Floer for Three: Symplectic Methods in Low-Dimensional Topology

这项资助下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。研究复杂的三维或四维空间的一种方法是将它们分解成更简单的部分；分解发生在低维空间，比如表面。这是用一把平面的二维刀切三维西瓜原理的复杂版本。尝试这样做会导致一个问题，以何种方式低维空间可以位于高维空间中。例如，一维的鞋带可以打结或不打结；最近发现，一个三维空间的西瓜放在四个空间里也可以打结。这涉及发展微分方程和抽象代数的各个方面。这个研究项目的一个长期目标是有一个计算机程序，可以通过计算理论物理学中微分方程的解来计算四维空间的某些微妙的不变量，即Seiberg-Witten方程。拨款还将支持培训这些主题的研究生，写一本书向研究生和高级本科生介绍其中的一些内容，与K-12学生分享几何学和拓扑学的推广活动，以及帮助其他研究人员使用非真实感光线追踪在他们自己的论文中绘制有用图形的网站。这项研究涉及若干具体项目。一是进一步发展有边Heegaard flower同调的“负”版Heegaard flower同调的扩展。有边Heegaard花同调是具有边界的3流形Heegaard花同调的一个版本；该理论的许多方面目前只定义了heeggaard flower同调的简单“帽子”版本。另一种方法是利用Khovanov同调上的变异体，在4空间中构造嵌入的非定向曲面的不变量。第三是将Floer同伦理论应用于等变三维拓扑问题，第四是研究Floer同伦与曲面映射类群之间的关系。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。

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.