The PI, Christopher Bishop, will study problems in two and three dimensional geometry which come from classical complex analysis, the theory of quasiconformal mappings, hyperbolic geometry and computational geometry, with a particular emphasis on interactions between these areas. In recent work the PI has shown that ideas from hyperbolic and computational geometry could help resolve a question of classical analysis: how to efficiently compute conformal maps onto planar domains. This work will be extended to new settings, and will be used to study problems arising in computational geometry, e.g., efficient generation of meshes with desirable properties. The PI will investigate the possible application of the carpenter's rule problem from computational geometry to an analysis problem about deforming chord-arc curves. He will also continue his work on the distortion properties of quasiconformal mappings and on the conformal welding problem.
The PI, Christopher Bishop, will continue his investigations into the theory and computation of conformal and quasiconformal maps. Conformal maps send one region to another so that angles (but not necessarily distances) are preserved. Quasiconformal maps may distort angles, but only by a limited amount. One of the oldest and most famous examples is the Mercator projection which maps the spherical Earth to flat piece of paper. This can't be done without some sort of distortion, and for navigation it is more convenient to have angles preserved than distances (so looking at a chart you know the correct direction to head, if not the exact distance to be covered). Conformal maps also appear in many applications where one wants to transfer a problem from a domain with complicated geometry to a simpler one (such as a disk) where it is easier to solve. Examples come from aerodynamics, fluid flow, vibrating membranes, heat flow, electrostatics and many other problems. Conformal maps also play a fundamental role within pure mathematics in areas such as dynamics, complex analysis, probability and geometry. Because of their importance, there are numerous techniques for computing conformal maps numerically, but different methods work best in different situations and often a method fails for sufficiently complicated regions. The PI has developed an algorithm that is guaranteed to work for a large class of regions, and is able to estimate the time needed to achieve a given accuracy. The algorithm is based on new connections between geometric concepts from computer science and non-Euclidean geometry arising in pure mathematics. Under the current proposal the PI will seek to turn this theoretical algorithm into a practical method, investigate generalizations more complicated regions and to higher dimensions, and apply the method to problems arising from computer science.
