Derived Symmetries and the Alekseev-Torossian Conjecture: From Algebraic Geometry to Knotted Objects in Dimension 4

对称性在整个数学科学中扮演着重要的角色。例如，在寻找方程组的所有可能解时，关键的一步是了解该系统的相关对称性的集合或“群”。这个项目的主要目标是研究两个特殊的，但在某种意义上，“泛”对称群：Grothendieck-Teichmueller群(GRT)和Kashiwara-Vergne群(KRV)。众所周知，这两个理论都与数学和数学物理的许多重要领域有着深刻的联系，包括：量子理论、数论、纽结和纠缠理论，以及在代数几何中被称为“动机”的通用几何不变量理论。尽管它们意义重大，但这两个群体的结构仍然相当神秘。关于它们之间的关系，以及它们与其他对称群之间的关系，长期以来都有一些猜测。特别是，A.阿列克舍夫和C.托罗辛证明了KRV包含GRT，并猜想他们实际上是相等的，这是一个悬而未决的问题。在这个项目中，PI和他的合作者将启动一种新的多学科方法来回答关于GRT和KRV的具体问题，从而更好地阐明阿列克舍夫和托罗西亚的问题。PI的重点将是通过它们对显式几何和拓扑对象的作用来研究这些群。该项目包括适合研究生研究的主题，并将有助于加速内华达大学里诺分校(UNR)新生的数学博士项目的发展。该项目的更广泛影响还包括与联合国妇女数学协会分会协调研究和职业发展活动。这个项目将通过建立PI以前在光滑复代数簇上的GRT的非平凡作用，以及在4维空间中使用轮式道具和结点曲面的KRV的拓扑特征来寻找Aleksev-Torossian猜想有效性的支持证据。这项工作所需的工具将使用同伦和形变理论方法来构建，这些方法在研究GRT和相关现象方面已经被证明是非常成功的。预期的结果包括形变量化中形式态射的“Kashiwara-Vergne提升”；对Duflo理论和Rozansky-Witten理论的新见解；以及代数几何中GRT和KRV作用的新例子。该项目由拓扑学和已建立的刺激竞争研究计划(EPSCoR)联合资助。该奖项反映了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。

Symmetry plays a fundamental role throughout the mathematical sciences. For example, a key step in finding all possible solutions to a system of equations is to understand the relevant collection, or "group", of symmetries of the system. The main goal of this project is to investigate two particular, yet, in a certain sense, "universal" symmetry groups: the Grothendieck-Teichmueller group (GRT), and the Kashiwara-Vergne group (KRV). Both are known to have deep connections to many important areas of mathematics and mathematical physics including: quantum theory, number theory, the theory of knots and tangles, and the theory of universal geometric invariants called "motives" in algebraic geometry. In spite of their significance, the structure of both groups remains quite mysterious. There are long-standing conjectures concerning their relationship to one another, as well as their relationship to other symmetry groups. In particular, A. Alekseev and C. Torossian proved that KRV contains GRT and conjectured that they are, in fact, equal, a question that remains unsolved. In this project, the PI and his collaborators will initiate a new multidisciplinary approach towards answering specific questions concerning GRT, KRV, and thus shed more light on the Alekseev and Torossian question. The PI’s focus will be on studying these groups via their actions on explicit geometric and topological objects. The project includes topics suitable for graduate student research and will help accelerate the growth of the nascent Mathematics Ph.D. program at the University of Nevada, Reno (UNR). The broader impacts of the project also include coordinating research and career development events with UNR's chapter of the Association for Women in Mathematics. This project will find supporting evidence for the validity of the Alekseev-Torossian conjecture by building on the PI's previous exhibiting non-trivial actions of GRT on smooth complex algebraic varieties, and a topological characterization of KRV using wheeled props and knotted surfaces in 4-dimensional space. The tools needed for this work will be constructed using homotopical and deformation-theoretic methods, which have already shown to be very successful in studying GRT and related phenomena. The anticipated outcomes include a "Kashiwara-Vergne lift" of formality morphisms in deformation quantization; new insights into Duflo theory and Rozansky-Witten theory; and new examples of GRT and KRV actions in algebraic geometry.This project is jointly funded by Topology, and the Established Program to Stimulate Competitive Research (EPSCoR).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.