Removability in Geometric Function Theory

Conformal maps are functions or transformations of space that, locally, preserve angles. The study of the geometric properties of such maps has led to the development of Geometric Function Theory and has proven over the years to be of fundamental importance to a wide variety of problems in analysis, geometry, probability, physics, and engineering. More recently, much attention has been devoted to the study of maps, or functions, that are a generalization of conformal maps, called of quasiconformal maps, where a controlled amount of angle distortion is allowed.

Quasiconformal mappings possess subtle properties, making them very useful in a wide variety of settings. In many of these applications, one must deal with maps that are (quasi)conformal in a given planar region except possibly for 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 analysis and other areas.

The principal investigator has no doubt that many more fruitful connections should come to light as progress is made toward a better understanding of removability. This project will also consider several other problems in Geometric Function Theory, some with applications in the field of numerical vision. An important portion of the proposed research involves numerical computations and constructive methods.

In addition to enhancing computational infrastructure, this has the potential to build interdisciplinary connections.

The first proposed activity deals with the study of the relationship between removability and the rigidity of circle domains in Koebe’s uniformization conjecture, building upon the pioneering work of He and Schramm. The principal investigator plans to pursue the study of this surprising connection. The second proposed activity is devoted to the study of conformal welding.

The principal investigator proposes to work on several constructive aspects as well as on the relationship between the non-injectivity of conformal welding and removability. Finally, the third proposed activity involves the study of the properties of analytic capacity. The principal investigator first suggests to further investigate the subadditivity of analytic capacity.

This part of the proposal involves numerical computations using a program developed by the principal investigator together with an undergraduate student. The principal investigator also plans to study the behavior of analytic capacity under holomorphic motions.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.