Abelianization of Connections in Two and Three Dimensions

PI使用从粒子物理学引进的方法研究几何问题。在与他的合作者的共同工作中，他最近开发了一种新的几何技术“阿贝尔化”——之所以这样称呼，是因为它将非阿贝尔问题（涉及我们进行操作的顺序重要的操作）简化为更简单的阿贝尔问题（顺序无关）。PI将与他的合作者和研究生一起研究阿贝尔化的几个新应用。一个应用是解决某些微分方程的新方法，包括支配某些量子系统物理的薛定谔方程。第二个应用是测量三维空间拓扑的新方法。这项工作的结果将在数学和高能物理界广泛传播，有助于将这两个领域更紧密地联系在一起。这项工作还将有助于培训这两个领域的研究生。PI最近与合作者的联合工作为平面连接理论引入了一个新成分：一种将平面上N阶复向量束上的平面连接“阿贝尔化”的方法，用N折分支覆盖表面上的线束上的几乎平坦连接取代它们。这个新理论的全部范围还不为人所知：看来阿贝尔化还有更多的用途有待发现。PI的目标是开发其中的一些。首先，他将研究表面上的一组特殊连接，称为“op”，它可以用规范的方式被化。一方面，这是对希钦系统中扭线的阿贝尔化的预热。另一方面，从经典的线性标量微分算子理论到拓扑弦理论的非微扰扩展，它提供了一种理解算子轨迹的新方法，从而为许多相关问题提供了新的视角。其次，他将考虑在3流形而不是曲面上的阿贝尔化。一个直接的应用是开发经典复陈氏-西蒙斯不变量的新公式。第三，在共切束上建立了阿贝尔化与花理论之间的新关系。

The PI studies problems of geometry using methods imported from particle physics. In joint work with his collaborators, he has recently developed a new geometric technique of "abelianization" -- so called because it reduces nonabelian problems (involving operations for which the order in which we do the operations matters) to simpler abelian ones (where the order does not matter). The PI, together with his collaborators and graduate students, will work on several new applications of abelianization. One application is a new approach to solving certain differential equations, including the Schrodinger equation which governs the physics of some quantum systems. A second application is a new way of measuring the topology of 3-dimensional spaces. The results of this work will be disseminated broadly both in the mathematics and high-energy physics communities, helping to bring these two areas closer together. The work will also contribute to the training of graduate students in both fields.The PI's recent joint work with collaborators introduced a new ingredient to the theory of flat connections: a way of "abelianizing" flat connections on a rank N complex vector bundle over a surface, replacing them by almost-flat connections on a line bundle over an N-fold branched covering surface. The full scope of this new theory is not yet known: it appears that there are many more uses of abelianization yet to be discovered. The PI aims to develop some of these. First, he will study a family of special connections on surfaces called "opers," which can be abelianized in a canonical way. On the one hand, this is a warmup for the abelianization of the twistor lines in the Hitchin system. On the other hand, it gives a new way of understanding the locus of opers and thus a new perspective on many related issues, from the classical theory of linear scalar differential operators to nonperturbative extensions of topological string theory. Second, he will consider abelianization on a 3-manifold instead of a surface. One immediate application is the development of new formulas for classical complex Chern-Simons invariants. Third, the PI aims to develop a new relation between abelianization and Floer theory on cotangent bundles.