Talks' abstracts > Rosales Ortiz Alejandro

We start by explaining how one can construct in a canonic way a metric space out of a deterministic tree indexed function, and we introduce a set of tools to study metric properties of these surfaces - namely, the cactus tree and metric nets. The construction can be thought of as the continuous version of classical bijections between planar maps and well-labeled trees, and has been used to construct Brownian surfaces such as the Brownian sphere and Brownian disk. We then apply these constructions and tools to Brownian motion indexed by alpha-stable trees, and related tree-indexed processes. In a first step, we explain how excursion theory for tree-indexed Markov processes [Riera, R-O 2024 & 2026] can be used to study these surfaces. If time permits, we then discuss aspects of a work in progress where we study various metric and topological questions. The content of the talk is taken from three works written in collaboration with Armand Riera.