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CONFERENCE

Real Analytic Geometry and Trajectories of Vector Fields
June 8 - 12, 2015

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.

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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 Sophia-Antipolis)
Jean-Philippe 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 Slow-Fast Darboux Systems
• Ralph Chill (TU Dresden)
  Gradient Systems Associated with j-Elliptic Functionals

• Georges Comte (Université Savoie Mont-Blanc)
  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)
  Classication of Spherical Quadrilaterals
• Lubomir Gavrilov (Université Toulouse III)
  Perturbations of Quadratic Hamiltonian Two-Saddle Cycles
• Tobias Kaiser (University of Passau)
  Lebesgue Measure and Integration Theory on Arbitrary Real Closed Fields
  
• Olivier Le Gal (Université Savoie Mont-Blanc)
  Realization of Formal Invariant Curves
• Pavao Mardesic (Université de Bourgogne)
  Unfoldings of Saddle-Nodes and their Dulac Time
• Jean-François Mattei (Université Toulouse III)
  Topological Classes and Topological Moduli Spaces of Marked Holomorphic Singular Foliations
• Chris Miller (Ohio State University)
  Expansions of the Real Field by Trajectories of Denable 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)
  Epsilon-Neighborhoods of Orbits and Classications 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)
  Classication of Higher Order Linear Differential Equations
• Yosef Yomdin (Weizmann Institute of Science)
  Smooth Parametrizations in Analysis, Dynamics, and Diophantine Geometry.