The trajectories of analytic vector fields, that is the solutions of ordinary differential equations with analytic coefficients, arise in many different areas of mathematics and they are being studied from various perspectives by analytic, algebraic, numerical, or geometric methods. The ANR project STAAVF combines all such approaches focusing in particular on the geometric behavior of trajectories of analytic vector fields. It is well known that most of the interesting differential equations cannot be solved exactly. Approximate solutions, obtained by numerical methods, often do not give satisfactory information on the qualitative behavior of the true solutions, such as stability, limit sets, limit cycles, or the phenomena of (non)-oscillation. The trajectories of real analytic vector fields are transcendental in general. However their geometry is often “tame”. The understanding of the qualitative geometric behavior of trajectories of analytic vector fields is the primary objective of our project. We want to determine in which cases the solutions are tame and to make precise the meaning of tameness in each case.

In recent years there has been substantial progress in understanding of the qualitative properties of trajectories of real analytic (and more general) vector fields by a large variety of geometric methods, such as: resolution of singularities, classification of real analytic function germs, stratifications and conormal geometry, gradient flow, ridge and valley lines, semi-algebraic and o-minimal geometry, and also by more analytic approaches such as: quasi-analytic classes, (pseudo)abelian integrals, formal series and asymptotic analysis, non-linear analysis, resurgent methods and resummation processes. The main goal of this meeting is to reunite the experts coming from different approaches, and the young researches, from our ANR project as well as the ones outside this project, to provide ground for the exposition of important recent results obtained during our ANR project, presentation of the underlying methods and free discussions.