Abelianization of Connections in Two and Three Dimensions

PI使用从粒子物理学引入的方法研究几何问题。在与他的合作者的联合工作中，他最近开发了一种新的几何技术“阿贝尔化”-所谓的，因为它减少了nonabelian问题（涉及操作的顺序，我们做的操作事项），以更简单的阿贝尔的（顺序无关紧要）。PI与他的合作者和研究生一起，将致力于阿贝尔化的几个新应用。一个应用是一种新的方法来解决某些微分方程，包括薛定谔方程，它控制着一些量子系统的物理。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，有助于使这两个领域更加紧密地联系在一起。PI最近与合作者的联合工作为平坦连接理论引入了一个新的成分：一种将曲面上秩为N的复向量丛上的平坦连接“阿贝尔化”的方法，将其替换为N重分支覆盖曲面上的线束上的几乎平坦连接。这个新理论的全部范围还不为人所知：似乎还有更多的阿贝尔化的用途有待发现。PI的目标是开发其中的一些。 首先，他将研究曲面上的一类特殊连接，称为“opers”，可以用规范的方式进行阿贝尔化。 一方面，这是希钦系统扭量线阿贝尔化的一个热身。另一方面，它提供了一种新的方式来理解opers的轨迹，从而对许多相关的问题，从线性标量微分算子的经典理论的拓扑弦理论的非微扰扩展的新视角。第二，他将考虑3流形上的阿贝尔化而不是曲面。一个直接的应用是发展新的公式为经典复杂的陈-西蒙斯不变量。第三，PI的目的是发展一个新的关系abelianization和Floer理论的余切丛。

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.