There exists a one-to-one correspond ence between the H\\"older continues matrix functi on and the holomorphic vector bundles described ab ove\, wherein the splitting type of vector bundles coincides with partial indices of matrix function s. It is known that every holomorphic vector bundl e equipped with meromorphic (in general) connecti on \; with logarithmic singularities at finite set of marked points and corresponding meromorphi c 1-from \; have first order poles in marked p oints and removable singularity at infinity. &nbs p\;

The Fucshian system of equations induced from this 1-form gives the monodromy representatio n of the fundamental group of Riemann sphere witho ut marked points. The monodromy representation ind uces trivial holomorphic vector bundles \; wit h connection. The extension of the pair (\\texttt{ bundle\, connection}) on the Riemann sphere is not unique and defines a family of holomorphically no ntrivial vector bundles. \; \;

In the talk we present about the following statement s: \; \; \;

1. From the solvability condition (in the sense Galois differe ntial theory) of the Fuchsian \; \; system \; follows formula for computation o f partial indices of piecewise constant matrix fun ction. \; \; \;

2. All e xtensions of \; vector bundle on noncompact Ri emann surface correspond to \;& nbsp\; rational matrix functions \; algorithmi cally computable by monodromy matrices of Fucshian system.

This work was supported\, i n part\, by the Shota Rustaveli National Science Foundation under Grant No 17-96. LOCATION:Seminar Room 1\, Newton Institute CONTACT:INI IT END:VEVENT END:VCALENDAR