Zeta Integrals, Discrete Number Theory and Geometry

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

This research project concerns interactions between number theory and geometry, with connections to physics. Number theory is discrete and geometry is continuous, but continuous structures can be fruitfully approximated by discrete structures through finer and finer approximations. Intermediate discrete models can exhibit both (discrete) number theoretic and (continuous) geometric properties. In this way, number theory connects with discrete geometric structures, via packings, leading to connections with problems in materials science. Certain finite discrete models in large sizes have parallels with exactly solvable models in physics that exhibit phase transitions when a temperature parameter is varied, as between water and ice. The projects in this proposal investigate a number of specific problems related to these analogies in order to bridge the discrete and the continuous in various limits. This work will lead to fruitful interactions with researchers in geometry, physics and material science and will support the training of graduate students in number theory and in discrete geometry. In more detail, the PI will investigate several topics relating number theory and geometry. The first, and main, topic continues the investigation of the Lerch zeta function, which is a function of three variables, that on specializing variables yields both the Hurwitz zeta function and the Riemann zeta function. This research is based on a recently discovered connection between these functions and representations of the real Heisenberg group and related solvable groups. The project will study generalizations of the Lerch zeta function to automorphic representations of various higher-dimensional Lie groups, via zeta integrals. A second topic concerns the study of various discrete finite models of arithmetic structures, typically with two variables, and the study of statistics of such models as their size increases, pursuing an analogy with integrable systems in statistical mechanics. A third topic concerns intra-universal Teichmuller theory introduced by Mochizuki. This new theory will be reformulated from a more accessible analytic perspective. A fourth topic concerns circle packings on Riemann surfaces. The PI will study scaling limits of perfect finite circle packings in two directions: a complex variables limit and a Diophantine approximation limit. These limits will be determined precisely. How they are related will also be determined.