Grant: Discovery Grants Program - Individual

I am interested in a variety of problems in continuum theory and dynamics. Some of these problems have their roots in analysis and set -theory. Studying the relation between the properties of boundaries of basic open sets of a topological space and the properties of the space itself is of great interest for me. In particular, I am interested in studying rim-metrizable locally connected continuua. Some of the problems are related to an important problem from set-theoretical topology concerning perfectly normal compact spaces. An important problem in Complex Dynamics is the classification of Julia sets of rational maps and understanding the structure of them. Knowing the topological structure of a Julia set gives insight to understanding the dynamical system associated with the Julia set. Many of the Julia sets are continua. I am interested in studying the topological structure of Julia sets as well as understanding the conditions under which indecomposable continua may appear in a Julia set. Another interesting and old problem concerns different measures on a space, such as volume. The problem is to find a characterization of spaces with a suitable metric so that the space has finite measure (or volume).