Sergei Yakovenko's blog: on Math and Teaching

Thursday, April 10, 2008

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.

Thursday, April 3, 2008

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 \iff local Fuchs property) and its reformulations.

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

Wednesday, March 26, 2008

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 \mathbb C P^1.

  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 4\times 4: the Bolibruch Counterexample.

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

Refresh your memory: Sections 16C16D (local theory), 17E-17I (degree of bundles)

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