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，找到一个方程系统的所有可能解的关键步骤是理解系统对称性的相关集合或“组”。这个项目的主要目标是研究两个特殊的，但在某种意义上，“普遍”的对称群：Grothendieck-Teichmueller群（GRT）和Kashiwara-Vergne群（KRV）。众所周知，两者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，结和缠结理论，以及代数几何中被称为“动机”的普遍几何不变量理论。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的关系，存在着长期存在的猜想。特别是a . Alekseev和C. Torossian证明了KRV包含GRT，并推测它们实际上是相等的，这个问题至今没有得到解决。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有关GRT， KRV的具体问题，从而更多地阐明Alekseev和Torossian问题。PI的重点将是通过这些群体对明确的几何和拓扑对象的作用来研究它们。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研究计划署妇女数学协会分会协调研究和职业发展活动。该项目将通过建立PI之前展示的GRT在光滑复杂代数变体上的非平凡作用，以及在四维空间中使用轮式支柱和结曲面的KRV的拓扑表征，为Alekseev-Torossian猜想的有效性找到支持证据。这项工作所需的工具将使用同调和变形理论方法来构建，这些方法在研究GRT和相关现象方面已经证明是非常成功的。预期结果包括变形量化中形式态态的“Kashiwara-Vergne抬升”；对Duflo理论和Rozansky-Witten理论的新认识；以及代数几何中GRT和KRV动作的新例子。该项目由拓扑学和促进竞争研究的既定计划（EPSCoR）共同资助。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。

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.