Special geometric structures and integrability

几何提供了代数（例如找到多项式方程的解）和分析（例如理解数学物理的微分方程）之间的桥梁。在这些领域中，有一种自然的划分，即一般问题和特殊问题，本项目涉及后者。在代数中，特殊结构涉及对称群的表示，而在分析中，特殊微分方程被称为可积系统。这里的桥梁是特别强大的，这个项目的目的是使用几何加深和扩展我们的理解的作用可积性发挥的几何结构和/或作用表示理论发挥在可积系统。这需要相当大的背景代数和微分几何，和主要的初始目标是掌握这一背景，通过研究计划，层和模空间代数几何，同调代数中的导出范畴，复几何和表示论的方法，如扭量理论和Kahler度量。在数学物理和控制工程以及代数几何，表示论和可积系统中有潜在的好处。

Geometry provides a bridge between algebra (e.g. finding solutions of polynomial equations) and analysis (e.g. understanding differential equations of mathematical physics). In these fields there is a natural division into generic problems and special problems, and this project concerns the latter. In algebra, the special structures involve representations of symmetry groups, while in analysis, the special differential equations are known as integrable systems. The bridges here are particularly strong, and the aim of this project is to use geometry to deepen and extend our understanding of the role integrability plays in geometric structures and/or the role representation theory plays in integrable systems.This requires considerable background in algebraic and differential geometry, and the main initial objectives are to master this background by studying schemes, sheaves and moduli spaces in algebraic geometry, derived categories in homological algebra, and methods of complex geometry and representation theory such as twistor theory and Kahler metrics.There are potential benefits in mathematical physics and control engineering as well as to algebraic geometry, representation theory and integrable systems.