Special geometric structures and integrability

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.