Problems in low-dimensional topology

拓扑学广义上指的是对形状的研究，低维拓扑学具体指的是对一维到四维形状的研究。从人择学的角度来看，这些维度是特别的，因为它们模拟了我们对物理世界的日常感知。 从数学的角度来看，它们也很特别，因为它们所表现出的现象以及用于研究它们的技术集合与更高维度的那些完全不同。 该项目的研究部分探讨了低维拓扑中的一系列重要问题。 一个具体的例子是一个著名的老问题，它问是否每个连续的封闭曲线在平面上包含一个正方形的顶点。贯穿研究的一条统一的主线是使用邻近领域的现代方法，如组合学（离散结构的数学）和辛几何（经典力学的几何）。 除了研究，PI还提出了教育和培训计划，从高中生到专业数学家都有受众。 PI将继续积极参与数学丰富的高中水平，通过汉普郡大学暑期数学研究和数学楼梯公司。 PI正在编辑一本书的过程中，这本书是基于他经营的低维拓扑学研究生暑期学校。 此外，PI目前为三名博士生提供咨询。该奖项为学生和博士后研究人员提供研究生支持和差旅支持。PI提议研究低维拓扑中的一系列问题，以延续既定计划。 主要主题是挂钩问题，使用辛方法;例外Dehn手术，使用图形的表面交叉;合理的同源配边，使用Floer同源和格;和丝带和谐，使用经典的拓扑方法。 组合和辛方法长期以来一直影响着该领域。 在低维拓扑学中涉及到的各种技术有弗洛尔同调、曲面相交图和格论方法。 每种技术都在低维拓扑学的主要问题上取得了令人瞩目的进展，并且它们在这个问题上提供了非常不同的观点。 该项目将更紧密地结合这些技术和低维拓扑结构。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Topology refers broadly to the study of shapes, and low-dimensional topology refers specifically to the study of shapes in dimensions one through four. These dimensions are special from an anthropic perspective, since they model our everyday perception of the physical world. They are also special from a mathematical perspective, since the phenomena they exhibit, and the collection of techniques used to study them are rather different from those in higher dimensions. The research component of the project explores a collection of important problems from across low-dimensional topology. A concrete example is a famous old problem which asks whether every continuous closed curve in the plane contains the vertices of a square. A unifying thread through the research is the use of modern methods from nearby fields, such as combinatorics (the mathematics of discrete structures) and symplectic geometry (the geometry of classical mechanics). Alongside the research, the PI proposes education and training initiatives reaching audiences from high schoolers to professional mathematicians. The PI will continue his active involvement with mathematics enrichment at the high school level through the Hampshire College Summer Studies in Mathematics and through Mathematical Staircase, Inc. The PI is in the process of editing a book based on a popular graduate summer school in low-dimensional topology that he ran. Moreover, the PI currently advises three PhD students. The award provides graduate student support and travel support for students and postdoctoral researchers.The PI proposes to study a collection of problems in low-dimensional topology, in continuation of an established program. The main themes are peg problems, using symplectic methods; exceptional Dehn surgery, using graphs of surface intersections; rational homology cobordism, using Floer homology and lattices; and ribbon concordance, using classical topological methods. Combinatorial and symplectic methods have long influenced the field. Amongst the various techniques that come to bear on low-dimensional topology are Floer homology, graphs of surface intersections, and lattice-theoretic methods. Each technique has led to sensational progress on the main problems in low-dimensional topology, and they lend very different perspectives on the subject. This project will more closely bind these techniques and low-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.