The 16th Problem of Hilbert is one of the most famous remaining unsolved problems of mathematics. It concerns whether a polynomial vector field on the plane has a finite number of limit cycles. There is a strong connection with divergent solutions of differential equations, where a central role is played by the Stokes Phenomenon, the change in asymptotic behaviour of the solutions in different sectors of the complex plane.

The contributions to these proceedings survey both of these themes, including historical and modern theoretical points of view. Topics covered include the Riemann-Hilbert problem, Painleve equations, nonlinear Stokes phenomena, and the inverse Galois problem.Contents:

• Stokes Phenomenon: Historical Background (J-P Ramis)
• Limit Cycles and Nonlinear Stokes Phenomena (Y Ilyashenko)
• Formal Solutions of Nonlinear Systems of Ordinary Differential Equations (W Balser)
• Holomorphic Bundles, Associated with Linear Differential Equations and the Riemann–Hilbert Problem (A A Bolibruch)
• Well Behaved Convolution Averages and the Non-Accumulation Theorem for Limit-Cycles (J Ecalle & F Menous)
• Passive and Active Resonance. Nonlinear Resurgence and Isoresurgent Deformations (J Ecalle & B Vallet)
• Connection Formulae for the Painlevé Transcendents (A R Its)
• On Invariants of Monodromy Groups (V P Kostov)
• The Inverse Problem in Differential Galois Theory (C Mitschi & M F Singer)
• Algorithmic Approach of the Multisummation of Formal Power Series Solutions of Linear ODE Applied to the Stokes Phenomena (F Naegele & J Thomann)
• Confluence of Turning Points in Exact WKB Analysis (F Pham)
• Galois Theory of Difference Equations. A Survey (M van der Put)
• About the Inverse Problem in Differential Galois Theory: The Differential Abhyankar Conjecture (J-P Ramis)
• Stokes Phenomenon in Dimension Two (C Sabbah)
• Exceptional Solutions of the Forced van der Pol Equation (R Schäfke & A Fruchard)
• Invariants for Singular Points of Holomorphic Vector Fields on the Complex Plane (S M Voronin)

Readership: Pure and applied mathematicians.