## Global Study of Differential Equations in the Complex Domain

### Lectures

#### List of lectures:

Galina Filipuk: On the connection of semiclassical orthogonal polynomials and the Painleve equations

Abstract: In this talk I shall explain the connection between the recurrence coefficients of the semi-classical q-Laguerre and little q-Laguerre orthogonal polynomials and certain discrete Painleve equations. This is a joint work with C. Smet (KULeuven, Belgium).

Yoshishige Haraoka: Global study of regular holonomic systems

Abstract: We consider linear holonomic systems of regular singular type. Many hypergeometric functions in several variables, say Appell’s ones and Lauricella’s ones, satisfy such systems. We note that such hypergeometric functions are defined by some expressions of solutions, and not by the holonomic systems. Thus these holonomic systems are obtained by expressions of solutions. It is very hard to construct a holonomic system without knowing solutions, which is a main reason of the difficulty of the study of holonomic systems.

Then we are interested in constructing holonomic systems explicitly. I will introduce several recent developments in this direction – construction by using finite monodromy groups, by using singular locus, by using deformation equations.

Kohei Iwaki: Stokes Matrices for the Quantum Cohomologies of Orbifold Projective Lines

Abstract: We prove a conjecture formulated by Boris Dubrovin in the cases of orbifold projective lines. The conjecture predicts that, for a projective variety X, Stokes matrix of an ordinary differential equation arising from quantum cohomology of X coincides with the Euler matrix of the derived category of coherent sheaves on X. Our proof is based on the homological mirror symmetry and Picard-Lefshetz theory. This is a joint work with Atsushi Takahashi.

Yoshitsugu Takei: On the exact WKB analysis for higher order differential equations - virtual turning points, new Stokes curves, and some recent results related to them

Abstract: As was first pointed out by Berk et al., new Stokes curves play a crucially important role in discussing connection problems and Stokes phenomena for higher order linear ordinary differential equations. Later, to clarify the meaning of new Stokes curves and their relationship with the Borel resummation technique, Aoki, Kawai and I introduced the notion of virtual turning points, that is, turning points from which new Stokes curves emanate. The introduction of virtual turning points is really successful, but still there remain some important open problems. For example, the Borel summability of WKB solutions of higher order linear ordinary differential equations outside (ordinary and new) Stokes curves is not established yet in general.

In this talk I will first give a brief survey on the fundamental part of the theory of virtual turning points and then discuss some open problems for them. In particular, I will explain some recent results related to virtual turning points and new Stokes curves (such as the Borel summability of formal solutions of inhomogeneous second order equations, and so on).

Mika Tanda: Parametric Stokes phenomena of Gauss hypergeometric differential equation with a large parameter

Abstract: We consider the hypergeometric differential equation with a large parameter from the viewpoint of the exact WKB analysis. One of main objects in the exact WKB analysis is the notion of WKB solutions which are formal solutions of the equations coming from solutions of the associated Riccati equation. Firstly explicit forms of the Voros coefficients and the Borel sums of them are given for the hypergeometric differential equation with a large parameter. The Voros coefficients are formal series in the negative powers of the large parameter which describes the discrepancy between two WKB solutions with different normalization of integration and they play a role in the analysis of Stokes phenomena with respect to parameters in the equation, which we call parametric Stokes phenomena. Secondly we give some formulas describing parametric Stokes phenomena of WKB solutions.

This work is a collaboration with T. Aoki.

Hiroshi Yamazawa: Summability of formal solutions of 1st order nonlinear ODE related to linearization problem

Abstract

Masafumi Yoshino: Stokes geometry and summability in the linearization problem for a singular vector field

Abstract: We study summability and a connection problem for a system of semi-linear equations - a transformation equation - related to the linearization problem of a singular vector field with an isolated singular point at the origin. The class of systems contains the so-called Noumi-Yamada system. Although the ODE case has still many interesting open questions, we will focus our attention to the PDE case in my presentation. Indeed, I will state some new results and open problems concerning how the Stokes geometry is defined in the multi dimensional situation and the way how the closely related asymptotic analysis is done. By using notions in Stokes geometry we will show the summability of formal solutions (a so-called 0-parameter solution). A new problem in removal of singularities of functions of several complex variables is proposed in relation with summability in the multi dimensional situation. The connection problem for our equation is mostly an open question, while we will show a recent result.

Alina Dobrogowska: Second order q-difference equations solvable by factorization method

Abstract: By solving an infinite nonlinear system of q-difference equations one constructs a chain of q-difference operators. The eigenproblems for the chain are solved and some applications, including the one related to q-Hahn orthogonal polynomials, are discussed. It is shown that in the limit q $\rightarrow$ 1 the present method corresponds to the one developed by Infeld and Hull.