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起着基础性的作用。例如，在寻找一个方程组的所有可能解的过程中，关键的一步是理解这个方程组的对称性的相关集合或“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，“普遍”的对称群：Grothendieck-Teichmueller群（GRT）和Kashiwara-Vergne群（KRV）。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、纽结和缠结理论，以及代数几何中称为“动机”的通用几何不变量理论。尽管它们的意义重大，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，存在着长期的争论。特别是A. Alekseev和C. Torossian证明了KRV包含GRT，并证明它们实际上是相等的，这是一个尚未解决的问题。在这个项目中，PI和他的合作者将启动一个新的多学科的方法来回答有关GRT，KRV的具体问题，从而揭示更多的阿列克谢耶夫和托罗西安问题。PI的重点将是研究这些群体通过他们的行动明确的几何和拓扑对象。该项目包括适合研究生研究的主题，并将有助于加速新生数学博士的成长。项目在内华达州，里诺（UNR）的大学。该项目更广泛的影响还包括与联合国大学数学界妇女协会分会协调研究和职业发展活动。 该项目将通过建立在PI之前展示的GRT在光滑复代数簇上的非平凡作用，以及使用4维空间中的轮式道具和打结表面的KRV拓扑表征，来为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伦和变形理论的方法，这已经被证明是非常成功的研究GRT和相关现象。预期成果包括形变量子化中形式态射的“柏原-韦尔涅提升”，对Duflo理论和Rozansky-Witten理论的新认识;以及代数几何中GRT和KRV作用的新例子。该项目由拓扑学，激励竞争研究计划（EPSCoR）该奖项反映了NSF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.