Floer for Three: Symplectic Methods in Low-Dimensional Topology

根据这项补助金的研究，通过关注这些不同维度之间的相互作用，进一步加深了我们对曲线，曲面，三维空间和四维空间的理解。不同维度之间的关系以两种方式出现。研究复杂的3维或4维空间的一种方法是将它们分解成更简单的部分;分解发生在沿着较低维的空间，如曲面。这些是复杂的版本的原则，你切一个三维的西瓜，用一个平面，二维刀。尝试这样做会导致人们问，低维空间可以以何种方式位于高维空间内。例如，一维鞋带可以打结或解开;最近发现，位于4空间的三维西瓜也可以打结。这涉及微分方程和抽象代数的发展方面。该研究项目的一个长期目标是拥有一个计算机程序，可以计算来自理论物理学微分方程（Seiberg-Witten方程）的某些微妙的四维空间不变量。该基金还将资助研究生在这些主题方面的培训，编写一本书，向研究生和高年级本科生介绍其中一些人，开展外联活动，与K-12学生分享几何和拓扑学的兴奋，并建立一个网站，帮助其他研究人员使用非真实感光线追踪在自己的论文中绘制有用的图形。一个是进一步发展扩展的边界Heegaard Floer同源性的“减”版本的Heegaard Floer同源性。加边Heegaard Floer同调是Heegaard Floer同调的一个版本，适用于有边界的三维流形;该理论的许多方面目前仅定义为Heegaard Floer同调的简单“帽子”版本。另一个是利用Khovanov同调的变式，在4-空间中建立嵌入不可定向曲面的新的不变量。第三个是应用弗洛尔同伦理论的问题，在等变三维拓扑结构，第四个是研究弗洛尔同源性和映射类groups的surfaces.This奖项之间的关系反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

Research under this grant furthers our understanding of curves, surfaces, 3-dimensional spaces, and 4-dimensional spaces, by focusing on the interplay between these different dimensions. Relationships between different dimensions arise in two ways. One way to study complicated 3- or 4-dimensional spaces is to decompose them into simpler pieces; the decomposition happens along lower-dimensional spaces, like surfaces. These are complicated versions of the principle that you cut a 3-dimensional watermelon by using a flat, 2-dimensional knife. Trying to do this leads one to ask in which ways low-dimensional spaces can sit inside high-dimensional ones. For example, 1-dimensional shoelaces can be knotted or unknotted; it was recently discovered that a 3-dimensional watermelon sitting in 4-space can also be knotted. This involves developing aspects of differential equations and abstract algebra. One long-term goal of the research project is to have a computer program that can compute certain subtle invariants of 4-dimensional spaces coming from counting solutions to differential equations from theoretical physics, the Seiberg-Witten equations. The grant will also support training graduate students in these topics, writing a book introducing some of them to graduate students and advanced undergraduates, outreach activities to share the excitement of geometry and topology with K-12 students, and a website to help other researchers use non-photorealistic raytracing to draw useful figures in their own papers.The research involves a number of specific projects. One is to further develop an extension of bordered Heegaard Floer homology to the "minus" version of Heegaard Floer homology. Bordered Heegaard Floer homology is a version of Heegaard Floer homology for 3-manifolds with boundary; many aspects of the theory are currently only defined for the simpler "hat" version of Heegaard Floer homology. Another is to develop new invariants of embedded non-orientable surfaces in 4-space, using variants on Khovanov homology. A third is to apply Floer homotopy theory to questions in equivariant 3-dimensional topology, and a fourth is to study relationships between Floer homology and the mapping class groups of surfaces.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.