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。由于流形都是局部不可区分的，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？一种方法是计算一个称为欧拉特征的不变量，它只是数字V-E + F（三角剖分中的顶点数减去边数加上面数）。这个数实际上与三角剖分无关，因此是一个不变量。 低维拓扑学中最重要的问题之一是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。申请人提出了新的不变量的3流形和结内，他们提出了新的方法来计算这些不变量。不变量是使用规范理论方法来定义的，并且有一些有趣的学生研究项目是提案中描述的培训计划的重要组成部分。这个研究计划的长期利益是双重的：获得的知识将有助于确定在何种程度上规范理论可以提供新的不变量，可用于帮助分类3-流形和结，和培训计划将产生高素质的人才在本科生，研究生和博士后水平与几何拓扑学的研究技能。

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 of knots inside them, and he proposes new methods for computing these invariants. The invariants are defined using gauge theoretic methods to character varieties, and there are a number of interesting student research projects that are an important part of the training program described in the proposal. The long-term benefits of this research program are two-fold: the knowledge gained will help determine to what extent gauge theory can deliver new invariants that can be used to help classify 3-manifolds and knots, and the training program will produce highly qualified personnel at the undergraduate, postgraduate, and postdoctoral levels with research skills in geometric topology.