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

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。该研究项目集中在三维流形上。三维流形是一个靠近每个点的空间，看起来就像我们生活的三维空间。从数学上讲，这样的空间可以从不同的角度来处理。其中之一是拓扑学：考虑通过连续变形保持的空间的性质。另一种观点是几何学：研究与空间相关的某些刚性结构。三维流形也可以用方程和称为群的代数对象来描述，它允许使用代数几何的工具。还有另一种观点是计算的：许多复杂的算法不仅有助于计算三维流形的不变量，而且还提出了关于各种数学问题的算法复杂性的问题。这个项目包括研究三维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。源自较难问题的有趣特殊情况的子项目适合职业生涯早期的数学家，使教育计划与研究目标紧密结合。PI将继续在所有阶段进行研究、培训和指导，从与本科生的项目到与博士后研究人员合作。通过跨学科研讨会，国际和平研究所旨在加强上述研究领域之间的联系。基于她之前通过数学妇女协会和花园州LS少数群体参与计划为代表不足群体的学生提供指导的经验，PI将继续通过参与研究来支持代表不足的社区。此外，为了促进数学中的性别多样性，国际数学联合会将在纽约罗格斯大学组织每季度一次的“女性拓扑学”讲座。在研究有限(双曲或单纯)体积的三维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维流形的子流形的长期悬而未决的问题。它们包括获得嵌入曲面数量的通用上限，这与Mirzakani关于曲线的工作的精神相同，但是一维向上的；灵感来自Menasco和Reid在1992年、Sakuma和Week从1995年开始、Finkelstein和Moriah从2000年开始的关于嵌入曲面和圆弧的公开猜想。其他令人感兴趣的问题包括代数几何和纽结理论的界面问题，以及关于众所周知的拓扑问题复杂性下界的猜想。这一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智力优势和更广泛的影响审查标准进行评估，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.