Geometry of Conformal and Quasiconformal Mappings

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AbstractAward: DMS-0405578Principal Investigator: Christopher BishopThe PI, Christopher Bishop, will study the geometric propertiesof conformal and quasiconformal mappings, focusing on theinteractions between conformal structures, hyperbolic geometry,low dimensional topology and numerical analysis. For example,Moore's theorem states that if we take a certain collection ofsets on the 2-sphere and topologically collapse them to pointsthen we obtain a new topological sphere. The PI will investigatewhen the quotient map can be conformal and will consider a numberof concrete problems from this general perspective, includingconformal welding, characterizations of John domains, Koebe'sconjecture, construction of Kleinian groups and other dynamicalobjects. The PI will continue his earlier work on the geometryof Kleinian limit sets and the behavior of the dimension as wedeform the limit set. The PI will also continue his work on theconnections between computational geometry, hyperbolic geometryand conformal mappings, and seek new algorithms which compute theRiemann mapping quickly and with rigorous error estimates. Inparticular he will investigate computing conformal maps using themedial axis (an object from computational geometry) which isclosely linked to 3-dimensional hyperbolic geometry via convexhulls.Conformal mappings are important both for their central role innumerous mathematical problems (complex analysis, dynamicalsystems,...) and in various applications (fluid flow, brainmapping, statistical physics, numerical analysis of differentialequations,...), so we must have a good theoretical understandingof these maps and good methods for computing them inpractice. The proposal deepens our theoretical understanding ofconformal maps by investigating new connections with other partsof mathematics and computer science (point set topology,3-dimensional hyperbolic geometry, Voronoi diagrams) and seeks touse these connections to invent new algorithms for computingconformal maps. For example, the medial axis is a widely studiedobject in computer science (it is a description of the shape onan object which has numerous applications in pattern recognition,robotic motion, biology,...). The PI discovered it can also beused to give a rough but fast approximation to conformal maps.The PI will seek to improve this method, giving better algorithmsfor conformal maps and also developing a better understanding ofthe medial axis which will impact its other applications (forexample, the medial axis can change drastically when the objectbeing described changes only a little; this is a seriouscomputational problem which can be addressed by thinking of themedial axis as an object in hyperbolic geometry instead of theusual Euclidean geometry). The connection between the medial axisand conformal maps may also lead to new ideas for studyingapplications in three dimensions (where conformal mappings do notexist, but maps based on the medial axis still do). Threedimensional problems are the most important for applications, butour understanding lags far behind the two dimensional case, sonew two dimensional ideas which generalize to higher dimensionsare important.