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.

拓扑泛指对形状的研究，低维拓扑特指对一维到四维形状的研究。从人类的角度来看，这些维度是特殊的，因为它们模拟了我们对物质世界的日常感知。 从数学角度来看，它们也很特别，因为它们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现象有很大不同。 该项目的研究部分探索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，该问题询问平面中的每条连续闭合曲线是否都包含正方形的顶点。研究的一个统一主线是使用邻近领域的现代方法，例如组合数学（离散结构的数学）和辛几何（经典力学的几何）。 除了研究之外，PI 还提出了教育和培训计划，覆盖从高中生到专业数学家的受众。 PI 将继续通过汉普郡学院数学暑期研究和 Mathematical Staircase, Inc 积极参与高中阶段的数学强化活动。PI 正在编辑一本基于低年级热门研究生暑期学校的书。他运行的维度拓扑。 此外，PI目前还为三名博士生提供指导。该奖项为研究生和博士后研究人员提供研究生支持和旅行支持。PI 提议研究低维拓扑中的一系列问题，以延续既定计划。 主要主题是使用辛方法的挂钩问题；出色的 Dehn 手术，使用表面交叉图；有理同调共边，使用Floer同调和格；和带状索引，使用经典的拓扑方法。 组合和辛方法长期以来影响着该领域。 适用于低维拓扑的各种技术包括Floer 同调、表面相交图和晶格理论方法。 每种技术都在低维拓扑的主要问题上取得了轰动的进展，并且它们为该主题提供了截然不同的视角。 该项目将更紧密地结合这些技术和低维拓扑。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。