On the Riemann-Hilbert Problem for Difference and q-Difference Systems

Student: Levin Roman

Educational Programme: Mathematics (Bachelor)

Year of Graduation: 2017

The Riemann——Hilbert problem is a classical inverse monodromy problem and is one of the major mathematical problems of the 20th century. An analogue of it arises in Birkhoff's theory of difference and $q$-difference equations. Birkhoff called that analogue a Generalized Riemann-Hilbert problem and proposed a positive solution for it. However, that solution could be insufficient in some cases. In this paper we introduce a method to generalize Birkhoff's solution to those cases. We consider a non-resonant case of the Generalized Riemann-Hilbert problem and, starting from Birkhoff's existence theorem, apply a sequence of isomonodromic gauge transformations which leads to the correct solution.

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