Analysis of Conformal and Quasiconformal Maps

Abstract Award: DMS-1006309 Principal Investigator: Christopher Bishop The principal investigator will study the geometric properties of conformal and quasiconformal maps, with an emphasis on the connections with other areas such as numerical analysis, computational geometry, complex dynamics and geometric measure theory. He will continue his work on fast conformal mapping algorithms, including the class of Schwarz-Christoffel iterations. These include Davis' method and the CRDT algorithm of Driscoll and Vavasis as special cases, and other methods involving the medial axis and quasiconformal mappings. One of the basic themes is to start with certain quasiconformal maps that approximate the desired conformal map, but are easier to compute and have better properties. These maps have natural connections to several well known problems such as Brennan's conjecture, and the PI will investigate these connections. The PI will also continue his work on various related problems involving optimal meshing algorithms, diffusion limited aggregation, the geometry of Brownian motion, the quasiconformal Jacobian problem and conformal welding. A fundamental problem of mathematics and engineering is to quickly and accurately solve various differential equations related to fluid flow, heat conduction and wave propagation on complex regions. For 2-dimensional surfaces, a standard method is to use a conformal mapping (i.e., angle preserving on small scales) to replace the region with a simpler one, such as a disk or rectangle. This approach has been studied for over 150 years, but not until the 1980's did computers made conformal mapping practical for highly complex regions. The PI will investigate how to improve existing methods and develop new ones that are faster and more reliable. He will also investigate the theoretical behavior of conformal maps on extremely complex domains such as fractals and the connections between conformal maps and probability theory. The PI has already applied the insights gained from these problems to meshing. This is the process of dividing a complex region into simple pieces such as triangles. Meshing is a basic part of most numerical methods and the usefulness of a mesh in applications depends on the number of pieces (fewer is better) and their shapes (better to avoid small and large angles). It is difficult to construct a mesh which is good in both respects, but PI has used non-Euclidean geometry to construct 2-dimensional quadrilateral meshes with optimal size and shapes for simple polygons and is working to extend this to more general domains. Greater generality is needed in various problems involving crack formation, interfaces between materials and computer learning. Efficient meshing also has numerous applications in high-performance computing such as modeling surfaces for engineering and computer graphics. This award is jointly funded by the programs in Analysis and Geometric Analysis.