**Data:**segunda-feira, 26 de fevereiro de 2018, às 14h

**Sala:**Sala B-144

**Palestrante:**Jesús M. F. Castillo (U. de Extremadura)

**Título:**Differentials of interpolation processes: A Tourbillon.

**Resumo:**The purpose of the talk is to present recent advances in the understanding of differentials of interpolation processes, as initiated by Kalton, and Rochberg and Weiss. Recall that a twisted sum of a Banach space $X$ is a Banach space $\Omega_X$ containing a subspace isomorphic to $X$ so that $\Omega_X/X$ is isomorphic to $X$. For our purposes, the simplest example is the Kalton-Peck space $Z_2$ that is a twisted Hilbert space (i.e., a twisted sum of Hilbert spaces $\ell_2$). On the other hand, it is well known that given a complex interpolation scale, a kind of derivation process produces twisted sums. To be more precise, let $\mathbb D$ be the complex unit disk, given an "interpolation family", say, $\{ X_\omega \}_{\omega \in \partial \mathbb D}$ for which complex interpolation produces spaces $\{X_\theta \}_{ \theta \in \mathbb D}$, the derivation process at $\theta$ produces twisted sums $\Omega_\theta$ of $X_\theta$. In this context, the space $\Omega_\theta$ is called the derived space (of the scale) at $\theta$. Again, the simplest example is the space $Z_2$, which appears as the derived space at $1/2$ of the scale of $\ell_p$-spaces when starting with the couple $(\ell_1, \ell_\infty)$.

Kalton did a tremendously deep work in the study of complex differential processes in a Köthe function space ambient, while Rochberg did a tremendous work towards the understanding of iterated derivation. Our purpose is to present a panorama of new results and directions of research in these topics. Which, in bird's-eye view could be:

Classical and new derivations.

Higher order derivations, and why.

Stability of the complex differential process.

Symmetries of differential processes.

Kalton vs. Rochberg derivation.

Real vs. complex derivation.

Aspects of this research have been jointly worked with Félix Cabello, Willian Correa, Valentin Ferenczi, Nigel Kalton, Manuel González, Daniel Morales and Jesús Súarez.