The principal investigator will use valuations, a tool from algebra, to study local problems in dynamics and analysis in two complex dimensions. Valuations can be visualized concretely as the different ways that a sequence of points can converge to the origin. In fact, they parameterize all the local data at a point. Local phenomena in complex variables, such as the singularity of a plurisubharmonic functions or the rate of convergence of an orbit to a fixed point, can be effectively studied by observing the behavior along different approaches to the point. Valuations constitute a very powerful tool for this study. The success of the approach depends on the fact that the set of all valuations at a point in two dimensions is a set that can be understood: it is a tree in the sense of an infinite collection of real intervals welded together at branch points so that no cycles appear. This tree is a complicated but manageable object and its one-dimensionality has striking applications.
Dynamical systems are mathematical models that are widely used in virtually every science for any situation that undergoes a time-evolution. Sir Isaac Newton invented calculus, or mathematical analysis, for the study of the planetary system, a prime example of a dynamical system. Much of mathematics has developed as a respond to the need to model and understand the world around us. In this proposal the principal investigator will use a tool from a different field of mathematics in order to understand certain dynamical systems. One example of a system that can be studied arises from numerical (iterative) algorithms, and the study of the dynamical systems in this proposal will give information on the performance of the algorithms.
