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

职业：有限体积的三流形、它们的几何形状、表示形式和复杂性

• 批准号: 2142487
• 负责人: Anastasiia Tsvietkova
• 金额: $ 46.72万
• 依托单位: Rutgers University Newark
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2027-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2142487&HistoricalAwards=false
• 关键词: CAREER; Three; manifolds; finite; volume

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.

该奖项全部或部分根据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。一个三维流形是一个空间，在每个点附近看起来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处理。其中之一是拓扑：考虑连续变形所保持的空间属性。另一种观点是几何学：研究与空间相关的某些刚性结构。一个三流形也可以用方程和一个叫做群的代数对象来描述，这允许代数几何的工具。然而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维流形的不变量，而且还提出了各种数学问题的算法复杂性问题。该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。子项目源于有趣的特殊情况下，更难的问题是适合早期的职业数学家，使教育计划与研究目标密切相关。PI将继续在所有阶段进行研究培训和指导，从本科生项目到与博士后研究人员合作。 通过跨学科研讨会，PI旨在加强上述研究领域之间的关系。建立在她以前的辅导经验，通过协会妇女在数学和花园州LS联盟少数民族参与计划的代表性不足的群体的学生，PI将继续通过研究参与支持代表性不足的社区。此外，为了促进数学中的性别多样性，PI将在纽瓦克的罗格斯大学组织季度“拓扑学中的女性”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内，该项目的目标包括关于三维流形的子流形的长期未决问题。它们包括获得通用上限的嵌入表面的数量，在Mirzakani的工作曲线的精神，但一个维度了;工作的灵感来自于开放的嵌入表面和弧Menasco和里德从1992年，佐久间和周从1995年，芬克尔斯坦和Moriah从2000年。在其他感兴趣的问题是问题的接口代数几何和纽结理论，并progratures有关下界的复杂性著名的拓扑问题。这些成果将对低维拓扑学和几何学做出重大贡献，对计算拓扑学产生积极影响，并深化几何学、拓扑学、代数几何学和理论计算机科学之间的联系。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响评审标准进行评估，被认为值得支持。