This proposal involves problems in a diverse array of topics including Berkovich spaces, tropical geometry, and complex dynamics. The primary intellectual merit of the proposal is that it will increase our understanding of each of these important areas of mathematics and unearth new relationships between them. The main unifying theme behind these problems is that our proposed strategies for solving them all involve potential theory, both in the classical and non-Archimedean setting. In recent years, a surprisingly robust non-Archimedean analog of classical complex potential theory has been developed by the PI and others. In addition, the PI has helped to develop a number of general techniques for comparing Berkovich analytifications and tropicalizations of algebraic varieties, showing that one can profitably view tropical geometry a `bridge' between Berkovich's theory of non-Archimedean analytic spaces and classical convex geometry. The PI proposes to develop new methods for constructing semistable models of curves via tropical geometry, to prove a non-Archimedean Berkovich space version of the Mumford-Neeman equidistribution theorem, to apply Berkovich's theory to the study of component groups of Neron models, and to explore arithmetic and geometric properties of post-critically finite rational maps within the moduli space of all rational maps.
The classical subject of complex potential theory first arose in physics, where it was used to describe gravitational and electromagnetic interactions. It has subsequently found a wealth of applications to various areas of mathematical research, including complex analysis and complex dynamics (where it is used to study fractals such as the celebrated Mandelbrot set). Non-Archimedean analysis is a crucial part of modern number theory which first arose in the early twentieth century work of Kurt Hensel on the famous 'p-adic numbers'. In non-Archimedean potential theory, one replaces the classical complex ``Riemann sphere'' by a p-adic counterpart, called the Berkovich projective line, which was introduced by Vladimir Berkovich in the 1980's. Berkovich's theory has since become an important tool in modern number theory and algebraic geometry. Tropical geometry is a relatively new and active area of research with applications to many fields of mathematics. One can think of tropical geometry as a piecewise linear approximation of classical algebraic geometry in which an ``algebraic variety'' (which is, roughly speaking, the set of common solutions to a system of polynomial equations) is replaced by a polyhedral complex (thought of as the set of common solutions to a system of linear inequalities). Surprisingly -- and rather mysteriously -- the tropical approximation remembers much more information about the original variety than one might originally expect.
