Knot theory and low-dimensional topology

纽结理论和低维拓扑

• 批准号: RGPIN-2021-04229
• 负责人: Boden, Hans
• 金额: $ 1.75万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=742940
• 关键词: Knot; theory; low; dimensional; topology

Knot theory is a branch of mathematics that studies an idealized version of a knotted string where the two ends are fused together. The central problem is to distinguish when two knots are the same and when they are different. The early pioneers of the subject developed heuristic methods which they used to tabulate prime knots to 10 crossings. It was only with the introduction of rigorous mathematical techniques that the subject was placed on a solid mathematical footing. With many more sophisticated techniques and tools, mathematicians have now tabulated knots up to 16 crossings.   Low-dimensional topology involves the study of manifolds in dimensions 2,3 and 4. The central problem is to classify 3 and 4-dimensional manifolds. Manifolds are geometric shapes that, near any point, look flat like Euclidean space but whose global structure may be twisted and curved. The surface of a ball and the surface of a doughnut provide concrete examples; imagine a beach ball or an inner tube. Neither is flat, nevertheless, any rupture to the ball or inner tube can be repaired with a small rectangular patch and a bit of glue. Since manifolds are locally indistinguishable, mathematicians search for invariants that reflect the manifold's global curving and twisting. For example, imagine a near-sighted insect living on either the surface of a ball or a flat sheet of paper. How could it tell the two apart? One method would be to compute the Euler characteristic V - E + F, the number of vertices minus the number of edges plus the number of faces in a triangulation, which is independent of the triangulation and is an example of a topological invariant. These two branches of mathematics are intimately related. One reason is that 3 and 4-dimensional manifolds can be constructed as Dehn surgeries along knots or links. Knot theory and low-dimensional topology have numerous applications to physics, chemistry and biology. The applicant proposes to introduce new invariants of 3-dimensional manifolds and knots inside them. He will develop new approaches for computing these invariants. The invariants will be defined and studied using algebraic and geometric methods, including gauge theory and quantum topology. There are many interesting student research projects related to these questions, and the applicant will promote participation from a diverse group of students and postdoctoral fellows. He will foster an inclusive research environment that encourages learning and advancement of women and underrepresented minorities. The long-term benefits of this research program are two-fold: the knowledge gained will help determine to what extent geometric methods can deliver new invariants of knots and 3-manifolds, and the training program will produce highly qualified personnel at all levels, with the necessary analytical and computational skills to take leadership roles in the information-based economy.

纽结理论是数学的一个分支，它研究的是一根打结的绳子的理想化形式，其中两端融合在一起。核心问题是区分两个结何时相同，何时不同。早期的先驱者的主题开发了启发式的方法，他们用来制表总理结10个路口。这是只有引进严格的数学技术，这一主题是放在一个坚实的数学基础。有了许多更复杂的技术和工具，数学家现在已经列出了多达16个交叉点的结。 低维拓扑学涉及到二维、三维和四维流形的研究。中心问题是对3维和4维流形进行分类。流形是几何形状，在任何点附近，看起来像欧几里得空间一样平坦，但其整体结构可能是扭曲和弯曲的。球的表面和甜甜圈的表面提供了具体的例子;想象一个海滩球或一个内胎。两者都不是平的，然而，任何破裂的球或内胎可以修复一个小矩形补丁和一点胶水。由于流形是局部不可区分的，数学家们寻找反映流形整体弯曲和扭曲的不变量。例如，想象一只近视的昆虫生活在一个球或一张平纸的表面。它怎么能分辨出来？一种方法是计算欧拉特征线V-E + F，即三角剖分中的顶点数减去边数加上面数，它独立于三角剖分，是拓扑不变量的一个例子。这两个数学分支密切相关。一个原因是三维和四维流形可以构造为沿沿着结或链的Dehn手术。纽结理论和低维拓扑在物理、化学和生物学中有着广泛的应用。申请人提出引入三维流形和其中的结的新不变量。他将开发新的方法来计算这些不变量。不变量将被定义和研究使用代数和几何方法，包括规范理论和量子拓扑。有许多有趣的学生研究项目与这些问题有关，申请人将促进来自不同群体的学生和博士后研究员的参与。他将促进包容性的研究环境，鼓励妇女和代表性不足的少数民族的学习和进步。这项研究计划的长期利益是双重的：所获得的知识将有助于确定几何方法在多大程度上可以提供新的不变量的结和3-流形，和培训计划将产生高素质的人员在各级，必要的分析和计算技能，以采取领导作用，在信息化经济。