08w5055 Classical Problems on Planar Polynomial Vector Fields

At the end of the 19th century Poincaré and Hilbert stated three problems which are still open today: the problem of the center and the problem of Poincaré, stated by Poincaré in 1885 and in 1891, and Hilbert’s 16th problem, stated in Hilbert’s address at the International Congress of Mathematicians in Paris in 1900. The first two and the second part of Hilbert’s 16th problem are on planar polynomial vector fields and were the focus of our Workshop at BIRS. These problems are very easy to state but very difficult to solve. All three problems are essentially of a global nature and this common feature is at the core of the difficulty: we are interested in whole classes of polynomial vector fields defined on the whole plane. These problems stir our interest because understanding this area necessarily involves using ideas and methods from a variety of fields: algebra, geometry, analysis, real and complex, numeric and symbolic computations. In the last two decades also logicians have become interested in this area and new connections were established. During the past twenty years things have been moving slowly but steadily ahead and a fresh new view is beginning to emerge highlighting a subtle unity among these classical problems. The algebraic nature implicit in the first two problems was known but work in the past twenty years brought deeper connections to light. This area is very much alive and growing. The purpose of this workshop was to share new knowledge, take a measure of the new developments and bring new people into the group who could help in confronting the new tasks ahead.