Gauge theory and low-dimensional topology

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。

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.