Sergei Yakovenko's blog: on Math and Teaching

Tuesday, February 26, 2008

Lecture 1 (Feb 27, 2008)

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 9:55
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Systems of Linear ODEs with complex time

  1. Total recall: on differential equations in the complex domain (for the newcomers, if any) and foliations.
  2. Linear systems: vector, matrix and Pfaffian form. Fyndamental solutions. Linearity of the transport maps.
  3. Holomorphic (gauge) equivalence of linear systems. Monodromy group.
  4. Linear systems with isolated singularities. Euler system and its properties.

Reading material: Section 15 from the Book (printing disabled)

Semester II: tentative programme

Filed under: Analytic ODE course — Sergei Yakovenko @ 9:03
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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the theory of linear systems, as exposed in Chapter III of the book (warning: the draft posted on my web page is really outdated.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 \mathbb C^n.

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

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