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会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：即使球不是平的，你也可以用一个小矩形补丁来修补任何小的穿孔。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球的表面或一张平坦的纸上。它怎么能区分这两个人呢？一种方法是计算欧拉特征V E+F，即三角剖分中的顶点数减去边数再加上面数，这与三角剖分无关，是拓扑不变量的一个例子。 低维拓扑学中最重要的问题之一是区分所有的三维流形。了解这些流形的全球结构在物理、化学和生物学中有许多应用。申请人提出了三维流形和其中的纽结的新的不变量，并提出了计算这些不变量的新方法。不变量是使用规范理论方法定义的，适用于字符变化，并且有许多有趣的学生项目是研究计划的重要部分。这项研究计划的长期好处是双重的：所获得的知识将有助于确定规范理论在多大程度上可以提供新的不变量，这些不变量可用于帮助对3-流形和纽结进行分类，以及培训计划将培养出具有几何拓扑学研究技能的本科生、研究生和博士后水平的高素质人才。

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. 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 knots inside them, and he proposes new methods for computing these invariants. The invariants are defined using gauge theoretic methods applied to character varieties, and there are a number of interesting student projects that are an important part of the research program. 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.