Triangulations in dimension three: algorithms and geometric structures

佩雷尔曼最近获得了菲尔兹奖，因为他解决了三维空间上的几何化和庞加莱猜想，使用了一种非常深的热流方法来找到这些空间上的最佳几何形状。该项目将开发一种新的建设性方法来构建这些最佳的几何结构。这将导致有效的算法方法来区分三维空间，并应用于结和链接的研究（例如，打结的DNA分子）和数学物理。该项目还将为研究三维空间分类中的重要问题提供新技术，例如虚拟哈肯猜想。

Perelman recently won a Fields medal for the solution of the geometrisation and Poincare conjectures on three-dimensional spaces, using a very deep heat flow method to find optimal geometries on these spaces. The project will develop a new constructive approach to building these optimal geometric structures. This will lead to effective algorithmic methods to distinguish three-dimensional spaces, with applications to the study of knots and links (for example, knotted DNA molecules) and to mathematical physics. The project will also provide new techniques to study important problems in the classification of three-dimensional spaces, such as the virtual Haken conjecture.