Mathematical Sciences: The Geometry and Topology of Instantons and Related Problems

首席研究员将扩大他最近的 复形上新同伦结构的发现 表面。 这个跨学科的项目涉及使用 代数拓扑学来理解解的结构 一类非线性偏微分方程组空间 微分几何中自然出现的方程， 数学物理 此外，他还将调查几个 涉及迭代循环空间结构的问题。 两个形状是同伦的，如果它们上的闭环有 相同的塌陷特性。 例如，一个圆和 一个实心咖啡杯（有一个手柄）的形状是同伦的 相当于 但是如果杯子有两个把手或者没有 如果它是一个圆，那么它就不是一个圆。 这 数学家们已经对这个概念进行了一百多年的探索。 多年来，但它在与数学物理有关的问题中的使用 是最近才出现的现象。 主要研究者将 为这一日益增长的跨学科活动做出贡献。

The principal investigator will expand upon his recent discovery of new homotopy-theoretic structures on complex surfaces. This interdisciplinary project involves the use of algebraic topology to understand the structure of the solution space of certain systems of nonlinear partial differential equations which appear naturally in differential geometry and mathematical physics. Additionally, he will investigate several problems involving iterated loop space structures. Two shapes are homotopic if closed loops on them have identical collapsing properties. For example, a circle and the shape of a solid coffee mug (with one handle) are homotopy equivalent. But if the mug were to have two handles or no handles, it would not be homotopy equivalent to a circle. This concept has been explored by mathematicians for over a hundred years, but its use in problems related to mathematical physics has been a recent phenomena. The principal investigator will contribute to this growing interdisciplinary activity.