CAREER: Branched Covers in Dimensions Three and Four

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。拓扑学是数学的一个领域，应用于许多领域，从蛋白质和DNA的打结到我们宇宙的结构。该项目关注三维和四维空间的拓扑结构，或流形，以及它们包含的打结物体。要对流形进行分类，就要研究它们之间的关系；其中一种关系是分支盖的关系。PI将开发组合和计算工具来研究三维和四维流形的分支覆盖。利用这些工具，PI将为具有各种数学背景和专业目标的本科生和学士后学生设计子项目，使该领域的前沿问题易于访问。该项目的教育部分扩展并支持PI目前在增加女性和代表性不足群体获得数学科学的项目中的工作，并通过为学业中断的学生提供支持来解决COVID对数学管道的影响。对此至关重要的是，PI在史密斯学院的女性数学后学士学位项目中心继续发挥领导作用，并为本科生研究人员组织会议。这些活动通过为学生在数学科学领域的研究生学习做准备，扩大了学生在该领域的参与。该项目探讨了三流形和四流形分支盖的地理和植物学问题，特别是当配备额外的几何结构时。地理问题要求在给定流形的覆盖程度或分支集的复杂性的约束下，哪些流形作为分支覆盖出现；植物学问题要求对给定的流形对之间的分支覆盖映射进行分类。一个关键的策略是开发组合和图解方法来计算分别由三流形和四流形分支覆盖导出的结和面的不变量。应用将包括解决几个活跃领域的开放问题，包括四流形的三切分，结一致性和接触拓扑。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。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Topology is an area of mathematics with applications to many fields, from the knotting of proteins and DNA to the structure of our universe. This project concerns the topology of three- and four-dimensional spaces, or manifolds, as well as the knotted objects they contain. To classify manifolds, one studies relationships between them; one such relationship is that of a branched cover. The PI will develop combinatorial and computational tools to study branched covers of three- and four-dimensional manifolds. Using these tools to make cutting-edge problems in the field accessible, the PI will design subprojects for undergraduate and post-baccalaureate students with a wide variety of mathematical backgrounds and professional goals. The educational component of this project expands and supports the PI’s current work in programs increasing access for women and underrepresented groups in the mathematical sciences, and addresses COVID impacts on the mathematical pipeline by providing support for students whose studies were disrupted. Crucial to this is the PI’s continued leadership role in the Center for Women in Mathematics Post-Baccalaureate Program at Smith College, as well as organization of conferences for undergraduate student researchers. These activities broaden participation in the field by preparing students for graduate study in the mathematical sciences. The project explores the geography and botany problems for branched covers of three- and four-manifolds, particularly when equipped with additional geometric structure. The geography problem asks which manifolds arise as branched covers of a given manifold, subject to constraints on the degree of the cover or complexity of the branching set; the botany problem asks for a classification of branched covering maps between a given pair of manifolds. A key strategy is the development of combinatorial and diagrammatic methods for computing invariants of knots and surfaces derived from branched covers of three- and four-manifolds, respectively. Applications will include resolution of open problems in several active areas of the field, including trisections of four-manifolds, knot concordance, and contact 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.