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

拟共形几何在复分析、微分方程、函数论和拓扑学中显示了它的能力和灵活性。 计算拟共形几何专注于拟共形几何理论的算法研究，将抽象数学的纯领域与具体的工程应用联系起来。 该项目解决了一些基本的工程问题，其中准共形几何可以提供关键的见解，包括建立一个计算具有一般拓扑结构的表面之间的最佳仿射的变分框架，建立一个理论上良好的框架来建模表面的形状空间，以及建立在无线传感器网络，计算机图形学，固体力学。 预期成果包括计算定理的探索，新的模型，和新的几何算法与可证明的性能guarantee.This跨学科的项目提供了桥梁之间的拟共形几何和应用在广泛的工程领域，通过确定重要的几何问题，在计算机图形学，计算机视觉，几何建模，和无线传感器网络，并提供基于拟共形几何理论的计算定理和有效的算法解决方案。 预计，探索将揭示这些领域的基本问题的关键见解。

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.