A dynamical system is a map of spaces $X \times S \to X$, and a map of dynamical systems $X \to Y$ over $S$ is an $S$-equivariant map. There is both an injective and a projective model structure for this category.

These model structures are special cases of injective and projective model structures for space-valued diagrams $X$ defined on a fixed category $A$ enriched in simplicial sets. Simultaneously varying the parameter category $A$ (or parameter space $S$) along with the diagrams $X$ up to weak equivalence is more interesting, and requires new model structures for $A$-diagrams having weak equivalences defined by homotopy colimits, as well as a generalization of Thomason's model structure for small categories to a model structure for simplicial categories.

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).

One of the most fascinating enigmas of modern art is the empty circle left at the centre of "Print Gallery", an engraving by Dutch artist M. C. Escher. In 1956, Escher challenged the laws of perspective with Print Gallery and found himself trapped behind an impossible barrier. This uncompleted masterpiece quickly became the most puzzling enigma of Modern Art, for both artists and scientists. Half a century later, mathematician Hendrik Lenstra took everyone by surprise by drawing a fantastic bridge between the intuition of the artist and his own, shattering the Infinity Barrier.

In this talk I will revisit the classical topic of polynomial equations in one variable and Tschirnhaus transformations. I will discuss 19th century theorems of Hermite, Joubert and Klein, recent results in this area, and several open problems.

The classical theorem of Privalov says that if $u$ is $Lip_\alpha$ with $0 < \alpha <1$, then its Hilbert transform $\tilde{u}$ is also $Lip_\alpha$. However, this result fails for $Lip_1$ functions. In this case, the modulus of continuity of $\tilde{u}$ behaves like $t \log 1/t$ as $t \to 0^+$. We introduce the “generalized Lipschitz class” $Lip_{\alpha(t)}$, which certainly coincides with the classical case when $\alpha(t) \equiv \alpha$, and then show that the above results, as well as some other classical results of Hardy—Littlewood, hold for $Lip_{\alpha(t)}$ functions.

Informally speaking, the essential dimension of an algebraic object is the minimal number of independent parameters one needs to define it. In the past 15 years this notion has been investigated in several contexts by a range of techniques, and has been found to have interesting and surprising connections to many problems in algebra and algebraic geometry. I will survey some of this research in my talk.