Knot theory and low-dimensional topology

打结理论是数学的一个分支，它研究打结的绳子的一个理想化版本，其中两端融合在一起。中心问题是区分两个结何时相同，何时不同。这门学科的早期先驱们开发了启发式方法，他们用这种方法将素数结列为10个交叉点。只是随着严格的数学技巧的引入，这门学科才建立在坚实的数学基础上。有了许多更复杂的技术和工具，数学家们现在已经计算出了多达16个交叉点的结点。低维拓扑学涉及研究2维、3维和4维流形。中心问题是对3维和4维流形进行分类。流形是几何形状，在任何一点附近，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；想象一个沙滩球或内胎。两者都不是平坦的，然而，球或内胎的任何破裂都可以用一个小矩形补丁和一点胶水来修复。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球的表面或一张平坦的纸上。它怎么能区分这两个人呢？一种方法是计算欧拉特征V-E+F，即三角剖分中的顶点数减去边数再加上面数，这与三角剖分无关，是拓扑不变量的一个例子。数学的这两个分支是密切相关的。其中一个原因是，三维和四维流形可以被构造为沿着纽结或链节的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。申请人提议引入新的三维流形不变量和其中的结点。他将开发计算这些不变量的新方法。不变量的定义和研究将使用代数和几何方法，包括规范理论和量子拓扑学。有许多与这些问题相关的有趣的学生研究项目，申请者将促进不同群体的学生和博士后研究员的参与。他将促进一个包容性的研究环境，鼓励妇女和代表性不足的少数群体的学习和提高。这项研究计划的长期好处是双重的：所获得的知识将有助于确定几何方法在多大程度上能够提供新的节点和3-流形不变量，以及培训计划将培养出具有必要的分析和计算技能的各级高素质人员，以在信息经济中发挥领导作用。

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.