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

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.