Algebraic and geometric topology

我们的物理宇宙被建模为三维几何对象，称为三维流形，类似于二维表面。 与曲面不同的是，三维流形的分类仍然是一个开放的和具有挑战性的问题。关于三维流形的许多信息都是用代数编码的，例如所谓的基本群。 尽管最近取得了一些引人注目的进展，如庞加莱猜想的解决，它确定了唯一的3-流形具有平凡的基本群，许多悬而未决的问题和奥秘仍然在3-D拓扑中关于流形和对象，如结和曲面。本建议是研究3-D拓扑和代数结构之间的深刻联系，反映他们的形状和结构在其中。 预计这将在宇宙学，动力系统和其他科学，如化学和生物学等不同领域中找到应用，其中理解3-D结构是重要的。

Our physical universe is modelled as a 3-dimensional geometric object called a 3-manifold, analogous to 2-D surfaces.  Unlike surfaces, classifying the possible types of 3-manifolds is still an open and challenging problem.Much of the information on 3-manifolds is encoded in algebra, for example the so-called fundamental group.  Despite spectacular recent progress, such as the solution of the Poincare conjecture, which identifies the only 3-manifold which has trivial fundamental group, many open questions and mysteries remain in 3-D topology regarding manifolds and objects in them such as knots and surfaces.This proposal is to study the deep connection between 3-D topology and the algebraic constructions which reflect their shapes and structures within them.  It is expected that this will find application in such diverse areas as cosmology, dynamical systems, and other sciences such as chemistry and biology in which understanding 3-D stuctures is important.