Special geometric structures and integrability

几何是代数（如多项式方程的解）和分析（如理解数学物理的微分方程）之间的桥梁。在这些领域中，有一般问题和特殊问题的自然划分，而本项目涉及后者。在代数中，特殊的结构涉及对称群的表示，而在分析中，特殊的微分方程被称为可积系统。这里的桥梁特别坚固，这个项目的目的是利用几何来加深和扩展我们对几何结构中的可积性和/或表征理论在可积系统中的作用的理解。这需要相当的代数和微分几何背景，主要的初始目标是通过学习代数几何中的格式、束和模空间，同调代数中的派生范畴，以及复杂几何和表示理论的方法，如扭转理论和Kahler度量，来掌握这一背景。在数学物理和控制工程，以及代数几何，表示理论和可积系统有潜在的好处。

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.