Gauge theory and low-dimensional topology

规范理论和低维拓扑

• 批准号: 238844-2011
• 负责人: Boden, Hans
• 金额: $ 1.09万
• 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: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=524470
• 关键词: Gauge; theory; low; dimensional; topology

Manifolds are geometric shapes that, near any point, look like Euclidean space even though their global structure may be twisted and curved in interesting ways. The surface of a ball provides a concrete example: even though it is not flat, one can repair any small puncture with a little rectangular patch. Since manifolds are all 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 is to compute an invariant called the Euler characteristic, which is simply the number V - E + F (the number of vertices minus the number of edges plus the number of faces in a triangulation). This number is actually independent of the triangulation and hence an invariant.

流形是几何形状，在任何一点附近，看起来都像欧几里得空间，尽管它们的整体结构可能以有趣的方式扭曲和弯曲。一个球的表面提供了一个具体的例子：即使它不是平坦的，人们也可以用一个小小的矩形补丁来修复任何小的穿孔。由于流形都是局部不可区分的，数学家们寻找反映流形全局弯曲和扭曲的不变量。例如，想象一只近视眼的昆虫生活在一个球的表面或一张平的纸上。它怎么能把两者区分开来呢？一种方法是计算一个称为欧拉特性的不变量，简单来说就是数字V - E + F（顶点数减去边数加上三角形中的面数）。这个数实际上与三角剖分无关，因此是不变量。