The PI, Christopher Bishop, will study the geometric properties of conformal and quasiconformal maps, with an emphasis on the connections with other areas such as dynamics, computational geometry, numerical analysis and geometric measure theory. The proposal describes several areas that the PI will investigate using quasiconformal mappings as a primary tool. The first is the iteration theory of complex analytic functions on the plane. The PI has previously used quasiconformal methods to solve a number of problems in the field and plans to refine these techniques to attack several remaining open problems. The second area is to use quasiconformal maps and hyperbolic geometry in computational geometry, mostly in problems related to algorithms for meshing planar regions with optimal complexity and geometry. Obtaining similar results in three dimensions is one of the most important goals of the field, and the proposal describes some ideas for attacking this problem too. The final part of the proposal describes the dimension distortion properties of quasiconformal maps. This is one of the most interesting problems intrinsic to study of quasiconformal maps, but, as the proposal describes, can also be linked to famous open problems in computational geometry.
The proposed work will investigate how combinatorial and discrete ideas yield new results in analysis and how ideas from conformal analysis and hyperbolic geometry can prove new theorems about discrete geometry and meshing. Conformal maps preserve angles; these maps have been intensively studied for over 150 years and are of fundamental importance to a wide variety of problems in analysis, geometry, probability, physics and engineering. Classically, conformal maps have been used in the study of various differential equations related to fluid flow, heat conduction and wave propagation. More recently, conformal maps have been fundamental to the study of statistical mechanics, percolation and random growth models. Quasiconformal maps allow a controlled amount of angle distortion. These maps are a flexible and extremely useful generalization of conformal maps that help us better understand the special case of conformal mappings, but also introduces many important new problems and techniques. The PI's previous work has used quasiconformal maps to give the best known algorithm for computing conformal maps onto polygons and the best known algorithms for meshing planar domains into triangles or quadrilaterals with optimal geometric properties. Such meshes play an important role in many numerical problems from computer graphs to finite element methods for PDEs and many of these methods work more effectively if the underlying mesh has good geometric properties. The proposed work will extend and sharpen the results already obtained and will use similar ideas to attack other problems that arise in dynamics and geometry.
