This Fall I will be teaching, among other courses, an introductory graduate course on Ordinary Differential Equations. This is actually one of my favorite courses, which, among other things, shows how actually mathematics is being developed, contrary to more abstract courses such as Analysis or Algebra, which develop various mathematical concepts without specific motivation. To proceed with the analysis of ODE I will need
- Notion of a Banach space
- Contraction mapping theorem
- Notion of a linear operator
- Matrix exponent
- Jordan normal form
- Derivative along a vector field
- Notion of a self-adjoint operator
- Hilbert spaces
and many other important things.
Anyway, as with many other my courses, I am not satisfied with any of the existing textbooks (but still think that Hirch and Smale is the best 100% rigorous graduate introduction, Arnold is indispensable for acquiring the geometric intuition, Hartman is a must have reference book, and Teschl is a comprehensive modern treatment), and planning to prove students with my own lecture notes. Here is the first introductory part of them
More can be found on the course web page.