Sergei Yakovenko's blog: on Math and Teaching

Thursday, May 29, 2008

Lecture 12 (May 29, 2008)

Logarithmic singularities

  1. De Rham division lemma (and its generalization)
  2. Definition of a logarithmic pole: (scalar case). Residues.
  3. Logarithmic complex: principal lemma on Λ-closedness.
  4. Principal example: logarithmic complex for the normal crossings. Saito theorem.
  5. Closed logarithmic 1-forms: complete description. Darbouxian foliations.
  6. Matrix casse. Conjugacy of the residues along the polar locus. Residues on the normal crossings.
  7. Schlesinger system: flat connexions with logarithmic poles along the diagonal.
  8. Flat connexions with first order poles are almost always logarithmic, yet resonances may spoil the pattern.

Recommended reading: the same notes, sect. 3-4.

Thursday, May 22, 2008

Lecture 11 (May 22, 2008)

Meromorphic flat connexions on holomorphic manifolds: Integrability, monodromy, classification

  1. Pfaffian systems and their integrability
  2. From local to global solutions: monodromy
  3. Geometric language: covariant derivative and its curvature
  4. Meromorphic functions, meromorphic forms
  5. Example: multidimensional Euler system
  6. Regular singularities
  7. Flat connexions vs. isomonodromic deformations

Recommended reading: D. Novikov & S.Y., Lectures on meromorphic flat connexions, sect. 1-2.

Thursday, May 15, 2008

Reminder

Filed under: Analytic ODE course,research seminar — Sergei Yakovenko @ 8:21

No classes today, as 50% of the students are speaking on a conference elsewhere.

Thursday, May 1, 2008

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

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