Multiple Polylogarithms and Algebraic K-theory Fields

Multiple polylogarithms are mathematical functions that appear in diverse areas such as algebra, geometry, and theoretical physics. Although studied since the 19th century, many of their most intriguing properties remain conjectural. Mathematicians Don Zagier and Alexander Goncharov proposed that these properties might be explained through deep connections with a highly abstract subject: algebraic K-theory of fields. Clarifying this connection is a central goal of the proposed project. The research aims to uncover new links between algebraic K-theory and multiple polylogarithms, with potential applications in number theory, topology, and mathematical physics. The project also provides training opportunities for graduate students and contributes to the development of an online database of 19th-century mathematical problems, a resource intended to support mathematical education at multiple levels. This project investigates the relationship between algebraic K-groups of fields, multiple polylogarithms, and cluster structures. These K-groups are believed to encode deep arithmetic and geometric information, as evidenced by their connections to the special values of zeta functions (Borel), mixed motives (Bloch), and hyperbolic geometry (Goncharov). Recent advances in homotopy theory have enabled the PI and collaborators to conceptualize how multiple polylogarithms can emerge from the topology of the K-theory spectrum. This framework offers new tools to address longstanding conjectures in the Goncharov program. Key objectives include proving the Goncharov conjecture in weight three, advancing a weak form of the Zagier conjecture, and proving a conjecture of Beilinson–MacPherson–Schechtman. The project also explores potential compatibilities between Goncharov’s and Rognes’ conjectures. In a parallel direction, it investigates new polylogarithmic equations derived from topological data, especially those linked to cluster algebras via homotopy-theoretic methods. 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.