Collaborative Research: CCF-TF: Computing Geometric Structures of 3-Manifolds

利用Poincar‘e猜想和瑟斯顿几何化猜想的解，从理论上证明了所有闭的3-流形都可以分解成具有8个正则几何之一的碎片。三维流形的几何结构在几何学和拓扑学中起着重要的作用。该提案的重点是发明实用的算法来计算三维流形的几何结构。几何算法结合了数值方法和符号方法来计算离散三维流形上的规范黎曼度量。这些算法将为解决工程领域中许多重要和持久的开放问题奠定基础。现实生活中的所有形状都是体积形状。关于形状的计算算法是基于三维流形的几何结构的，或显式或隐式。重要的是要了解三维流形上的各种几何结构，并设计严格的计算框架来逼近它们。该方案着重于利用三角剖分、角度结构和体积泛函计算三维流形上的正则黎曼度量。在离散环境中，黎曼度量用边长表示，曲率用二面角表示。四面体的体积对称性导致了特殊的体积能形式。体积能的临界点对应于所需的正则度规。对于双曲三维流形，体积能是凸的。全局极小点是唯一的，用牛顿方法是可达的。在一般的三维流形中，体积能量比较复杂，可能存在拓扑障碍。该方案研究了障碍物的形成，并设计了不同的策略来修改三角剖分，以消除障碍物并获得规范度量解。三维流形的几何结构可以直接应用于计算机图形学、计算机视觉、几何建模和医学成像等诸多领域。该实用计算工具将有助于数学家和物理学家研究低维拓扑。可视化工具对于教授和传播知识将是有价值的。

0830550Gu, XianfengWith the solution of Poincar´e's conjecture and Thurston's geometrization conjecture, it has been proven in theory that all closed 3-manifolds can be decomposed to pieces which admit one of eight canonical geometries. Geometric structures of 3-manifolds play fundamental roles in geometry and topology. The proposal focuses on inventing practical algorithms to compute geometric structures of 3-manifolds. The geometric algorithms combine both numerical and symbolic methods to compute the canonical Riemannian metrics on discrete 3-manifolds. These algorithms will lay down the foundations to tackle many important and long lasting open problems in engineering fields.All shapes in real life are volumetric. The computational algorithms on shapes are based on geometric structures of 3-manifolds, either explicitly or implicitly. It is important to understand various geometric structures on 3-manifolds and to design rigorous computational framework to approximate them. The proposal focuses on computing canonical Riemannian metrics on 3-manifold using triangulations, angle structures and the volume functional. In discrete setting, Riemannian metrics are represented as edge lengths, the curvatures are represented as dihedral angles. The symmetry of volumes of tetrahedra induces special volumetric energy form. The critical points of the volume energies correspond to the desired canonical metrics. For hyperbolic 3-manifolds, the volume energy is convex. The global minimal point is unique and reachable using Newton's method. In general 3-manifolds, the volumetric energy is more complicted, there may exist topological obstructions. The proposal studies the formation of the obstructions, and designs different strategies to modify the triangulation to remove the obstruction and reach the canonical metric solutions. The geometric structure of 3-manifolds can be directly applied in comuter graphics, computer vision, geometric modeling and medical imaging and many other fields. The practical computational tool will be helpful for mathematicians and physists in studying low dimensional topology. The visualization tools will be valuable for teaching and propogating the knowledge.