简单的部分，以及在给定的四维空间中可以容纳什么样的表面的问题。通过利用与数学和高能物理的其他部分的关系，这项提议寻求通过开发工具来回答其中一些基本问题来促进人类的理解。这个项目专注于辛拓扑、代数拓扑学和表示论在低维拓扑学中的应用，有四个相互关联的目标。第一个目标是将Lipshitz-Ozsváth-瑟斯顿的具有边界的三维流形的“有边Heegaard Floer同调”不变量从“帽子”变量推广到更丰富的“负”变量，用于具有环面边界的三维流形。第二个目标是证明Heegaard Floer同调的一个更高的自然性结果，即给出一个从适当的装饰余边范畴到链复合体的拟范畴的函子。第三个目标是证明一个空间的Heegaard Floer同调与它的分支覆盖的Heegaard Floer同调有关的新的局部化结果。四是将Lipshitz-Sarkar对Khovanov同调的稳定同伦精化推广到纽结之间的余弦和Khovanov同调的其他变体，包括Lee同调。这些工具有望在三维和四维拓扑的界面上应用于协调、同源协边和其他问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。