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).

Abstract: I will discuss configuration spaces of complex and real spheres modulo the inversive space transformations. A classical construction of S. Lie and F. Klein allows one to treat these spaces as configuration spaces of points in the projective space of one dimension higher modulo the orthogonal transformations. Some special cases can be described very explicitly using subvarieties in toric varieties.

We introduce a completion tower for algebras over operads in unbounded chain complexes (resp. symmetric spectra) and prove that under appropriate connectivity conditions this tower interpolates between Quillen homology and the identity functor. This talk will focus on the chain complex version of these results, beginning with a short introduction to Quillen's notion of derived abelianization, and followed by a sketch of the proof with an emphasis on several of the homotopical arguments. We will illustrate the tower results in the special case of commutative differential graded algebras and André-Quillen homology. These results are joint work with K. Hess.