Removable Sets and Questions in Geometric Function Theory

University Of Hawaii, Honolulu

Investigators

Abstract

Conformal maps are planar changes of coordinates that locally preserve angles. The study of the properties of such maps led to the development of the subject known as geometric function theory. A cornerstone of this discipline is the remarkable fact that every planar region without holes can be conformally transformed into the round disk in an essentially unique fashion. Such coordinate changes have proven over the years to be of great value in a wide variety of applications in physics and engineering. In many of these applications, one has to deal with coordinate changes that are conformal in a given planar region except possibly some "exceptional set" of points inside the region. The question whether the exceptional set is negligible and small enough to be ignored leads to the notion of conformal removability, central to this research project and closely related to fundamental questions in complex dynamics and in random surfaces. The project will also consider several other questions in geometric function theory, such as conformal welding (a correspondence between curves in the plane and functions on the circle, recently observed to have important applications in the field of numerical vision), shapes of Julia sets (fractal sets arising from the iteration of polynomials), and the subadditivity of analytic capacity. Each of these investigations has a numerical aspect, and successful completion of the research project will enhance computational structure and build interdisciplinary connections with applied sciences such as finance and pattern recognition. The first part of the research project deals with the study of the geometric properties of conformally removable sets. More precisely, the investigator will further study various settings where removability appears naturally, including the rigidity of circle domains. This first component of the research project also includes problems related to conformal welding and fingerprints of lemniscates. The second part is devoted to the study of the possible shapes of Julia sets. More specifically, the investigator plans to explore various questions revolving around the constructive approximation of planar sets by polynomial Julia sets in the Hausdorff distance, such as sharp rates of approximation, dynamical properties of the approximating polynomials, and an efficient numerical implementation of the approximation scheme. Finally, the last part of the research project concerns the subadditivity problem for analytic capacity. The investigator will further explore a conjecture that, if true, would imply analytic capacity is indeed subadditive. This involves an improvement of a numerical method for the computation of the analytic capacity of finite unions of disks.