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 6 (Thu, Mar 27, 2008)

Bolibruch Impossibility Theorem

Revealing an obstruction for realization of a matrix group as the monodromy of a Fuchsian system on .

1. Degree (Chern class) of a complex bundle vs. that of a subbundle. The total trace of residues of a meromorphic connexion.
2. Linear algebra: Monoblock operators and their invariant subspaces.
3. Local theory revisited: local invariant subbundles of a (resonant) Fuchsian singularity in the Poincaré–Dulac–Levelt normal form.
4. Bolibruch connexions on the trivial bundle: theorem on the spectra of residues.
5. Three Matrices : the Bolibruch Counterexample.

Reading: Section 18E from the book (printing disabled).

Refresh your memory: Sections 16C-16D (local theory), 17E-17I (degree of bundles)