Volume Optimization on Triangulated 3-Manifolds.

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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.