Topics in number theory, dynamical systems and discrete geometry

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

Investigators

Abstract

There have recently been many important interactions between number theory and other fields, including dynamical systems and discrete geometry. The connection of number theory with discrete geometry structures via packings leads to connections with problems in materials science. Problems in number theory involving zeta functions and distribution of primes have parallels with problems in mathematical physics. There are further connections of number theory with physical theories such as conformal field theory through modular forms. Number theory structure appears in some exactly solvable models of phase transitions in physical models. The proposal investigates several possible areas of contact between fields, which may lead to fruitful interactions with researchers in physics and material sciences. The grant will support the training of graduate students in these areas. Dr. Lagarias will investigate several independent topics interacting among these fields. 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 the Hurwitz zeta function and Riemann zeta function. A connection is made with the representation theory of the Heisenberg group and related groups, and the project aims to connect it more closely to automorphic representations on various groups. A second topic concerns two exploratory projects; one concerns a certain family of virtual representations of the symmetric groups, which arises as a degenerate limit of splitting properties of polynomials (modulo p), associated to properties of the discriminant locus of the polynomial. It asks if there is an underlying geometric structure explaining the observed limit properties, possibly with an associated dynamical system. Another exploratory project studies polyharmonic modular forms, which are functions with modular invariance that are annihilated by a power of the Laplacian. A third topic is to investigate of circle packings on Riemann surfaces, a topic in the area of discrete geometry. It considers formulation of scaling limits of such packings in two directions, a complex variables limit and a Diophantine approximation limit. Whether or not this can be done precisely, special questions are proposed to initially investigate these areas, including study of surfaces having rigid packings with a finite number of circles.