Volume Optimization on Triangulated 3-Manifolds.

本课题从三角测量的角度探讨了三维流形的几何与拓扑之间的联系。这与SL(2，C) chen - simon理论在三维中的离散化密切相关。PI提出将圆值角结构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化3-流形或伪3-流形，有Haken的法线曲面理论和Thurston的代数粘接方程。哈肯的理论是拓扑学的，研究3流形中的曲面，瑟斯顿的方程是几何的，试图从三角剖分构造双曲度量。哈肯方程的解很容易理解。然而，瑟斯顿方程没有已知的存在性定理。本文的主要目的是在三角剖分上建立保证Thurston方程解存在的条件。PI将重点关注以下关于Haken方程和Thurston方程的猜想。指出对于任何封闭的最小三角化不可约定向3流形，要么存在Thurston代数方程的一个解，要么存在Haken法曲面方程的三个特解，该特解恰好具有一个或两个非零四边形坐标，且均支持在四面体中。PI最近利用体积优化建立了该猜想的一种弱形式。Futer-Gueritaud， Segerman-Tillmann和Luo-Tillmann最近的工作表明，在单连通3-流形情况下的猜想与三维的庞加莱猜想等效（不使用里奇流）。我们的宇宙是三维的。为了理解宇宙和其他三维固体的形状，数学家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）的一个革命性思想是，人们应该使用几何和几何工具来理解空间。瑟斯顿的这一方案被称为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的研究。G.佩雷尔曼最近的工作，利用R.汉密尔顿发展的里奇流方法，建立了瑟斯顿猜想，并彻底改变了这一领域。佩雷尔曼的工作被广泛认为是数学历史上的重要里程碑之一。然而，如何找到Thurston， Perelman和Hamilton在理论上预测的几何结构仍然是一个问题。该提案的目标之一是开发在三维空间中找到这些几何形状的算法。

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.