Topics in Number Theory and Geometry: Zeta Functions and Circle Packings

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Dr. Lagarias will investigate three problem areas in number theory and geometry. The first topic concerns study of generalizations of the Lerch zeta function and their interpretation in representation theory terms. The second topic concerns the study of Hilbert spaces of entire functions and associated operators related to automorphic L-functions. The third topic concerns exploration of analogies between circle packings, conformal geometry and Diophantine approximation.The subjects of number theory and geometry interact in several areas, including packing problems and the geometry of numbers. In recent years these connections have led to interactions across mathematics, physics and materials sciences. Number theory has also provided many useful applications in communications, coding theory and cryptography. Zeta functions encode both number theoretic and geometric invariants. The proposal will study one class of such functions to explore connections of this kind. There have also been connections noted between zeta functions and operators in mathematical physics, and Dr.Lagarias will investigate connections of this kind from the operator side. Thirdly, circle packings have recently been intensively studied, as they seem to provide a discrete bridge between conformal geometry and number theory. The problems considered may stimulate interactions between researchers in the fields: conformal geometry,mathematical physics and number theory. Dr. Lagarias is training graduate students in both number theoretic and geometric topics.