Related papers: Monodromies of Second Order $q$-difference Equatio…

We study a second-order linear differential equation known as the deformed cubic oscillator, whose isomonodromic deformations are controlled by the first Painlev{\'e} equation. We use the generalised monodromy map for this equation to give…

Multisummation provides a transparent description of Stokes matrices which is reviewed here together with some applications. Examples of moduli spaces for Stokes matrices are computed and discussed. A moduli space for a third Painlev\'e…

The second Painlev\'e equation with a large parameter ($P_{II}$) is analyzed by using the exact WKB analysis. The purpose of this study is to investigate the problem of the degeneration of $P$-Stokes geometry of ($P_{II}$), which relates to…

In this paper, we study the isomonodromy deformation equations for the $n\times n$ system of first order meromorphic linear ordinary differential equations with two second order poles. We analyze the asymptotic behaviour of the solutions at…

We consider an $n\times n$ system of ODEs on $\mathbb{P}^1$ with a simple pole $A$ at $z=0$ and a double pole $u={\rm diag}(u_1, \dots, u_n)$ at $z=\infty$. This is the simplest situation in which the monodromy data of the system are…

In this paper, we compute the Stokes matrices of a special quantum confluent hypergeometric system with Poincar\'e rank one. The sources of the interests in the Stokes phenomenon of such system are from representation theory and the theory…

We study N = 2 supersymmetric gauge theories with gauge group SU(2) coupled to fundamental flavours, covering all asymptotically free and conformal cases. We re-derive, from the conformal field theory perspective, the differential equations…

The first part of these lecture notes is devoted to an introduction to the theory of exact WKB analysis for second-order Schr\"odinger-type ordinary differential equations. It reviews the construction of the WKB solution, Borel summability,…

This article studies a confluence of a pair of regular singular points to an irregular one in a generic family of time-dependent Hamiltonian systems in dimension 2. This is a general setting for the understanding of the degeneration of the…

We compute Stokes matrices and monodromy for the quantum cohomology of projective spaces. We prove that the Stokes' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves.

A brief review of a self-contained genuinely three-dimensional monodromy-matrix based non-perturbative covariant path-integral approach to {\it polynomial invariants} of knots and links in the framework of (topological) quantum Chern-Simons…

In a paper published by the Annales de la Facult\'e de Sciences de Toulouse, with Yousuke Ohyama, we defined and studied a space of monodromy data underlying the well known derivation of q-Painlev\'e VI equation from q-isomonodromy…

Extensions of standard one-dimensional supersymmetric quantum mechanics are discussed. Supercharges involving higher order derivatives are introduced leading to an algebra which incorporates a higher order polynomial in the Hamiltonian. We…

For more than a century, the Painlev\'e I equation has played an important role in both physics and mathematics. Its two-parameter family of solutions was studied in many different ways, yet still leads to new surprises and discoveries. Two…

In this paper we study the Gauss and Kummer hypergeometric equations in depth. In particular, we focus on the confluence of two regular singularities of the Gauss hypergeometric equation to produce the Kummer hypergeometric equation with an…

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 formulate a geometric Riemann-Hilbert correspondence that applies to the derivation by Jimbo and Sakai of equation $q$-PVI from ``isomonodromy'' conditions. This is a step within work in progress towards the application of…

Recently, a remarkable correspondence has been unveiled between a certain class of ordinary linear differential equations (ODE) and integrable models. In the first part of the report, we survey the results concerning the 2nd order…

In this paper we study the map associating to a linear differential operator with rational coefficients its monodromy data. The operator has one regular and one irregular singularity of Poincare' rank 1. We compute the Poisson structure of…

The elliptic quantum Knizhnik-Zamolodchikov-Bernard (qKZB) difference equations associated to the elliptic quantum group $E_{\tau,\eta}(sl_2)$ is a system of difference equations with values in a tensor product of representations of the…