Lecture 10 (Thu, May 1st, 2008)

Stokes phenomenon for irregular singularities of linear systems

1. Irregular singularities: total recall. Formal diagonalizability of non-resonant systems.
2. Sectorial gauge equivalence: formal, holomorphic, asymptotic series.
3. Separation rays. Sibuya theorem on sectorial normalization (statement).
4. Sectorial authomorphisms. Rigidity of the normal form in large sectors.
5. Stokes matrix cochain and Stokes matrix multipliers as complete invariants of holomorphic classification of irregular singularities.
6. Stokes phenomenon. Realization theorem (Birkhoff). Generic divergence of the formal gauge normalizing transformations.

Recommended reading: Sections 20F-20I from the Book

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: