Algebraic and geometric topology

我们的物理宇宙被建模为一个被称为3-流形的三维几何对象，类似于2-D表面。与表面不同的是，对3-流形的可能类型的分类仍然是一个开放和具有挑战性的问题。关于3-流形的许多信息是用代数编码的，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，它标识了唯一具有平凡基本群的3-流形，关于流形和流形中的对象，如纽结和曲面，在三维拓扑学中仍有许多悬而未决的问题和谜团。本文旨在研究三维拓扑学和反映其内部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、动力学系统以及其他科学领域中得到应用，在这些领域中，理解三维结构是非常重要的。

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.