represents a new direction, which has rapidly matured in the past few years. The proposed collaborative research project is devoted to applying the methods and tools of noncommutative geometry to specific topics in number theory, pertaining to the study of the explicit class field theory problem (Hilbert's 12th problem), of the Riemann zeta function and of the L-functions of algebraic varieties. One anticipated outcome will be a novel understanding of the Weil explicit formulae as Lefschetz trace formulae in the context of cyclic cohomology. Another central aspect of the project involves supplementing Manin's approach to Stark's conjectures for real quadratic fields (via noncommutative tori with real multiplication) with ideas stemming from the recent investigation of the quantum statistical mechanical properties of noncommutative spaces of Q-lattices modulo commensurability. New results on modular forms and Hecke operators are expected, arising from the transfer of transverse geometry concepts and constructions to the setting of modular forms. The formalism of spectral triples together with the local index formula in noncommutative geometry will be exploited to investigate rigid analytic spaces more general than Mumford curves. Significant progress is also anticipated in the uncovering of the relationship between residues of Feynman graphs in quantum field theory and periods of mixed Tate motives.
This collaborative research project aims to shed light on a number of important topics pertaining to the rich and largely untapped interconnection between the fields of noncommutative geometry, number theory and mathematical physics. These topics address central aspects and open problems, that involve some of the key mathematical objects in the latter fields, such as the celebrated Riemann zeta function and its generalizations called L-functions in number theory and Feynman integrals in perturbative quantum field theory. Their investigation will be approached in a novel and unified manner, through the methods of noncommutative geometry, a discipline which grew out of the fusion between one of the oldest branches of mathematics -- geometry, and one of the youngest -- quantum mechanics.
