The inverse monodromic (or isomonodromic) transformation (IMT) method [1, 2] is a powerful tool for studying the class of nonlinear ordinary differential equations (ODE’s) representable as the compatibility condition of the overdetermined linear system (P–Q pair). The IMT method reduces the initial value problem for the system of nonlinear ODE’s to solving the inverse problem for associated iso...

(1.2) the ranges of f and g are identical on almost all OK/p. The most trivial cases are where g(x) = f(ax + b) for some a, b ∈ Q̄, the algebraic numbers. When K = Q, mostly that relation forces a, b∈K. For example this holds when f is indecomposable (not a composite of lower degree polynomials). With the indecomposability assumption, solutions to Davenport’s and Schinzel’s problems were essenti...

One of the central problems of mathematics in the second half of the 19th century and at the beginning of the 20th century was the problem of uniformization of Riemann surfaces. The classics, Klein [1] and Poincaré [2], associated it with studying second-order ordinary differential equations with regular singular points. Poincaré proposed another approach to the uniformization problem [3]. It c...

Qualitative features of the Manakov top are discussed for the classical and quantum versions of the problem. Energy-momentum diagram for this integrable classical problem and quantum joint spectrum of two commuting observables for associated quantum problem are analyzed. It is demonstrated that the evolution of the specially chosen quantum cell through the joint quantum spectrum can be defined ...

We explore the relation between cohomology jump loci in a finite Galois cover, formality properties and algebraic monodromy action. We show that the jump loci of the base and total space are essentially the same, provided the base space is 1-formal and the monodromy action in degree 1 is trivial. We use reduced multinet structures on line arrangements to construct components of the first charac...

The parameter q is called an accessory parameter. Although the local monodromy (local exponent) is independent of q, the global monodromy (e.g. the monodromy on the cycle enclosing two singularities) depends on q. Some properties of Heun’s equation are written in the books [21, 23], but an important feature related with the theory of finite-gap potential for the case γ, δ, ǫ, α−β ∈ Z+ 12 (see [...

The ‘monodromy conjecture’ for a hypersurface singularity f predicts that a pole of its topological (or related) zeta function induces one of its monodromy eigenvalues. However, in general only a few eigenvalues are obtained this way. The second author proposed to consider zeta functions associated with the hypersurface and with a differential form and raised the following question. Can one fin...

In this paper we study the monodromy action on the first Betti and de Rham nonabelian cohomology arising from a family of smooth curves. We describe sufficient conditions for the existence of a Zariski dense monodromy orbit. In particular we show that for a Lefschetz pencil of sufficiently high degree the monodromy action is dense.

We study the monodromy operators on the Betti cohomologies associated to a good degeneration of irreducible symplectic manifold and we show that the unipotency of the monodromy operator on the middle cohomology is at least the half of the dimension. This implies that the “mildest” singular fiber of a good degeneration with non-trivial monodromy of irreducible symplectic manifolds is quite diffe...

H. Davenport’s Problem asks: What can we expect of two polynomials, over Z, with the same ranges on almost all residue class fields? This stood out among many separated variable problems posed by Davenport, D.J. Lewis and A. Schinzel. By bounding the degrees, but expanding the maps and variables in Davenport’s Problem, Galois stratification enhanced the separated variable theme, solving an Ax a...