Volume Optimization on Triangulated 3-Manifolds.

三角 3 流形的体积优化。

• 批准号: 1105808
• 负责人: Feng Luo
• 金额: $ 9.59万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105808&HistoricalAwards=false
• 关键词: Volume; Optimization; Triangulated; Manifolds.

This project investigates the connection between geometry and topology of 3-manifolds from the point of view of triangulations. This is closely related to the discretization of SL(2,C) Chern-Simon theory in 3-dimensions. The PI proposes to use the volume functional on the finite dimensional space of circle valued angle structures as the basic tool. Given a closed triangulated 3-manifold or pseudo 3-manifold, there are Haken's theory of normal surfaces, and Thurston's algebraic gluing equation associated to the triangulation. Haken's theory is topological and studies surfaces in 3-manifolds, and Thurston's equation is geometric and tries to construct hyperbolic metrics from triangulations. Solutions to Haken's equation are well understood. However, there is no known existence theorem for Thurston's equation. The main objective of the proposal is to establish conditions on the triangulation to guarantee the existence of solutions to Thurston's equation. The PI will focus on the following conjecture relating Haken's equation with Thurston's equation. It states that for any closed minimally triangulated irreducible oriented 3-manifold, either there exists a solution to Thurston's algebraic equation, or there exist three special solutions to Haken's normal surface equation which has exactly one or two non-zero quadrilateral coordinates all supported in a tetrahedron. A weaker form of the conjecture has been established by the PI recently using volume optimization. Recent work of Futer-Gueritaud, Segerman-Tillmann, and Luo-Tillmann shows that the conjecture in the case of simply connected 3-manifolds is equivalent to the Poincare conjecture in dimension three (without using the Ricci flow).Our universe is 3-dimensional. To understand the shapes of the universe and other 3-dimensional solids, mathematicians developed the theory of 3-manifolds using topology and geometry. To investigate these 3-dimensional spaces, one of the revolutionary ideas of William Thurston says that one should use geometry and geometric tools to understand the space. This program of Thurston is called the geometrization of 3-manifolds and has dominated the study of 3-dimensional topological investigation for the past 40 years. Recent work of G. Perelman, using the Ricci flow method developed by R. Hamilton, established the conjecture of Thurston and revolutionized the field. Perelman's work is widely considered to be one of the major mile-stones in the history of mathematics. However, there remains the problem of how to find those geometric structures theoretically predicated by Thurston, Perelman and Hamilton. One of the goals of the proposal aims at developing algorithms to find these geometries on 3-dimensional spaces.

本计画从三角剖分的观点探讨三维流形的几何与拓扑之间的关系。这与SL（2，C）Chern-Simon理论在三维空间中的离散化密切相关。PI提出使用圆值角结构的有限维空间上的体积泛函作为基本工具。给定一个封闭的三角剖分三维流形或伪三维流形，有哈肯的法曲面理论和瑟斯顿的代数胶合方程与三角剖分相关联。哈肯的理论是拓扑和研究表面的3流形，和瑟斯顿的方程是几何和试图构建双曲度量三角。 哈肯方程的解是很好理解的。 然而，没有已知的存在定理瑟斯顿方程。该提案的主要目标是建立三角剖分的条件，以保证瑟斯顿方程的解的存在性。 PI将专注于以下关于哈肯方程与瑟斯顿方程的猜想。它指出，对于任何封闭的极小三角化不可约定向3-流形，要么存在Thurston代数方程的解，要么存在Haken法面方程的三个特解，这些特解恰好有一个或两个非零四边形坐标都支持在四面体中。PI最近使用体积优化建立了一个较弱形式的猜想。Futer-Gueritaud，Segerman-Tillmann和Luo-Tillmann最近的工作表明，在单连通3-流形的情况下的猜想等价于3维的庞加莱猜想（不使用Ricci流）。我们的宇宙是3维的。为了理解宇宙和其他三维固体的形状，数学家们利用拓扑学和几何学发展了三维流形理论。为了研究这些三维空间，威廉·瑟斯顿（William Thurston）的一个革命性想法是，人们应该使用几何和几何工具来理解空间。Thurston的这个程序被称为3-流形的几何化，在过去的40年里一直主导着三维拓扑研究。最近的工作G。Perelman提出的Ricci流方法，汉密尔顿，建立了猜想瑟斯顿和革命性的领域。佩雷尔曼的工作被广泛认为是一个主要里程碑在数学史上。然而，如何找到瑟斯顿、佩雷尔曼和汉密尔顿理论上预言的那些几何结构仍然是一个问题。该提案的目标之一旨在开发算法，以在三维空间中找到这些几何形状。