Problems in low-dimensional topology

低维拓扑问题

• 批准号: 2304856
• 负责人: Joshua Greene
• 金额: $ 43.87万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2304856&HistoricalAwards=false
• 关键词: Problems; low; dimensional; topology

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.

拓扑学泛指对形状的研究，而低维拓扑学专门指对1到4维形状的研究。从人类的角度来看，这些维度是特别的，因为它们模拟了我们对物质世界的日常感知。从数学的角度来看，它们也很特别，因为它们所展示的现象，以及用于研究它们的技术集合，与高维的现象大不相同。该项目的研究部分探索了一系列跨低维拓扑的重要问题。一个具体的例子是一个著名的老问题，它问平面上的每条连续封闭曲线是否包含一个正方形的顶点。贯穿整个研究的一条统一线索是使用来自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）。在进行研究的同时，PI还提出了教育和培训计划，目标受众从高中生到专业数学家。PI将通过汉普郡学院夏季数学研究和数学阶梯公司继续积极参与高中水平的数学丰富。PI正在根据他开办的一个受欢迎的低维拓扑学研究生暑期学校编辑一本书。此外，PI目前有3名博士生。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提议研究低维拓扑中的一系列问题，作为既定计划的延续。主要的主题是钉住问题，使用辛方法；特殊的Dehn手术，使用曲面相交图；利用flower同调和格的有理同调配；并采用经典拓扑方法进行条带一致性分析。组合方法和辛方法长期影响着这一领域。研究低维拓扑的各种技术包括花同调、曲面交点图和格理论方法。每一种技术都在低维拓扑的主要问题上取得了令人瞩目的进展，它们为这个主题提供了非常不同的视角。本项目将更紧密地将这些技术与低维拓扑结合起来。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。