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

0830550顾先锋随着Poincar ′ e猜想和Thurston几何化猜想的解决，从理论上证明了所有闭三维流形都可以分解为八个标准几何中的一个.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算法来计算三维流形的几何结构。几何算法联合收割机结合数值和符号方法计算离散三维流形上的正则黎曼度量。这些算法将为解决工程领域中许多重要而持久的开放问题奠定基础。形状的计算算法是基于三维流形的几何结构，无论是显式或隐式。理解三维流形上的各种几何结构并设计严格的计算框架来近似它们是很重要的。该建议的重点是使用三角剖分，角结构和体积泛函计算3-流形上的典型黎曼度量。在离散情况下，黎曼度量表示为边长，曲率表示为二面角。四面体体积的对称性导致了特殊的体积能量形式。体积能量的临界点对应于所需的正则度量。对于双曲三维流形，体积能量是凸的。全局极小点是唯一的，并可达到使用牛顿法。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，并达到规范的度量解决方案。 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。这一实用的计算工具将有助于数学家和物理学家研究低维拓扑。这些可视化工具对知识的教学和传播具有重要的价值。

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.