ILYA VYUGIN (LOMONOSOV UNIV.MOSCOW)
On the Inverse Monodromy Problems
We consider the problem of constructing the fuchsian system (or equation) with
prescribed monodromy representation on $\mathbb{CP}^1$ (the
Riemann-Hilbert problem). These problems can be interpreted as
problems of constructing of the holomorphic vector bundles with logarithmic
connections. Any fuchsian system can be interpreted as a logarithmic
connection in a trivial bundle.
Scalar fuchsian equation is equivalent to the
logarithmic connection in the special non-trivial bundle.
Using this approach we can prove a new results and reprove some known facts about
fuchsian equations.
The inverse monodromy problem has a positive solution in the class of regular systems (result of I.Plemelj).
We give an estimation of orders of
poles of such system. We give an estimation of the number of apparent singularities for the scalar fuchsian equations.
This approach can be applied to the study of Painleve VI equations.