Problems in low-dimensional topology

拓扑学泛指对形状的研究，而低维拓扑学专门指从一维到四维的形状研究。从人类学的角度来看，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，它们也是特殊的，因为它们所展示的现象和用于研究它们的技术集合与更高维度的那些相当不同。该项目的研究部分探索了跨低维拓扑的一系列重要问题。一个具体的例子是一个著名的老问题，即平面上的每条连续闭合曲线是否都包含正方形的顶点。贯穿整个研究的一个统一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(经典力学的几何学)。除了这项研究，PI还提出了教育和培训倡议，覆盖了从高中生到专业数学家的受众。PI将通过汉普郡学院数学暑期研究和数学阶梯公司继续积极参与高中阶段的数学充实。PI正在编辑一本基于他经营的低维拓扑结构的热门研究生暑期学校的书。此外，PI目前还为三名博士生提供咨询。该奖项为研究生提供支持，并为学生和博士后研究人员提供旅行支持。PI建议继续现有计划，研究低维拓扑中的一系列问题。主要主题是使用辛方法的peg问题；使用曲面交点图的例外Dehn手术；使用Floer同调和格的有理同调余边法；以及使用经典拓扑方法的带状协调。组合方法和辛方法长期以来一直影响着这个领域。在用于低维拓扑的各种技术中，有Floer同调、曲面交点图和格论方法。每一种技术都在低维拓扑学的主要问题上取得了轰动的进展，它们为这个主题提供了非常不同的视角。这个项目将更紧密地将这些技术和低维拓扑结构联系在一起。这个奖项反映了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。

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.