Wilmer Smilde, Exterior Differential Systems
Once upon a time there was a mathematician called Élie Cartan. He was no regular mathematician, and it was no regular time. It was at the height of the reign of the Coordinate Chad, dominating most of regions analysis and geometry. While Coordinate Chad was a very efficient governor, it limited the peoples vision by spreading a thick mist all over the area. Everyone at the time was forced to work with Coordinate Chad and not allowed to look further. For example, the work of Sophus Lie, sophusticated enough to see the global symmetries in differential equations, was largely discarded by Chad as mere Lies.
Élie Cartan was born with a superpower. His eyes could penetrate the mist spread by Coordinate Chad. With his power he sought to free the lands from Coordinate. The first aim was to free differential Calculus from coordinates. This was completed by the discovery of Cartan’s magic formula: Lie’s derivative was now Coordinate-free!
To free PDEs from Coordinate’s reign, Cartan developed a new theory of differential systems, nowadays known as Exterior Differential Systems. It completely recasts a system of PDEs as a graded ideal in the ring of differential forms that is closed under the exterior derivative. It’s geometric nature makes it extremely suitable for geometric problems. For instance, Cartan used it to solve Lie’s integration problem of Lie algebras. Now that’s pretty cool :)
In this two-part series I will give an introduction to Exterior Differential systems. I will go over the basic definitions, some examples and state the Cartan-Kähler theorem, which is a widely applicable “existence of solutions” result that basically relies on a linear algebra computation. I will go over many applications of this theorem.
In the second talk, I will finish the applications of the Cartan-Kähler theorem and discuss another topic that I have yet to prepare.
There will be pizza!
