Studies in the hyperbolic circle problem

PhD defense: Giacomo Cherubini

 The study of negatively curved Riemann surfaces is an open field of research in mathematics. In certain cases when we have an arithmetic construction, then we can relate the geometric objects to the investigation of specific problems in number theory. As an example, one of the most fascinating example of such connection is the modularity theorem, which relates modular curves to elliptic curves with assigned data.

A totally different aspect of the study of hyperbolic surfaces is that of the spectral theory, which is the one we investigated in our studies: the hyperbolic Laplace operator has a spectrum that consists of a discrete part and a continuous part for surfaces with cusps, and working with such spectrum provides new information on number theoretical problems. Surprisingly, relatively little is known about eigenvalues and eigenfunctions of the Laplacian, in contrast to the Euclidean case where everything is explicit.

The hyperbolic circle problem is the analogous of the Gauss circle problem in the hyperbolic plane. To study the problem, methods from spectral theory of automorphic forms as well as more classical techniques are used. I will give an overview of the problem and present the results of my thesis regarding the existence of asymptotic moments and limiting distribution, and I will mention the possibility of using the methods of fractional calculus. The results also extend to a more general class of functions that behave like almost periodic functions.

Supervisor: Ass. Prof. Morten S. Risager, Math, University of Copenhagen

Assessment committee:

Prof. Henrik Schlichtkrull (chairman), MATH, University of Copenhagen

Reader, Yiannis Petridis ,University College London

Prof. Andreas Strömbergsson, Uppsala University