Fernando Sanz participa en el GAiA 2021 - Geometriy, Analysis and Applications que se realiza en Santander del 25 de enero al 5 de febrero de 2021.  Organizado por la Universidad de Cantabria: Carlos Beltrán, Rafael Granero y Nuria Corral (Investigadora vinculada al CTRI). 


Fernando Sanz: On the problem of the gradient of an analytic function



In the 70's, Lojasiewicz an Thom promotd the investigation of geometrical properites of trajectories of a gradient vector field of a real analytic function. By means of a celebrated inequality of Lojasiewicz, bounded trajectories have finite lengh and accumulate to a single point. Thom conjectured then that they possess a well-defined tangent at the limit point. The problem was revived aroung 20 years later wih the proof of Thom's conjecture by Kurdyka, Mostowski and  Parusinski and also with Moussu's stronger Non-oscillation Conjeture: a trajectory of a gradient cannot oscillate. The answer to this last conjecture is only known in dimension two, but still open in general. In this talk, we give a panorma of the achievements concerning this problem, a subject that reveals a particuar interplay between analysis of ODEs, real analytic and subanalytic geometry and qualitative theory of dynamical systems.