# 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.

# 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 $\iff$ local Fuchs property) and its reformulations.

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

# Riemann–Hilbert Problem: positive results

1. Formulation of the problem and its tautological solution on an abstract holomorphic vector bundle
2. Meromorhic trivialization and Plemelj theorem (solvability of the problem if one of the monodromies is diagonalizable).
3. Invariant subbundles, (ir)reducibility of a regular connexion.
4. Lemma on too different orders. Bounds on the splitting type of a bundle with irreducible Fuchsian connexion.
5. Bolibruch–Kostov theorem: solvability of the Riemann–Hilbert problem for irreducible representations.

Reading: Sections 18A-18D from the book (printing disabled).

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