Constructing solutions of a triangular Schlesinger system via periods of meromorphic differentials on Riemann surfaces

Our starting point is the fact that two periods of an elliptic curve
$$w^2=z(z-1)(z-t)$$
as functions of a local parameter $t$, are basic solutions of some hypergeometric equation, a second order linear ordinary differential equation on the $t$-plane (sometimes called the Picard–Fuchs equation).
Considering algebraic curves of a more general form, with several local parameters, and not only holomorphic but meromorphic differentials on them, we show that periods of such differentials help one to construct a family of solutions of the Schlesinger equation arising in the theory of isomonodromic deformations (the particular case of which the Picard–Fuchs equation is). If one takes small loops encircling poles of the differentials as cycles determining their periods, the computation is reduced to that of the residues of the differentials and allows one to obtain explicit expressions for solutions of the Schlesinger equation in terms of elementary functions, such as polynomial, rational or algebraic (represented in radicals).
The talk is based on joint works with Vladimir Dragovic and Vasilisa Shramchenko, the published one and that in progress.