  Interpolation inequalities play an essential role in Analysis with
fundamental consequences in Mathematical Physics, Nonlinear Partial
Differential Equations (PDEs), Markov Processes, etc., and have a wide range of
applications in various other areas of Science. Research interests have evolved
over the years: while mathematicians were originally focussed on abstract
properties (for instance appropriate notions of functional spaces for the
existence of weak solutions in PDEs), more qualitative questions (for instance,
bifurcation diagrams, multiplicity of the solutions in PDEs and their
qualitative behaviour) progressively emerged. The use of entropy methods in
nonlinear PDEs is a typical example: in some cases, the optimal constant in the
inequality can be interpreted as an optimal rate of decay of an entropy for an
associated evolution equation. Much more has been learned by adopting this
point of view.
  This paper aims at illustrating some of these recent aspect of
entropy-entropy production inequalities, with applications to stability in
Gagliardo-Nirenberg-Sobolev inequalities and symmetry results in
Caffarelli-Kohn-Nirenberg inequalities. Entropy methods provide a framework
which relates nonlinear regimes with their linearized counterparts. This
framework allows to prove optimality results, symmetry results and stability
estimates. Some emphasis will be put on the hidden structure which explain such
properties. Related open problems will be listed.
