Combinatorial Methods in Low-Dimensional Topology

拓扑学广义上指的是对形状的研究，而低维拓扑学具体指的是它们在一维到四维中的研究。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，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中的那些完全不同。 组合学是指对离散结构（如网络和流）的研究，它对代码和算法的开发有影响。 组合方法长期以来一直影响着拓扑结构。 在低维拓扑学的各种技术中，有表面相交图、格论方法和极值组合学。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。 这个研究项目的统一目标是推进低维拓扑问题的组合方法。除了研究部分，PI将开展活动，将他的研究兴趣与教育和培训计划相结合，这些计划将覆盖从高中到博士后研究人员的受众。例如，PI通过汉普郡学院暑期数学研究和数学阶梯公司积极参与高中数学课程。在这些项目和其他指导活动的背景下，他试图激发发现过程，并帮助展示美丽的数学。他开办了一所暑期研究生学校，重点关注低维拓扑学的一个中心主题--德恩手术，并正在编辑一本基于该主题的书。他还为《AMS通知》撰写了一篇关于Heegaard Floer同调的调查文章，这是针对专业数学家的最广泛阅读的大众兴趣期刊，他致力于针对广泛的数学受众进行更多解释性工作。该奖项为研究生提供资金支持。PI将继续发展低维拓扑学的组合方法，以延续既定的计划。主要项目集中在研究图形的表面交叉应用到特殊的Dehn手术和嵌入的表面在四个领域;格嵌入和他们的改进应用到研究的合理同源球，切片链接，表面合痕;和极值组合学应用到性能的曲线表面。曲面求交技术更直接，依赖于图论工具的发展，以得出拓扑结论。 弗洛尔同源性方法不那么直接，但适用于重型机械的大量收集的问题。 这个项目将更紧密地结合组合学和低维拓扑。PI目前为三名博士生提供咨询，并刚刚毕业一名学生。 PI在各种主题上运行原始研究生课程，包括高级组合学和结补问题，并将继续这样做。 该奖项将在夏季支持研究生，并协助他们的旅行资金。在博士后阶段，PI指导并与博士后研究员合作。 该奖项将为他们提供旅行资金。在学术推广层面，PI打算开展更多的临时项目，旨在接触和教育更广泛的受众。该奖项反映了NSF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。

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.