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。低维拓扑中最重要的问题之一是如何区分所有的三维流形。了解这些流形的整体结构在物理、化学和生物学上有许多应用。申请人提出了新的3流形不变量及其计算的新方法。长期目标是使用不变量来帮助分类3流形以及其中包含的结和链接。

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.     One of the most important problems in low-dimensional topology is that of distinguishing all 3-dimensional manifolds. Understanding the global structure of these manifolds has numerous applications to physics, chemistry and biology. The applicant proposes new invariants of 3-manifolds and new methods for computing them. The long-term goal is to use the invariants to help in classifying 3-manifolds and knots and links contained in them.