Floer for Three: Symplectic Methods in Low-Dimensional Topology

这项拨款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、3维空间和4维空间的理解。不同维度之间的关系以两种方式产生。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；分解沿着较低维的空间进行，比如表面。这些都是用扁平的二维刀具切割三维西瓜这一原理的复杂版本。试图做到这一点会让人问，低维空间可以以何种方式位于高维空间中。例如，一维鞋带可以打结也可以解开；最近发现，坐在四维空间中的三维西瓜也可以打结。这涉及到发展微分方程和抽象代数的各个方面。该研究项目的一个长期目标是拥有一个计算机程序，该程序可以计算4维空间的某些微妙不变量，这些不变量来自于理论物理中的微分方程式--塞伯格-威腾方程。这笔赠款还将支持对研究生进行这些主题的培训，撰写一本书，向研究生和高级本科生介绍他们中的一些人，开展推广活动，与K-12学生分享几何和拓扑的兴奋，以及建立一个网站，帮助其他研究人员使用非照片真实感光线跟踪在他们自己的论文中绘制有用的图形。其一是进一步发展边界Heegaard Floer同调的扩展到Heegaard Floer同调的“减号”版本。带边Heegaard Floer同调是具有边界的3-流形的Heegaard Floer同调的一个版本；该理论的许多方面目前仅被定义为Heegaard Floer同调的更简单的“帽子”版本。另一类是利用Khovanov同调的变体，发展了4维空间中嵌入的不可定向曲面的新的不变量。第三项是将Floer同伦理论应用于等变三维拓扑中的问题，第四项是研究Floer同调与曲面映射类群之间的关系。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。

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.