Algebraic and geometric topology

我们的物理宇宙被建模为一个三维几何物体，称为三维流形，类似于二维表面。与曲面不同，对3-流形的可能类型进行分类仍然是一个开放且具有挑战性的问题。关于3-流形的大部分信息都是用代数编码的，例如所谓的基本群。尽管最近取得了惊人的进展，例如庞加莱猜想的解决，它确定了唯一具有平凡基本群的3-流形，但关于流形及其中的物体（如结和表面）的许多悬而未决的问题和谜团仍然存在于3-D拓扑中。这一建议是研究三维拓扑和反映其形状和结构的代数结构之间的深层联系。预计这将在宇宙学、动力系统和其他科学（如化学和生物学）等不同领域得到应用，在这些领域中，理解三维结构是很重要的。

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.