Combinatorial Methods in Low-Dimensional Topology

拓扑学泛指对形状的研究，而低维拓扑学具体指的是从一维到四维的研究。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，因为它们展示的现象和用于研究它们的技术集合与更高维度的现象相当不同。组合学指的是对离散结构的研究，如网络和流，它对代码和算法的开发具有影响。组合方法长期以来一直影响着拓扑学。在研究低维拓扑的各种技术中，有曲面交叉图、格子理论方法和极值组合学。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。本研究项目的统一目标是将组合方法推广到低维拓扑问题。除了研究部分，PI还将开展活动，将他的研究兴趣与教育和培训倡议结合起来，覆盖从高中到博士后研究人员的受众。例如，PI通过汉普郡学院数学暑期学习和数学楼梯公司，积极参与高中阶段的数学丰富。在这些计划和其他指导活动的背景下，他试图启发发现过程并帮助展示美丽的数学。他开办了一个研究生暑期班，专注于低维拓扑学中的一个中心主题--德恩手术，目前正在编辑一本以此为基础的书。他还为AMS的通告写了一篇关于Heegaard Floer同源性的调查文章，AMS是一本读者最多的一般兴趣期刊，面向专业数学家，他致力于针对广泛的数学受众进行更多的说明性工作。该奖项为研究生提供资助。PI将在现有计划的基础上，开发低维拓扑学的组合方法。主要项目集中在研究曲面交图在例外Dehn运算和曲面在四面体中的嵌入中的应用；格嵌入及其改进在有理同调球、切片环和曲面同伦研究中的应用；极值组合学在曲面上的曲线性质中的应用。曲面求交技术更加直接，并且依赖于图论工具的发展来得出拓扑结论。Floer同调方法不那么直接，但将重型机械应用于大量问题。这个项目将把组合数学和低维拓扑更紧密地结合在一起。PI目前为三名博士生提供咨询，刚刚毕业一名学生。PI开设了各种主题的原创研究生课程，包括高级组合学和纽结互补问题，并将继续这样做。该奖项将在暑假期间支持研究生，并帮助他们支付旅行费用。在博士后层面，PI指导并与博士后研究员合作。这项奖励将用旅行资金支持他们。在学术推广层面，国际和平协会打算进行更多的说明性项目，目的是接触和教育广泛的受众。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。

Topology refers broadly to the study of shapes, and low-dimensional topology refers specifically to their study in dimensions one through four. These dimensions are special from an anthropic perspective, since they model our everyday perception of the physical world, and 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. Combinatorics refers to the study of discrete structures, such as networks and flows, and it is influential in the development of codes and algorithms. Combinatorial methods have long influenced topology. Amongst the various techniques that come to bear on low-dimensional topology are graphs of surface intersections; lattice-theoretic methods; and extremal combinatorics. Each technique has led to sensational progress on the main problems in low-dimensional topology, and they lend very different perspectives on the subject. The unifying goal of this research project is to advance combinatorial methods towards problems in low-dimensional topology. Alongside the research component, the PI will conduct activities that integrate his research interests with education and training initiatives that reach audiences from the high school level to postdoctoral researchers. For instance, the PI is actively involved with mathematics enrichment at the high school level through the Hampshire College Summer Studies in Mathematics and Mathematical Staircase, Inc. In the context of these programs and in other mentoring activities, he seeks to inspire the discovery process and aid in the exposition of beautiful mathematics. He ran a graduate summer school focused on a central theme in low-dimensional topology, Dehn surgery, and is in the process of editing a book based on it. He has also written a survey article on Heegaard Floer homology for the Notices of the AMS, the most widely-read general interest periodical targeted at professional mathematicians, and he is committed to more expository work aimed at a wide mathematical audience. The award provides funds for supporting graduate students.The PI will develop combinatorial methods in low-dimensional topology, in continuation of an established program. The main projects focus on studying graphs of surface intersections in application to exceptional Dehn surgery and embeddings of surfaces in the four-sphere; lattice embeddings and their refinements in application to the study of rational homology balls, slice links, and surface isotopy; and extremal combinatorics in application to properties of curves on surfaces. The surface intersection techniques are more direct and rely on the development of graph theoretic tools in order to draw topological conclusions. Floer homology methods are less direct but apply heavy machinery to a vast collection of problems. This project will more closely bind combinatorics and low-dimensional topology. The PI currently advises three PhD students and has just graduated one student. The PI runs original graduate courses on various themes, including advanced combinatorics and the knot complement problem, and will continue to do so. This award will support graduate students during summer months and assist them with travel funds. At the postdoctoral level, the PI mentors and collaborates with post-doctoral fellows. This award will support them with travel funds. At the scholarly outreach level, the PI intends to undertake more expository projects with the aim of reaching and educating a wide audience.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.