**06.09.2015 - 12.09.2015 | Będlewo**

The Organizers offer some grants for PhD students, which cover costs of participation in the school (excluding travel costs). The applicants are kindly asked to send their CV and a letter of recommendation from their supervisors to the Organizers (e-mail: aagade@impan.pl) by May 31, 2015. The decisions will be sent promptly.

During the school we plan a series of talks (4-5) on each topic, (possibly) followed by some problem or discussion session. For younger participants we shall organize either the poster session or the short communications session. Although aimed primarily on younger researchers, some of the topics will be of interest to more experienced researchers.

** **

**Topics of the school:**

**WKB analysis and Stokes geometry of differential equations** (Yoshitsugu Takei - Kyoto University)

Abstract: In physics, since the very beginnings of the quantum mechanics, the WKB approximation has been employed to obtain approximate eigenfunctions and solve the eigenvalue problems of Schrödinger equations. The (full-order) WKB approximations provide formal solutions (with respect to the Planck constant) of Schrödinger equations but, as they are divergent in almost all cases, they were not so often used in rigorous mathematical analysis. Around 1980, using the Borel resummed WKB solutions, A. Voros (*Ann. Inst. H. Poincaré*, 39 (1983), 211-338) studied successfully some spectral functions of quartic oscillators. After the pioneering work of Voros, F. Pham, E. Delabaere and others (cf., e.g., E. Delabaere and F. Pham: *Ann. Inst. H. Poincaré*, 71 (1999), 1-94) have developed this new kind of WKB analysis (sometimes called "exact WKB analysis'' or "complex WKB analysis'') based on the Borel resummation technique and, in particular, the theory of Ecalle's resurgent functions. At present it turns out that the exact WKB analysis is very efficient not only for eigenvalue problems of Schrödinger equations but also for the global study of differential equations in the complex domain.

In the lectures, mainly using some concrete and illuminating examples, we explain the basic theory of the exact WKB analysis, its application to the global study of differential equations in the complex domain, and some recent developments of the theory. In the former part of lectures, we consider the exact WKB analysis for second order linear ordinary differential equations (cf., T. Kawai and Y. Takei: *Algebraic Analysis of Singular Perturbation Theory*, AMS, 2005): Starting from the definition of WKB solutions, we first introduce the Stokes geometry and Voros' connection formulas for Borel resummed WKB solutions, which play a crucially important role in the theory. Then we discuss its application to the computation of monodromy groups of Fuchsian equations and wall crossing formulas for WKB solutions with respect to the change of parameters contained in the equation. In the latter part of lectures, we consider generalizations of the exact WKB analysis in various directions such as WKB analysis for higher order linear ordinary differential equations, generalization to completely integrable systems, and so on. New Stokes curves introduced by Berk-Nevins-Roberts (*J. Math. Phys.*, 23 (1982), 988-1002) are central problems in discussing such generalizations.

Contents:

- Exact WKB analysis for second-order linear ODEs

- WKB solutions, Stokes geometry, connection formulas

- Application to the computation of monodromy groups of Fuchsian equations

- Voros coefficients and wall-crossing formulas

- New Stokes curves and virtual turning points for higher-order ODEs

- Generalization to completely integrable systems

**Sub-Riemannian geometry and sub-elliptic operators** (Irina Markina - Bergen University)

Abstract: The main aim of my course is to introduce the geometry of smooth manifolds with distinguished smooth subbundle of the tangent bundle, that we will call sub-Riemannian manifolds if additionally a Riemannian metric is given. The typical examples are contact manifolds and the Heisenberg group. Such geometries describes mechanical systems with kinematic constrains, differential equations with controls, quantum systems with magnetic fields, Yang-Mills magnetic fields and have many other applications. It is well known that the geometry of smooth manifold with a Riemannian metric are closely related to the properties of the elliptic operators on that manifolds, think about the close relations of the geometry of the euclidean plane and the Laplace operator or the heat operator. In the same fashion the sub-Riemanian manifolds are intimately related to degenerate elliptic operators, that under some conditions can be called sub-elliptic operators. We will reveal this relation, explaining the main difference between the Riemannian and sub-Riemannian geometries and it influence on the behavior of the differential operators.

Contents:

- Riemannian geometry and elliptic operators.

- Sub-Riemannian manifolds.

- Hypoelliptic and sub-elliptic operators.

- Show-Rashevskii and Hörmander theorems.

- Group of diffeomorphisms of the unit circle and geodesic equations.

- Applications of the infinite dimensional sub-Riemannian geometry to the vision theory.

**Asymptotic analysis and summability of formal power series** (Javier Sanz - Valladolid University)

Abstract: For many problems (ODEs, PDEs, difference equations, etc.) it makes sense to look for formal power series solutions which, if found, could well be divergent. However, these formal solutions will frequently have an asymptotic meaning, being representations, in a precise sense, of actual, analytic solutions of the corresponding problem. Summability techniques aim at reconstructing such proper solutions from the formal ones. We will present a slight extension of the successful and well-known technique of k-summability in a direction of the complex plane, put forward by J.-P. Ramis and which was the building block for multisummability, a procedure able to sum any formal solution to a system of meromorphic ordinary differential equations at an irregular singular point. The extension concerns the consideration of Carleman ultraholomorphic classes in sectors, more general than the Gevrey classes appearing in Ramis' theory, and which consist of holomorphic functions whose derivatives' growth is governed in terms of a sequence of real numbers, say M. Whenever M is subject to standard conditions, flat functions in the class are constructed on sectors of optimal opening and, resting on the work of W. Balser on moment summability methods, suitable kernels and Laplace and Borel-type transforms are introduced which lead to a tractable concept of M-summability. We will comment on some applications of this tool to the study of the summability properties of formal solutions to some classes of ordinary and partial differential equations.

Contents:

- Introduction. Asymptotic expansions and ultraholomorphic classes in sectors.

- Strongly regular sequences. Associated functions and growth indices.

- Injectivity of the Borel map: Watson's Lemma, Korenbljum's result and proximate orders.

- Summability in a direction. First properties.

- Kernels of summability for a strongly regular sequence. Formal and analytic Laplace and Borel transforms. The sum as a Laplace-type integral.

- Applications.

**Holonomic systems** (Yoshishige Haraoka - Kumamoto University)

Abstract: Holonomic systems are natural extensions of ordinary differential equations, and appear in physics, representation theory, theory of automorphic functions, and so on. The deformation theory of linear differential equations is in a sense a study of holonomic systems. Thus holonomic systems concern various branches of physics and mathematics, and then seem to be substantial objects. There are many similarities between holonomic systems and ordinary differential equations, and also several differences. Recently, in the theory of Fuchsian ordinary differential equations, a big progress is caused by N. M. Katz ("Rigid Local Systems", Princeton Univ. Press, Princeton, NJ, 1996.) and T. Oshima ("Fractional calculus of Weyl algebra and Fuchsian differential equations", MSJ Memoirs, 28. Mathematical Society of Japan, Tokyo, 2012.). These results can be applied to the study of holonomic systems, and will bring a new development.

In this lecture, we first explain the new understanding of Fuchsian ordinary differential equations given by Katz and Oshima. Fuchsian ordinary differential equations are classified by using the spectral types, and in each class equations are connected by two operations - addition and middle convolution. Analytic properties of solutions are transmitted by these operations. Next we apply these results to the study of regular holonomic systems. We can define the spectral type and the middle convolution similarly as in ODE case. These notions will become powerful tools. Finally we focus on the difference between Fuchsian ordinary differential equations and regular holonomic systems. For the global analysis of regular holonomic systems, geometry of the singular locus plays decisive role. We explain the mechanism, and also show that topology of hypersurfaces works for constructing holonomic systems. Relation to algebraic solutions of deformation equations will also be discussed.

Contents:

- Spectral types of Fuchsian ordinary differential equations.

- Middle convolution.

- Existence problem for Fuchsian ordinary differential equations.

- Basis of the theory of linear holonomic systems.

- Spectral types and middle convolution for regular holonomic systems.

- Construction and rigidity of regular holonomic systems

**An introduction to Dunkl theory (from harmonic analysis on symmetric spaces and integrable systems related to quantum many body problems to Hecke algebras and special functions related to root systems)** (Jean-Philippe Anker - Orleans University)

Abstract: Dunkl theory is a far reaching generalization of Fourier analysis and special function theory related to root systems. During the sixties and seventies, it became gradually clear that radial Fourier analysis on rank one symmetric spaces was closely connected with certain classes of special functions in one variable:

- Bessel functions in connection with radial Fourier analysis on Euclidean spaces,

- Jacobi polynomials in connection with radial Fourier analysis on spheres,

- Jacobi functions (i.e. the Gauss hypergeometric function 2F1) in connection with radial Fourier analysis on hyperbolic spaces.

See T.H. Koornwinder ("Jacobi functions and analysis on noncompact semisimple Lie groups", in "Special functions (group theoretical aspects and applications)", R.A. Askey,T.H. Koornwinder, W. Schempp (eds.), Reidel (1984), 1--84) for a survey. During the eighties, several attempts were made, mainly by the Dutch school (Koornwinder, Heckman, Opdam), to extend these results in higher rank (i.e. in several variables), until the discovery of Dunkl operators in the rational case and Cherednik operators in the trigonometric case. Together with q-special functions introduced by Macdonald, this has led to a beautiful theory, developed by several authors, which encompasses in a unified way harmonic analysis on all Riemannian symmetric spaces and spherical functions thereon:

- generalized Bessel functions on flat symmetric spaces, and their asymmetric version, known as the Dunkl kernel,

- Heckman-Opdam hypergeometric functions on positively or negatively curved symmetric spaces, and their asymmetric version, due to Opdam,

- Macdonald polynomials on affine buildings.

Beside Fourier analysis and special functions, this theory has also deep and fruitful interactions with

- algebra (double affine Hecke algebras),

- mathematical physics (Calogero-Moser-Sutherland models, quantum many body problems),

- probability theory (Feller processes with jumps).

In this series of lectures, we aim at giving an updated overview of Dunkl theory, mostly of its analytic and algebraic aspects.