CAREER: Three-manifolds with finite volume, their geometry, representations, and complexity

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。该研究项目的重点是三维流形。三流形是一个空间，在每个点附近看起来都像我们生活的三维空间。在数学上，这样的空间可以从不同的角度来看待。其中之一是拓扑学：考虑连续变形所保留的空间性质。另一种观点是几何：研究与空间相关的某些刚性结构。三流形也可以用方程和称为群的代数对象来描述，这允许使用代数几何的工具。另一个观点是计算性的：许多复杂的算法不仅有助于计算三流形的不变量，而且也提出了各种数学问题的算法复杂性问题。本项目包括研究3流形的内在几何和拓扑性质，以及所有这些方法之间的丰富相互作用。从更困难的问题的有趣的特殊案例中产生的子项目适合早期职业数学家，使教育计划与研究目标紧密地交织在一起。PI将继续在所有阶段进行研究培训和指导，从本科生项目到博士后研究人员的工作。通过跨学科研讨会，PI旨在加强上述研究领域之间的关系。基于她之前通过妇女数学协会和花园州LS少数民族参与联盟项目指导代表性不足群体学生的经验，PI将继续通过参与研究来支持代表性不足的社区。此外，为了促进数学领域的性别多样性，PI将在纽瓦克的罗格斯大学每季度组织一次“拓扑学中的女性”讲座。在研究有限体积（双曲或简单）三流形的内在几何和拓扑性质的总体主题中，该项目的目标包括关于三流形子流形的长期开放问题。它们包括获得嵌入曲面数量的普遍上界，在米尔扎卡尼曲线研究的精神中，但是是一维的；作品灵感来自1992年的Menasco和Reid， 1995年的Sakuma和Weeks， 2000年的Finkelstein和Moriah对嵌入式表面和弧的公开猜想。在其他感兴趣的问题中，代数几何和结理论的界面问题，以及关于已知拓扑问题复杂性下界的猜想。这些成果将对低维拓扑和几何有重要贡献，对计算拓扑有积极影响，并加深几何、拓扑、代数几何和理论计算机科学之间的联系。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). The research project focuses on three dimensional manifolds. A three-manifold is a space that near each point looks like the three-dimensional space we live in. Mathematically, such spaces can be approached from different viewpoints. One of them is topological: considering properties of the space that are preserved by continuous deformations. Another viewpoint is geometric: studying certain rigid structures associated to the space. A three-manifold can also be described by equations and by an algebraic object called a group, which allows tools from algebraic geometry. Yet another point of view is computational: many sophisticated algorithms not only help calculate invariants of three-manifolds, but also raise questions about algorithmic complexity of various mathematical problems. This project includes a study of intrinsic geometric and topological properties of 3-manifolds, as well as the rich interplay between all these approaches. Subprojects stemming from interesting special cases of harder problems are suitable for early-career mathematicians, allowing the educational program to be strongly intertwined with the research goals. The PI will continue research training and mentoring at all stages, from projects with undergraduates to working with postdoctoral researchers. Through cross-disciplinary workshops, the PI aims to strengthen relations between the above mentioned fields of research. Building on her prior mentoring experience with students from underrepresented groups through the Association for Women in Mathematics and the Garden State LS Alliance for Minority Participation programs, the PI will continue to support underrepresented communities through research involvement. Additionally, to promote gender diversity in mathematics, the PI will organize quarterly “Women in Topology” lectures at Rutgers, Newark.Within the overarching theme to study intrinsic geometric and topological properties of three-manifolds with finite (hyperbolic or simplicial) volume, the project's goals encompass long-standing open questions about submanifolds of three-manifolds. They include obtaining universal upper bounds on the number of embedded surfaces, in the spirit of Mirzakani’s work on curves, but one dimension up; work inspired by open conjectures about embedded surfaces and arcs by Menasco and Reid from 1992, Sakuma and Weeks from 1995, Finkelstein and Moriah from 2000. Among other questions of interest are problems on the interface of algebraic geometry and knot theory, and conjectures about lower bounds on complexity of well-known topological problems. The outcomes will significantly contribute to low-dimensional topology and geometry, positively impact computational topology, and deepen the connections between geometry, topology, algebraic geometry and theoretical computer science.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.