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.

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.

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 9 (Thu, Apr 17, 2008)

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:

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 1 (Feb 27, 2008)

Systems of Linear ODEs with complex time

1. Total recall: on differential equations in the complex domain (for the newcomers, if any) and foliations.
2. Linear systems: vector, matrix and Pfaffian form. Fyndamental solutions. Linearity of the transport maps.
3. Holomorphic (gauge) equivalence of linear systems. Monodromy group.
4. Linear systems with isolated singularities. Euler system and its properties.

Reading material: Section 15 from the Book (printing disabled)

Semester II: tentative programme

The course on Analytic and Geometric Theory of Differential Equations resumes on Wed Feb 27, 2008.

Note the change in the schedule: in the second semester classes will be in Room 261, on Wednesdays (instead of Thursdays), still between 9:00 and 10:50 (apologies before those who suffer from the pre-dawn wake-up). Thursdays, from 14:00 till 16:00.

The second semester will be centered on the theory of linear systems, as exposed in Chapter III of the book (warning: the draft posted on my web page is really outdated. I will provide links to individual sections of the printed edition, with printing option disabled, as before, to protect the copyright).

More precise (albeit still provisional) plan is as follows.

1. General properties of systems of linear ordinary differential equations in the complex domain. Gauge equivalence. Monodromy and holonomy.
2. Local theory of singular points. Fuchsian, regular and irregular singularities.
3. Towards the global theory of  linear systems: holomorphic vector bundles.
4. Towards the global theory of linear systems: meromorphic connexions on vector bundles.
5. Reconstruction of a linear system from its monodromy group. The Riemann–Hilbert problem.
6. Positive results on solvability of the Riemann–Hilbert problem. Bolibruch–Kostov theorem.
7. Negative results and the Bolibruch counterexample.
8. Scalar high order linear ordinary differential equations and associated geometric structures. Hypergeometric equations.
9. Irregular singularities: formal theory.
10. Irregular singularities: elements of analytic theory. Stokes phenomenon.
11. Elements of multidimensional theory: meromorphic flat connexions on .

The second semester is intended to be as independent from the first semester, as possible, so that newcomers may join at this junction.