The PI proposes to study several problems in geometric function theory that have as common underlying theme "conformal invariants" such as the hyperbolic and Kobayashi metric, harmonic measure, Green's functions, and modulus of path families. One such problem consists in studying dimensionality properties of p-Harmonic measure on domains beyond simply-connected ones. A second question deals with generalizations of the Chang-Marshall theorem in space, namely with exponential integrability properties for the trace of analytic functions, and their quasiregular counterparts in higher dimensions, when restricted to the boundary. A third problems studies iteration of analytic functions in one and several dimensions with a focus on the interplay between complex dynamics and the hyperbolic geometry of the unit disk in the complex plane and of the unit ball in higher dimensions.
This research will also draw on the properties of conformal invariants mentioned above to obtain concrete applications in the study of large networks. This is an area that has become more salient with the advent of the internet and the need to analyze large databases (so-called massive data-sets). One example that most people are familiar with is search-engines. The way internet searches work is through random processes that continually sample the web and periodically return averages and other statistics. The simplest such process is called a random crawler or walker and the mathematics that governs its behavior is derived from the study of conformal invariants in geometric function theory. The PI is conducting research that is expected to bring new tools to the task of comparing the behavior of such random processes to the geometry of the data-set. Because of the large applicability of such results the PI will also study the problem of epidemic outbreaks. In this context the PI has already obtained initial funding from the Center for Engagement and Community Development at Kansas State University for a joint project with Professor Scoglio in the Department of Electrical and Computing Engineering and Professor Schumm in the Department of Family Studies. Our team collected data in the city of Chanute, Kansas, and has already built a "contact" network, which is now being analyzed using the conformal invariants mentioned above. The ultimate goal is to provide the city of Chanute with a concrete set of directions that could help its city officials mitigate and manage an epidemic outbreak, especially one of zoonotic nature, originating on a farm.
