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

导出的对称性和 Alekseev-Torossian 猜想：从代数几何到 4 维中的结物体

• 批准号: 2305407
• 负责人: Christopher Rogers
• 金额: $ 36.74万
• 依托单位: Board of Regents, NSHE, obo University of Nevada, Reno
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-15 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2305407&HistoricalAwards=false
• 关键词: Derived; Symmetries; Alekseev; Torossian; Conjecture

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.

对称性在整个数学科学中发挥着基础作用，例如，找到方程组的所有可能解的关键步骤是了解系统对称性的相关集合或“群”。该项目的主要目标。的目的是研究两个特殊的、但在某种意义上“普遍”的对称群：格洛腾迪克-泰克穆勒群（GRT）和柏原-维尔涅群（KRV），两者都与对称群有着深刻的联系。数学和数学物理的许多重要领域，包括：量子论、数论、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，尽管它们很重要，但这两个群的结构仍然存在。关于它们之间的关系以及它们与其他对称群的关系长期以来一直存在猜想，特别是 A. Alekseev 和 C. Torossian 证明了 KRV 包含 GRT 并猜想。事实上，它们是平等的，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种新的多学科方法来回答有关 GRT、KRV 的具体问题，从而进一步阐明 Alekseev 和 Torossian 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体，该项目包括适合研究生研究的主题，并将有助于加速新成立的数学博士项目的发展。该项目的更广泛影响还包括与 UNR 女性数学协会分会协调研究和职业发展活动。该项目将为 Alekseev-Torossian 猜想的有效性找到支持证据。通过建立在 PI 之前展示的 GRT 对光滑复杂代数簇的非平凡行为以及使用轮式道具和结曲面的 KRV 拓扑表征的基础上4 维空间。这项工作所需的工具将使用同伦和变形理论方法来构建，这些方法在研究 GRT 和相关现象方面已被证明非常成功，预期的结果包括形式上的“Kashiwara-Vergne lift”。形变量子化中的态射；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。该项目由拓扑学和刺激竞争性研究的既定计划 (EPSCoR)。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。