Algebraic and geometric topology

代数和几何拓扑

• 批准号: 8082-2007
• 负责人: Rolfsen, Dale
• 金额: $ 1.78万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=363761
• 关键词: Algebraic; geometric; topology

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.

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。与表面不同，对三维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。尽管最近取得了惊人的进展，例如庞加莱猜想的解决方案，它确定了唯一一个具有平凡基本群的三维流形，在三维拓扑学中，关于流形和流形中的物体，如结点和曲面，还有许多悬而未决的问题和谜团。三维拓扑和代数结构，反映了他们的形状和结构，其中。预计这将找到应用在这样的不同宇宙学、动力系统和其他科学领域，如化学和生物学，在这些领域中理解三维结构是很重要的。