This volume consists of invited lecture notes, survey papers and original research papers from the AAGADE school and conference held in Bedlewo, Poland in September 2015. The contributions provide an overview of the current level of interaction between algebra, geometry and analysis and demonstrate the manifold aspects of the theory of ordinary and partial differential equations, while also pointing out the highly fruitful interrelations between those aspects. These interactions continue to yield new developments, not only in the theory of differential equations but also in several related areas of mathematics and physics such as differential geometry, representation theory, number theory and mathematical physics.
The main goal of the volume is to introduce basic concepts, techniques, detailed and illustrative examples and theorems (in a manner suitable for non-specialists), and to present recent developments in the field, together with open problems for more advanced and experienced readers.
It will be of interest to graduate students, early-career researchers and specialists in analysis, geometry, algebra and related areas, as well as anyone interested in learning new methods and techniques.
Preface.- An introduction to Dunkl theory and its analytic aspects.- Holonomic Systems.- Sub-Riemannian geometry and hypoelliptic operators.- Asymptotic analysis and summability of formal power series.- WKB analysis and Stokes geometry of differential equations.- Transcendental Meromorphic Solutions of P34and Small Targets.- Towards the convergence of generalized power series solutions of algebraic ODEs.- Connection problem for regular holonomic systems in several variables.- On k-summability of formal solutions for certain higher order partial differential operators with polynomial coefficients.- On Stokes phenomena for the alternate discrete PI equation.- Flat structures and algebraic solutions toPainlevé VI equation.- Relation of Semi-classical orthogonal polynomials to General Schlesinger systems via Twistor theory.- Some notes on the multi-level Gevrey solutions of singularly perturbed linear partial differential equations.- Reducibility of hypergeometric equations.- Parametric Borel summability of partial differential equations of irregular singular type.