RUI: Link Homology Theories and Other Quantum Invariants

RUI：链接同源理论和其他量子不变量

• 批准号: 2204386
• 负责人: Carmen Caprau
• 金额: $ 22.63万
• 依托单位: California State University-Fresno Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204386&HistoricalAwards=false
• 关键词: RUI; Link; Homology; Theories; Other

Low-dimensional topology is a branch of mathematics that studies spaces of low dimensions, including knots, singular links, and virtual knots, as well as knotted surfaces. Invariants of knots and links are efficient tools that allow us to distinguish between such mathematical objects. Study of link invariants has yielded powerful new invariants in the form of homology theories that arise through categorification, the process of replacing a known invariant for links with a family of algebraic objects that generalize and enrich the original invariant. One focus of the project is to better understand some existing link homologies and investigate their properties and applications, as well as to construct new link homologies. Another goal of the project is to extend known quantum invariants to other knot-like objects, including singular links. The award will support research experiences in which students are charged with investigating and working on publication-worthy open questions. The PI is committed to increasing the participation of women in mathematics and will also continue to organize the university's annual Sonia Kovalevsky Math Day, an event designed for students in grades 7-12, with the goal of empowering the next generation of female mathematicians, scientists, engineers, and innovators. Moreover, the PI organizes a series of talks aimed for college students, celebrating the achievements of young mathematicians, in particular women and underrepresented groups in mathematical sciences; speakers are encouraged to talk not only about math and their mathematical achievements but also about their path and endeavors to a career in mathematical sciences. The range of topics in this project will establish connections between various areas of mathematics, including low-dimensional topology, combinatorics, abstract algebra, and representation theory. The project aims to construct and study new Khovanov-type homology theories for classical knots and singular knots via webs and foams modulo relations and to investigate applications of these theories to questions about braids and cobordisms, in particular to questions related to concordance, ribbon distance, and invariants of surface-links and surface-knots. Another goal of the project is to extend classical quantum invariants to singular links and virtual links by means of skein modules, Markov traces, and Birman-Murakami-Wenzl algebras, as well as combinatorial techniques involving graphical calculus. There are manageable problems for students stemming from the project.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.

低维拓扑学是数学的一个分支，研究低维空间，包括纽结、奇异链、虚拟纽结以及纽结曲面。纽结和链环的不变量是有效的工具，使我们能够区分这样的数学对象。对链接不变量的研究已经产生了强大的同调理论形式的新不变量，这些不变量是通过分类产生的，分类是用一族代数对象来替换已知的链接不变量的过程，这些代数对象是对原始不变量的推广和丰富。该项目的一个重点是更好地了解一些现有的链接同调，并研究它们的性质和应用，以及构建新的链接同调。该项目的另一个目标是将已知的量子不变量扩展到其他类似纽结的物体，包括奇异链接。该奖项将支持学生负责调查和研究值得发表的公开问题的研究经验。PI致力于增加女性对数学的参与，并将继续组织该大学一年一度的索尼娅·科瓦列夫斯基数学日，这是为7-12年级的学生设计的活动，目的是赋予下一代女性数学家、科学家、工程师和创新者权力。此外，国际数学联合会组织了一系列针对大学生的讲座，庆祝年轻数学家的成就，特别是女性和在数学科学中代表性不足的群体；鼓励演讲者不仅谈论数学及其数学成就，而且还谈论他们在数学科学领域的道路和努力。这个项目的主题范围将在数学的不同领域之间建立联系，包括低维拓扑学、组合学、抽象代数和表示论。该项目旨在通过网络和泡沫模关系来构建和研究经典纽结和奇异纽结的新的Khovanov型同调理论，并研究这些理论在辫子和边线问题上的应用，特别是关于相容、带距以及表面环和表面纽结的不变量的问题。该项目的另一个目标是通过skein模块、马尔可夫迹、Bman-Murakami-Wenzl代数以及涉及图形演算的组合技术，将经典量子不变量扩展到奇异链接和虚拟链接。该项目给学生带来了一些可管理的问题。这一奖项反映了NSF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，被认为值得支持。