Lecture 9 (Thu, Apr 17, 2008)

Irregular singularities of linear systems

1. One-dimensional case: complete classification.
2. Polynomial “normal forms”: Birkhoff theorem and its “uselessness”.
3. Local reducibility: similarities and differences with the regular (Fuchsian) case.
4. Polynomial “normal form” for irreducible irregular singularity: Bolibruch theorem
5. First steps of the “genuine” normal forms theory.
   • Resonances.
   • Formal diagonalizability of nonresonant systems
   • Divergence of the normalizing transformations

Recommended reading: Section 20 from the Book

Notice

The next week there will be no classes for this reason. Expect the end of the story on May 1, 2008. In the meantime I wish to everybody חג פסח שמח and merry holidays.

Recommended reading:

Lecture 8 (Thu, Apr 10, 2008)

Geometric and global theory of linear ordinary differential equations

1. Global theory of linear equations. Jet bundles, Cartan distribution. Meromorphic connexion associated with a linear equation.
2. “Natural bundle” for a globally Fuchsian equation. Sum of characteristic exponents.
3. Riemann–Hilbert problem for Fuchsian equations. Hypergeometric equation.

Lecture 7 (Thu, Apr 3, 2008)

Linear ordinary differential equations of order n

1. Construction of the Weyl algebra (noncommutative “differential polynomials of one independent variable”). Division with remainder, factorization, solutions.
2. Reconstruction of differential equations from their solutions. Riemann theorem.
3. Regular and Fuchsian operators. Complete local reducibility. Fuchs theorem (local regularity local Fuchs property) and its reformulations.

Recommended reading: Section 19 from the book (printing disabled)