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We review recent developments in the method of algebro-geometric integration of integrable systems related to deformations of algebraic curves. In particular, we discuss the theta-functional solutions of Schlesinger system, Ernst equation and self-dual SU(2)-invariant Einstein equations.
By constructing suitable Borcherds forms on Shimura curves and using Schofers formula for norms of values of Borcherds forms at CM-points, we determine all the equations of hyperelliptic Shimura curves $X_0^D(N)$. As a byproduct, we also address the
Given a hyperelliptic curve $C$ of genus $g$ over a number field $K$ and a Weierstrass model $mathscr{C}$ of $C$ over the ring of integers ${mathcal O}_K$ (i.e. the hyperelliptic involution of $C$ extends to $mathscr{C}$ and the quotient is a smooth
We study the Schlesinger system of partial differential equations in the case when the unknown matrices of arbitrary size $(ptimes p)$ are triangular and the eigenvalues of each matrix form an arithmetic progression with a rational difference $q$, th
The linear Einstein-Boltzmann equations describe the evolution of perturbations in the universe and its numerical solutions play a central role in cosmology. We revisit this system of differential equations and present a detailed investigation of its
The Raychaudhuri equations for the expansion, shear and vorticity are generalized in a spacetime with torsion for timelike as well as null congruences. These equations are purely geometrical like the original Raychuadhuri equations and could be reduc