This project concerns research at the boundary of Number Theory and Algebraic Geometry and extends to several other branches of mathematics. In particular it concerns work on polynomial equations and their solutions. Associated to the solutions of a collection of polynomial equations is a mathematical object, their zeta function. This research will investigate a deep collection of conjectures concerning these zeta functions, made by Alexander Beilinson, Don Zagier and others 30-40 years ago. These investigations also have consequences for theoretical Physics. Another aspect of the project is that it will support the training of graduate students in this area of research.
Any system of polynomial equations with integral coefficients gives rise to a function of one complex variable, the Zeta Function Z(s). It encodes in a very mysterious way the most important characteristics of the space of solutions. In particular, the special values of the zeta function Z(s) at the integral values of s should be expressed as periods, that is integrals of certain specific type. The PI is going to study the special values of the zeta functions, relating them to the very classical mathematical objects, going back to Euler - the classical polylogarithm functions, which generalize the logarithm function. The technique the PI is using came from several parts of Mathematics and Mathematical Physics, such as the theory of motives, algebraic K-theory on one hand, and cluster varieties and their quantization on the other. The very existence of deep connections between polylogarithms and cluster varieties is surprising. It will have many applications far beyond Algebraic Geometry and Number Theory, e.g. in the investigation of Scattering Amplitudes in Theoretical Physics. The PI will continue his work on quantum geometry of various moduli spaces and their applications in Algebraic Geometry, Mathematical Physics and Number Theory, including the study of classical and quantum polylogarithms, special values of L-functions, motivic Galois groups, Quantum Hodge Field Theory, cluster structure and quantization of moduli spaces of local systems. The main priorities of the project are the following: a) To use the relationship between cluster varieties and polylogarithms to prove Zagier's conjecture on the special values of the Dedekind zeta function at least for s=5, and relate the motivic cohomology to the cohomology of the polylogarithmic motivic complexes. b) To develop the theory of quantum multiple polylogarithms, providing a quantum deformation of the periods of the prounipotent completion of the motivic fundamental group of the punctured projective line. c) To give a comprehensive treatment of the cluster structure of various moduli related to the moduli spaces of G-local systems on surfaces. Apply this to quantization of these moduli spaces, representation theory of quantum groups, mirror symmetry and Mathematical Physics. d) Develop Quantum Hodge Field Theory. Its tree level gives a Feynman integral approach to Hodge theory. Derive it as a Hodge-theoretic analog of Chern-Simons theory.
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.
