Mathematical Sciences: Gauge Theory and Holomorphic Bundles with Sections

首席研究员将继续他在规范理论方面的研究，并探索一组被称为涡旋方程的方程。这些方程是为Kahler流形上的全纯束定义的，并且与Hermitian-Yang- Mills方程密切相关。在这个项目中，首席研究员将研究规范理论的三个领域：Yang-Mills型泛函，与规范不变泛函相关的模空间，以及全纯向量束的性质。规范论是几何学的一个分支，与数学物理密切相关。在这个领域中，物理学的思想被用来发展和丰富几何学，数学和几何学的技术被用来探索物理学的思想。许多不同的方法，有些是非常分析的，有些是代数的，正在成功地用于创建一个强大而美丽的物理理论。这是当今数学领域跨学科研究中令人兴奋的例子之一。

The principal investigator will continue his research in Gauge theory and explore a set of equations known as the Vortex equations. These equations are defined for holomorphic bundles over Kahler manifolds and are closely related to the Hermitian-Yang- Mills equations. The principal investigator will study three areas of Gauge theory in this project: Yang-Mills type functionals, the moduli spaces associated with gauge invariant functionals, and properties of holomorphic vector bundles. Gauge theory is a branch of geometry very closely connected with mathematical physics. It is an area in which ideas from physics are serving to develop and enrich geometry and techniques of mathematics and geometry are used to explore ideas in physics. A number of different methods, some very analytic and others algebraic, are successfully being used to create a powerful and beautiful theory in physics. It is one of the exciting examples of cross disciplinary research being done in mathematics today.