Symmetric structures in L_p spaces

In this talk, we will review some classical results on the
classification of symmetric structures in Lebesgue spaces. This will
lead to the introduction of some embedding problems in the theory of
$L_p$ spaces, and its metric characterization via the Ribe program,
from which we will remark the so-called $\mathrm{X}_p$ inequalities
introduced by Naor and Schechtman in 2016. Moreover, we will briefly
comment some anologous results on Schatten classes and some other
noncommutative spaces.