Lecture 5 (Thu, Mar 20, 2008)

Riemann–Hilbert Problem: positive results

1. Formulation of the problem and its tautological solution on an abstract holomorphic vector bundle
2. Meromorhic trivialization and Plemelj theorem (solvability of the problem if one of the monodromies is diagonalizable).
3. Invariant subbundles, (ir)reducibility of a regular connexion.
4. Lemma on too different orders. Bounds on the splitting type of a bundle with irreducible Fuchsian connexion.
5. Bolibruch–Kostov theorem: solvability of the Riemann–Hilbert problem for irreducible representations.

Reading: Sections 18A-18D from the book (printing disabled).

Lecture 4 (Thu, Mar 13, 2008)

Piecemeal remarks on rational matrix functions of a complex variable

The global theory of rational linear systems on requires the study of (rational) gauge transformations which are holomorphic and holomorphically invertible except for a single point. If this point is at infinity, then the matrix of such transformation is necessarily polynomial with constant nonzero determinant. Such matrix functions are provisionally referred to as monopoles, .

Multiplication of a rational matrix function from the left by a monopole matrix corresponds to adding the second row of , multiplied by , to the first row. Thus manipulations with rows of , which aim at Gauss-type elimination of certain monomials from matrix elements, can be represented as gauge actions of the monopole group. The principal result that will be used throughout the next few lectures, is the following Bolibruch Permutation Lemma.

Lemma. Let be the germ of a matrix function, holomorphic and invertible at . Then for any ordered tuple of integer numbers the product , , is monopole equivalent to a product of the form , where is also holomorphic and invertible at , and is a permutation of the tuple .

The proof of this result is not difficult, yet is too technical to be delivered in the classroom.

Lecture 3 (Thu, Mar 6, 2008)

Global theory of linear systems: holomorphic vector bundles

1. Definitions. Gluing bundles from cylindrical charts.
2. Matrix cocycles and their equivalence.
3. Operations on bundles vs. operations with cocycles.
4. Example: linear bundles over . Degree.
5. Sections (holomorphic and meromorphic) of holomorphic bundles.
6. Triviality of holomorphic vector bundles over and classification of bundles over : Cartan and Birkhoff–Grothendieck theorems.

Recommended reading: the subject is treated in various sources with accent on analytic, geometric or algebraic side of it. You can choose your favorite textbook or one of the following expositions.

1. O. Forster, Riemann surfaces, §§29-30 (analytic treatment).
2. P. Griffiths & M. Harris, Principles of Algebraic Geometry, §0.5 (algebraic “neoclassical”).
3. R. O. Wells, Differrential Analysis on Complex Manifolds, §2.
4. A. Hatcher, Vector bundles and -theory (rather advanced topological exposition).
5. Parts 17A-17F and 17J from Section 17 from the Book (printing disabled).

Lecture 2 (Sun, Mar 2, 2008)

Local theory of regular singular points of linear systems

This lecture, in an exceptional way, will take place on Sunday, 16:00-18:00, in the Room 261.

1. Regular and irregular singularities: growth matters.
2. Local gauge equivalence (holomorphic, meromorphic, formal). Meromorphic classification of regular singularities.
3. Fuchsian singularities as a particular class of regular singularities (Sauvage lemma).
4. Formal classification of Fuchsian singularities (Poincaré-Dulac theorem revisited). Resonances. Levelt upper triangular normal form.
5. Coincidence of formal and holomorphic classification in the Fuchsian case.
6. Integrability of the normal form.
7. Towards global theory of Fuchsian systems on : Monopoles as special classes of rational matrix functions.

Reading: Sections 16A-16E from the Book (printing disabled).

Reminder: Today (actually, on Friday) was the deadline for submission of the home exam

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.

Exam for the Winter Semester 2007/8

Contrary to the pessimistic expectations voiced in the previous post, the strike is over and everybody can come out of the closet.

The course “Geometric and Analytic Theory of Differential Equations” is declared a guided reading course based on this weblog. The winter semester for this course is over: the classes will resume some time on the last week of February, 2008.

 Those interested in grades or in controlling how well they digested  the material, are welcome to pass the exam. The rules of the game are simple: the exam is take-home, the deadline for submission is February 28.

Problems for Semester I are available online.  Any questions (if they appear)  can be left in the comments to this post.

End of non-semester

Announcement

The non-lecture today was the last one in the first non-semester of the academic year that perhaps will also be declared non-existing (see the Disclaimer).

There will be no meetings on January {24, 31} and February {7, 14}, as I am going to participate in the Carnival in Rio-de-Janeiro. The preliminary date for the next meeting is set February 21, 2008, unless there will be a complete closure of all universities, mass dismissal of the academic staff etc. Follow the announcements!

Andrei Gabrielov @ W.I.S. - 2nd seminar

Andrei will continue his story on complexity of various classes of problems in tame (e.g., semialgebraic) geometry.

Venue: Room 261,

Tuesday, January 22, 2008, 

14:00-16:00 (the ordinary time for the seminar).

Andrei Gabrielov @ W.I.S.

Mini-programme on real analytic/algebraic/o-minimal geometry

Andrei Gabrielov (Purdue U.) is visiting us for a month (until February 8). Among other things, he will explain his recent work with N. Vorobjov on topology of o-minimal sets via approximation.

The exposition, split into several lectures, will serve also as an initiation to the field of semianalytic/subanalitic geometry, accessible to newcomers.

Recommended reading:

1. M. Coste, Introduction to O-minimal Geometry,
2. E. Bierstone & P. Milman, Semianalytic and subanalytic sets.

The first meeting:

Tuesday, January 15, 2008, 16:00-18:00 (note the unusual time), Room 261 (unless suddenly changed).