Gauge theory and low dimensional topology

流形是几何形状，在任何点附近，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。球的表面提供了一个具体的例子：即使它不是平的，也可以用一个小矩形补丁来修复任何小的穿孔。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。

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 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.