My research spans across some (but not all!) parts of dynamics, geometry and analysis. In analysis and geometry one usually works with real or complex numbers, but it is also possible to use, for instance, p-adic numbers. Doing so leads to non-Archimedean analysis and geometry, in honor (dishonor?) of Archimedes of Syracuse.
One of my main interests is in how non-Archimedean objects, such as Berkovich spaces, can be used to study problems where the original problem is phrased in terms of complex or rational numbers. Examples include singularities (of psh functions) in complex analysis and the growth of the arithmetic complexity (height) of orbits of certain polynomial, discrete-time, dynamical systems. I am also interested in developing non-Archimedean geometry in a way parallel to complex geometry.