In this project, the Principal Investigators study various aspects of analysis on p-adic spaces, especially those arising in connection with canonical heights, capacity theory, and algebraic dynamical systems. The project involves the development of both theory and applications, and comprises three main topics: adelic equidistribution theorems for points of small height; arithmetic aspects of the dynamics of iterated rational functions; and analysis on Berkovich spaces with applications to dynamical systems, arithmetic intersection theory, and higher-dimensional capacity theory.
The motivation for this work is to understand and generalize certain recently discovered properties of "height functions". Height functions first arose in connection with elliptic curves, and are ubiquitous in modern number theory. One of the goals of the project is to establish properties of heights for arbitrary curves and arbitrary dynamical systems which are similar to those for elliptic curves. This will be approached by using methods from potential theory. Another goal is to establish "p-adic" analogues of results which are known to hold in the theory of manifolds. When completed, the project will reveal new connections between number theory, dynamical systems, and potential theory. Funding for this project will support the infrastructure of the University of Georgia's number theory group, which has historically been very strong. The project will impact both the graduate and undergraduate programs at UGA: some of the questions raised by this research will lead to PhD dissertation topics for graduate students, and others will provide an opportunity to involve undergraduate students in cutting edge mathematical research.
