Related papers: Differential Equations and Monodromy

We discuss some features of the so-called Zariski's multiplicity problem especially the application of the work of A'Campo on the zeta function of a monodromy of an isolated singularity of a complex hypersurface to the problem.

We study Cauchy problems associated to elliptic operators acting on vector-valued functions and coupled up to the first-order. We prove pointwise estimates for the spatial derivatives of the semigroup associated to these problems in the…

In this paper we study the Fuchsian Riemann-Hilbert (inverse monodromy) problem corresponding to Frobenius structures on Hurwitz spaces. We find a solution to this Riemann-Hilbert problem in terms of integrals of certain meromorphic…

We consider the quantum difference equation of the Hilbert scheme of points in $\mathbb{C}^2$. This equation is the K-theoretic generalization of the quantum differential equation discovered by A. Okounkov and R. Pandharipande. We obtain…

We prove that a generic differential operator of type DN is irreducible, regular, (anti)self-adjoint, and has quasiunipotent local monodromies. We prove that the defining matrix of a DN operator can be recovered from the expression of the…

In this paper, we study the algebraic form of the symmetric generalized Lam\'e equations which have finite projective monodromy groups. In particular, we consider equations with $3$ regular singular points on a flat torus $T$ which takes…

The conjecture of Fuchs, Schellekens and Schweigert on the relation of mapping class group representations and fixed point resolution in simple current extensions is investigated, and a cohomological interpretation of the untwisted…

We prove that, for first-order, fully nonlinear systems of partial differential equations, under an hypothesis of ellipticity for the principal symbol, the Cauchy problem has no solution within a range of Sobolev indices depending on the…

One of the most natural and challenging issues in discrete complex analysis is to prove the convergence of discrete holomorphic functions to their continuous counterparts. This article is to solve the open problem in the general setting. To…

We relate two parameter solutions of the sixth Painlev\'e equation and finite-gap solutions of the Heun equation by considering monodromy on a certain class of Fuchsian differential equations. In the appendix, we present formulae on…

This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and…

We show that for every second order Fuchsian linear differential equation $E$ with $n$ singularities of which $n-3$ are apparent there exists a hypergeometric equation $H$ and a linear differential operator with polynomial coefficients…

Via Carleman estimates we prove uniqueness and continuous dependence results for lateral Cauchy problems for linear integro-differential parabolic equations without initial conditions. The additional information supplied prescribes the…

Let $(E)$ a homogeneous linear differential equation of order $n$ Fuchsien over $\mathbb{P}^{1}(\mathbb{C}) $. The idea of Riemann (1857) was to obtain the properties of solutions of (E) by studying the local system. Thus, he obtained some…

We study finite-dimensional representations of hyper loop algebras over non-algebraically closed fields. The main results concern the classification of the irreducible representations, the construction of the Weyl modules, base change,…

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups and we use structure theorems for these groups to…

The article presents a rather surprising Floquet-type representation of time-varying transition matrices associated with a class of nonlinear matrix differential Riccati equations. The main difference with conventional Floquet theory comes…

We prove maximal Schauder regularity for solutions to elliptic systems and Cauchy problems, in the space $C_b(\mathbb{R}^d;\mathbb{R}^m)$ of bounded and continuous functions, associated to a class of nonautonomous weakly coupled…

A rather complete phenomenology of the singularities is developed according to a new algebraic point of view in the frame of Langlands global correspondences. That is to say,a process of: -singularizations and versal deformations of these,…

The Shafarevich conjecture for a class of varieties over a number field posits the finitude of those with good reduction outside a finite set of primes. In the case of hypersurfaces in the torus $\mathbb{G}_m^n$, a natural class to consider…