A method for solving certain nonlinear ordinary and partial differential equations is developed. The central idea is to study monodromy preserving deformations of linear ordinary differential equations with regular and irregular singular points. The connections with isospectral deformations and with classical and recent work on monodromy preserving deformations are discussed. Specific new resul...

We study the rational potentials V (x), with sextic growth at infinity , such that the corresponding one-dimensional Schrödinger equation has no monodromy in the complex domain for all values of the spectral parameter. We investigate in detail the subclass of such potentials which can be constructed by the Darboux transformations from the well-known class of quasi-exactly solvable potentials V ...

We define monodromy maps for tropical Dolbeault cohomology of algebraic varieties over non-Archimedean fields. We propose a conjecture of Hodge isomorphisms via monodromy maps, and provide some evidence.

Using Mellin-Barnes integrals we give a method to compute a relevant subgroup of the monodromy group of an A-hypergeometric system of differential equations. Presumably this group is the full monodromy group of the system

To a plane algebraic curve of degree n, Moishezon associated a braid monodromy homomorphism from a finitely generated free group to Artin’s braid group Bn. Using Hansen’s polynomial covering space theory, we give a new interpretation of this construction. Next, we provide an explicit description of the braid monodromy of an arrangement of complex affine hyperplanes, by means of an associated “b...

The elliptic quantum Knizhnik–Zamolodchikov–Bernard (qKZB) difference equations associated to the elliptic quantum group Eτ,η(sl2) is a system of difference equations with values in a tensor product of representations of the quantum group and defined in terms of the elliptic R-matrices associated with pairs of representations of the quantum group. In this paper we solve the qKZB equations in te...

In this paper, Hamiltonian monodromy is addressed from the point of view of geometric quantization, and various differential geometric aspects thereof are dealt with, all related to holonomies of suitable flat connections. In the case of completely integrable Hamiltonian systems with two degrees of freedom, a link is established between monodromy and (2-level) theta functions, by resorting to t...

In this paper we investigate the monodromy groups of the truncated simplices of all possible ranks and investigate related structures. In particular, we demonstrate that the monodromy group of a truncated simplex is the direct product of symmetric groups.

The purpose of this paper is to show that the monodromy of action variables of the Lagrange top and its generalizations can be deduced from the monodromy of cycles on a suitable hyperelliptic curve (computed by the Picard-Lefschetz formula).