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 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)

Seminar on Khovanskii-Varchenko theorem (II)

Topological properties of Abelian integrals

The second “learning in groups” meeting will be devoted to the study of the Gauss–Manin connexion in homology, which will ultimately result in a local representation of Abelian integrals as linear combinations of real powers and logarithms with analytic coefficients analytically depending on parameters.

This representation already suffices to produce local uniform bounds for the number of isolated zeros, as was explained on the previous Tuesday.

Recommended reading: Section 26 from the book (printing disabled), esp., subsections F and I-K.

Time and location: Tuesday Dec. 18, 2007, 14:00 (in place of the usual Geometry & Topology seminar time), Pekeris Room.

What it will be about: