Geometry of Conformal and Quasiconformal Mappings

Abstract Award: DMS-0405578 Principal Investigator: Christopher Bishop The PI, Christopher Bishop, will study the geometric properties of conformal and quasiconformal mappings, focusing on the interactions between conformal structures, hyperbolic geometry, low dimensional topology and numerical analysis. For example, Moore's theorem states that if we take a certain collection of sets on the 2-sphere and topologically collapse them to points then we obtain a new topological sphere. The PI will investigate when the quotient map can be conformal and will consider a number of concrete problems from this general perspective, including conformal welding, characterizations of John domains, Koebe's conjecture, construction of Kleinian groups and other dynamical objects. The PI will continue his earlier work on the geometry of Kleinian limit sets and the behavior of the dimension as we deform the limit set. The PI will also continue his work on the connections between computational geometry, hyperbolic geometry and conformal mappings, and seek new algorithms which compute the Riemann mapping quickly and with rigorous error estimates. In particular he will investigate computing conformal maps using the medial axis (an object from computational geometry) which is closely linked to 3-dimensional hyperbolic geometry via convex hulls. Conformal mappings are important both for their central role in numerous mathematical problems (complex analysis, dynamical systems,...) and in various applications (fluid flow, brain mapping, statistical physics, numerical analysis of differential equations,...), so we must have a good theoretical understanding of these maps and good methods for computing them in practice. The proposal deepens our theoretical understanding of conformal maps by investigating new connections with other parts of mathematics and computer science (point set topology, 3-dimensional hyperbolic geometry, Voronoi diagrams) and seeks to use these connections to invent new algorithms for computing conformal maps. For example, the medial axis is a widely studied object in computer science (it is a description of the shape on an object which has numerous applications in pattern recognition, robotic motion, biology,...). The PI discovered it can also be used to give a rough but fast approximation to conformal maps. The PI will seek to improve this method, giving better algorithms for conformal maps and also developing a better understanding of the medial axis which will impact its other applications (for example, the medial axis can change drastically when the object being described changes only a little; this is a serious computational problem which can be addressed by thinking of the medial axis as an object in hyperbolic geometry instead of the usual Euclidean geometry). The connection between the medial axis and conformal maps may also lead to new ideas for studying applications in three dimensions (where conformal mappings do not exist, but maps based on the medial axis still do). Three dimensional problems are the most important for applications, but our understanding lags far behind the two dimensional case, so new two dimensional ideas which generalize to higher dimensions are important.