Actually, I forgot to tell the last time that the academic year is over. Soon I will post the exam problems here, - watch for the updates!

And congratulations to all the survivors who made it till the end. Hope you don’t regret.

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.

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.

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

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

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:

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.

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)

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)

If a smooth curve  embedded in the plane  is deformed allowing self-intersections but remaining smooth, then there is a natural integral invariant, the rotation number, which prevents eversion of a circle (deformation of the oriented circle into another circle with an opposite orientation). For two-dimensional surfaces smoothly embedded in a similar invariant of deformations exists, yet this invariant does not preclude eversion of the sphere inside out.

 The possibility of such deformation was discovered bt S. Smale in 1958. Relatively recently W. Thurston invented a general algorithm of smoothening, which yields an explicit sphere eversion. All these spectacular things are discussed on the level accessible to high school students in the most fascinating animation (21 min.) discovered on the web by Dmitry Novikov (thanks!). A much shorter animation (mere 22 sec.) does not easily reveal the mistery, so the longer one is really worth its time!