CAREER: Theorem, Algorithm, and Applications of Computational Quasiconformal Geometry

University Of Louisiana At Lafayette, Lafayette LA

Investigators

Abstract

Quasiconformal geometry has showed its power and flexibility in complex analysis, differential equations, function theory, and topology. Computational quasiconformal geometry focuses on algorithmic study of quasiconformal geometry theory, which links very pure areas of abstract mathematics to concrete engineering applications. This project addresses a number of fundamental engineering problems where quasiconformal geometry can provide a key insight, including building a variational framework of computing the optimal diffeomorphism between surfaces with general topologies, building a theoretically well sound framework to model shape space of surfaces, and building anisotropic models which are widely observed in various areas including wireless sensor networks, computer graphics, and solid mechanics. Expected results include the exploration of computational theorems, new models, and novel geometric algorithms with provable performance guarantee. This interdisciplinary project provides the bridge between quasiconformal geometry and applications in broad engineering fields by identifying important geometric problems in computer graphics, computer vision, geometric modeling, and wireless sensor networks as well as supplying computational theorems and efficient algorithmic solutions based on quasiconformal geometry theory. It is expected that the exploration will reveal key insights of fundamental problems in those fields.