RUI: Link Homology Theories and Other Quantum Invariants

低维拓扑是研究低维空间的一个数学分支，包括结、奇异连接、虚结以及结曲面。结点和连杆的不变量是我们区分这类数学对象的有效工具。对连杆不变量的研究已经产生了强大的新不变量，其形式是同调理论，这些同调理论是通过分类产生的，用一组代数对象代替已知的连杆不变量的过程，这些代数对象推广和丰富了原来的不变量。该项目的一个重点是更好地了解一些现有的链接同源，研究它们的性质和应用，以及构建新的链接同源。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结的对象，包括奇异链接。该奖项将支持学生负责调查和研究有出版价值的开放性问题的研究经验。PI致力于提高女性在数学领域的参与度，并将继续组织该校每年一次的索尼娅·科瓦列夫斯基数学日，这是一个为7-12年级学生设计的活动，目的是增强下一代女数学家、科学家、工程师和创新者的能力。此外，PI组织了一系列针对大学生的讲座，庆祝年轻数学家的成就，特别是女性和数学科学中代表性不足的群体；演讲者不仅要谈论数学和他们的数学成就，还要谈论他们在数学科学领域的道路和努力。本项目的主题范围将建立数学各个领域之间的联系，包括低维拓扑、组合学、抽象代数和表示理论。本项目旨在通过网和泡沫模关系构建和研究经典结和奇异结的新的khovanov型同调理论，并研究这些理论在辫状结和协同问题上的应用，特别是在与面链和面结的一致性、丝带距离和不变量有关的问题上。该项目的另一个目标是通过串模、马尔可夫迹、Birman-Murakami-Wenzl代数以及涉及图形微积分的组合技术，将经典量子不变量扩展到奇异链路和虚链路。这个项目给学生带来了一些可管理的问题。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。

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.