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)

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)

Lecture 3 (Thu, Mar 6, 2008)

Global theory of linear systems: holomorphic vector bundles

1. Definitions. Gluing bundles from cylindrical charts.
2. Matrix cocycles and their equivalence.
3. Operations on bundles vs. operations with cocycles.
4. Example: linear bundles over . Degree.
5. Sections (holomorphic and meromorphic) of holomorphic bundles.
6. Triviality of holomorphic vector bundles over and classification of bundles over : Cartan and Birkhoff–Grothendieck theorems.

Recommended reading: the subject is treated in various sources with accent on analytic, geometric or algebraic side of it. You can choose your favorite textbook or one of the following expositions.

1. O. Forster, Riemann surfaces, §§29-30 (analytic treatment).
2. P. Griffiths & M. Harris, Principles of Algebraic Geometry, §0.5 (algebraic “neoclassical”).
3. R. O. Wells, Differrential Analysis on Complex Manifolds, §2.
4. A. Hatcher, Vector bundles and -theory (rather advanced topological exposition).
5. Parts 17A-17F and 17J from Section 17 from the Book (printing disabled).