Sergei Yakovenko's blog: on Math and Teaching

Wednesday, March 26, 2008

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 \mathbb C P^1.

  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 4\times 4: the Bolibruch Counterexample.

Reading: Section 18E from the book (printing disabled).

Refresh your memory: Sections 16C16D (local theory), 17E-17I (degree of bundles)

Sunday, March 23, 2008

2-Sphere eversion in 3D-space

Filed under: links — Sergei Yakovenko @ 12:23
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If a smooth curve  embedded in the plane \mathbb R^2  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 \mathbb R^3 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!

Thursday, March 20, 2008

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).

Wednesday, March 19, 2008

Lecture 4 (Thu, Mar 13, 2008)

Piecemeal remarks on rational matrix functions of a complex variable

The global theory of rational linear systems on \mathbb C P^1 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, H(t)\in\textrm{GL}(n,\mathbb C[t]),\ \text{det}H=\text{const}\ne 0.

Multiplication of a rational matrix function H(t) from the left by a monopole matrix \begin{pmatrix}1 & t\\ & 1\end{pmatrix} corresponds to adding the second row of H, multiplied by t, to the first row. Thus manipulations with rows of H, 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 H(t) be the germ of a matrix function, holomorphic and invertible at t=\infty. Then for any ordered tuple of integer numbers D=\{d_1,\dots,d_n\} the product t^D\,H(t), t^D=\text{diag}(t^{d_1},\dots,t^{d_n}), is monopole equivalent to a product of the form H'(t)\,t^{D'}, where H'(t) is also holomorphic and invertible at t=\infty, and D' is a permutation of the tuple D.

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

Monday, March 3, 2008

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 \mathbb C P^1. Degree.
  5. Sections (holomorphic and meromorphic) of holomorphic bundles.
  6. Triviality of holomorphic vector bundles over \mathbb D,~\mathbb C and classification of bundles over \mathbb C P^1: 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.

Saturday, March 1, 2008

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 \mathbb C P^1: 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 😦

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