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, semialgebraic and ominimal geometry, and also by more analytic approaches such as: quasianalytic classes, (pseudo)abelian integrals, formal series and asymptotic analysis, nonlinear 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.

Scientific Committee
Edward Bierstone (University of Toronto) Daniel Panazzolo (Université de Haute Alsace) Patrick Speissegger (MacMaster University) Yosef Yomdin (Weizmann Institute of Science)
Organizing Committee
Krzysztof Kurdyka (Université Savoie Mont Blanc) Adam Parusinski (Université Nice SophiaAntipolis) JeanPhilippe Rolin (Université de Bourgogne) Fernando Sanz (University of Valladolid)
Speakers
 André Belotto (University of Toronto)
Monomialization of Differential Forms on an Algebraic or Analytic Variety
 Gal Binyamini (University of Toronto)
Counting Solutions of Differential Equations
 Marcin Bobienski (University of Warsaw)
Finite Cyclicity of SlowFast Darboux Systems
 Ralph Chill (TU Dresden)
Gradient Systems Associated with jElliptic Functionals
 Georges Comte (Université Savoie MontBlanc)
Sets with Few Rational Points
 Raf Cluckers (Université Lille 1)
Lebesgue Integration of Oscillating and Subanalytic Functions, Part I
 Aris Daniilidis (University of Chile)
Trajectory Length of the Tame Sweeping Process
 Andrei Gabrielov (Purdue University)
Classication of Spherical Quadrilaterals
 Lubomir Gavrilov (Université Toulouse III)
Perturbations of Quadratic Hamiltonian TwoSaddle Cycles
 Tobias Kaiser (University of Passau)
Lebesgue Measure and Integration Theory on Arbitrary Real Closed Fields
 Olivier Le Gal (Université Savoie MontBlanc)
Realization of Formal Invariant Curves
 Pavao Mardesic (Université de Bourgogne)
Unfoldings of SaddleNodes and their Dulac Time
 JeanFrançois Mattei (Université Toulouse III)
Topological Classes and Topological Moduli Spaces of Marked Holomorphic Singular Foliation
s
 Chris Miller (Ohio State University)
Expansions of the Real Field by Trajectories of Denable Vector Fields
 Laurentiu Paunescu (University of Sydney)
Nuij Type Pencils of Hyperbolic Polynomials
 Armin Rainer (University of Vienna)
Optimal Regularity of Roots of Smooth Polynomials
 Maja Resman (University of Zagreb)
EpsilonNeighborhoods of Orbits and Classications of Parabolic Germs
 Tamara Servi (University of Pisa)
Lebesgue Integration of Oscillating and Subanalytic Functions, Part II
 David Sauzin (Fibonacci Laboratory of Pisa)
Nonlinear Analysis with Resurgent Functions
 Stanislaw Spodzieja (University of Lodz)
Convexifying Positive Polynomials and a Proximity Algorithm
 Joris Van der Hoeven (Ecole polytechnique)
Towards a Model Theory for Transseries
 Sergei Yakovenko (Weizmann Institute of Science)
Classication of Higher Order Linear Differential Equations
 Yosef Yomdin (Weizmann Institute of Science)
Smooth Parametrizations in Analysis, Dynamics, and Diophantine Geometry.
