Abstract: Consider $\mathbb{R}^n$ equipped with a real analytic Riemannian metric ${\bf g}$. Let $f : \mathbb{R}^n\to\mathbb{R}$ be a real analytic function singular at $O$ the origin. We would like to understand the dynamics of $\nabla f$ in a neighbourhood of the critical point $O$, where $\nabla f$ stands for the gradient vector field of the function $f$ associated with the metric ${\bf g}$. We are particularly interested in the oscillating/non-oscillating behaviour in a neighbourhood of $O$ of any gradient trajectory accumulating on $O$.

We prove that if a trajectory lies in a real analytic surface with an isolated singularity at $O$, then it cannot oscillate at $O$.

In the first talk, I will recall elementary and well known facts and ideas about the gradient problem. In the second one, I will sketch the proof of our theorem.

This is joint work with Fernando Sanz (Valladolid).