Semester II: tentative programme

The course on Analytic and Geometric Theory of Differential Equations resumes on Wed Feb 27, 2008.

Note the change in the schedule: in the second semester classes will be in Room 261, on Wednesdays (instead of Thursdays), still between 9:00 and 10:50 (apologies before those who suffer from the pre-dawn wake-up). Thursdays, from 14:00 till 16:00.

The second semester will be centered on the theory of linear systems, as exposed in Chapter III of the book (warning: the draft posted on my web page is really outdated. I will provide links to individual sections of the printed edition, with printing option disabled, as before, to protect the copyright).

More precise (albeit still provisional) plan is as follows.

1. General properties of systems of linear ordinary differential equations in the complex domain. Gauge equivalence. Monodromy and holonomy.
2. Local theory of singular points. Fuchsian, regular and irregular singularities.
3. Towards the global theory of  linear systems: holomorphic vector bundles.
4. Towards the global theory of linear systems: meromorphic connexions on vector bundles.
5. Reconstruction of a linear system from its monodromy group. The Riemann–Hilbert problem.
6. Positive results on solvability of the Riemann–Hilbert problem. Bolibruch–Kostov theorem.
7. Negative results and the Bolibruch counterexample.
8. Scalar high order linear ordinary differential equations and associated geometric structures. Hypergeometric equations.
9. Irregular singularities: formal theory.
10. Irregular singularities: elements of analytic theory. Stokes phenomenon.
11. Elements of multidimensional theory: meromorphic flat connexions on .

The second semester is intended to be as independent from the first semester, as possible, so that newcomers may join at this junction.