Meromorphic flat connexions on holomorphic manifolds: Integrability, monodromy, classification
- Pfaffian systems and their integrability
- From local to global solutions: monodromy
- Geometric language: covariant derivative and its curvature
- Meromorphic functions, meromorphic forms
- Example: multidimensional Euler system
- Regular singularities
- Flat connexions vs. isomonodromic deformations
Recommended reading: D. Novikov & S.Y., Lectures on meromorphic flat connexions, sect. 1-2.
Riemann–Hilbert Problem: positive results
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Formulation of the problem and its tautological solution on an abstract holomorphic vector bundle
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Meromorhic trivialization and Plemelj theorem (solvability of the problem if one of the monodromies is diagonalizable).
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Invariant subbundles, (ir)reducibility of a regular connexion.
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Lemma on too different orders. Bounds on the splitting type of a bundle with irreducible Fuchsian connexion.
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Bolibruch–Kostov theorem: solvability of the Riemann–Hilbert problem for irreducible representations.
Reading: Sections 18A-18D from the book (printing disabled).
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, ![H(t)\in\textrm{GL}(n,\mathbb C[t]),\ \text{det}H=\text{const}\ne 0](http://l.wordpress.com/latex.php?latex=H%28t%29%5Cin%5Ctextrm%7BGL%7D%28n%2C%5Cmathbb+C%5Bt%5D%29%2C%5C+%5Ctext%7Bdet%7DH%3D%5Ctext%7Bconst%7D%5Cne+0&bg=ffffff&fg=000000&s=0 "H(t)\in\textrm{GL}(n,\mathbb C[t]),\ \text{det}H=\text{const}\ne 0")
.
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.
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.
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Regular and irregular singularities: growth matters.
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Local gauge equivalence (holomorphic, meromorphic, formal). Meromorphic classification of regular singularities.
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Fuchsian singularities as a particular class of regular singularities (Sauvage lemma).
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Formal classification of Fuchsian singularities (Poincaré-Dulac theorem revisited). Resonances. Levelt upper triangular normal form.
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Coincidence of formal and holomorphic classification in the Fuchsian case.
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Integrability of the normal form.
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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 
