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SUMMARY:On the partial indices of piecewise constant matrix functions - Gr
 igori Giorgadze (Tbilisi State University\; Georgian Technical University)
DTSTART:20190812T153000Z
DTEND:20190812T160000Z
UID:TALK128383@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:Every holomorphic vector bundle  &nbsp\;&nbsp\; on Riemann sph
 ere&nbsp\; splits into the direct sum of line bundles and the total Chern 
 number of this vector bundle&nbsp\; is equal to sum of Chern numbers of li
 ne bundles. The integer-valued vector with components Chern number of line
  bundles is called splitting type of holomorphic vector bundle and is anal
 ytic invariant of complex vector bundles.  &nbsp\;  <br>There exists a one
 -to-one correspondence between the H\\"older continues matrix function and
  the holomorphic vector bundles described above\, wherein the splitting ty
 pe of vector bundles coincides with partial indices of matrix functions. I
 t is known that every holomorphic vector bundle equipped with meromorphic 
 (in  general) connection&nbsp\; with logarithmic singularities at finite s
 et of marked points and corresponding meromorphic 1-from&nbsp\; have first
  order poles in marked points and removable singularity at infinity.  &nbs
 p\;  <br>The Fucshian system of equations induced from this 1-form gives t
 he monodromy representation of the fundamental group of Riemann sphere wit
 hout marked points. The monodromy representation induces trivial holomorph
 ic vector bundles&nbsp\; with connection. The extension of the pair (\\tex
 ttt{bundle\, connection}) on the Riemann sphere is not unique and defines 
 a family of holomorphically nontrivial vector bundles.  &nbsp\;  &nbsp\;  
 <br>In the talk we present about the following statements:  &nbsp\;  <span
 >&nbsp\;&nbsp\; <br>1. From the solvability condition (in the sense Galois
  differential theory) of the Fuchsian</span>  &nbsp\;&nbsp\; system&nbsp\;
  follows formula for computation of partial indices of piecewise constant 
 matrix function.  &nbsp\;  <span>&nbsp\;&nbsp\; <br>2. All extensions of&n
 bsp\; vector bundle on noncompact Riemann surface correspond to</span>  <s
 pan>&nbsp\;&nbsp\; rational matrix functions&nbsp\; algorithmically comput
 able by monodromy matrices of Fucshian system.<br><br></span>This work was
  supported\, in part\, by the Shota Rustaveli National  Science Foundation
  under Grant No 17-96.
LOCATION:Seminar Room 1\, Newton Institute
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