While functional inequalities play an important role in
mathematical physics, the focus was usually onSchr\"odinger
operators with some notable exceptions like Hardy inequalities
for Dirac operators.
In this talk I present some preliminary results for the sharp
spinorial versionn of the Caffarelli-Kohn-Nirenberg
inequalities, i.e., Sobolev inequalities with weights. 
This is joint work with Jean Dolbeault, Maria Esteban
and Rupert Frank.

Minimizing the first eigenvalue of systems
Antoine Henrot (Nancy)

In this talk, we ask whether a Faber-Krahn type inequality is true
for the first eigenvalue of classical systems in physics: Stokes,
Lam\'e (linear elasticity) or Maxwell. We will give partial results
showing that the ball is not always the minimizer, by contrast with
the scalar case of the Laplacian.
This is a joint work with Antoine Lemenant, Idriss Mazari
and Yannick Privat

The sharp sigma_2 - curvature inequality on the sphere in quantitative form
Jonas Peteranderl (LMU Munich)

In case a sharp functional inequality admits optimizers, we are
interested in improving the inequality by adding terms that
involve a distance to the set of optimizers. As is well known,
among all metrics on the sphere that are conformal to the
standard metric and have positive scalar curvature, the total
sigma_2 -curvature, normalized by the volume, is uniquely
(up to Möbius transformations) minimized by the standard
metric. Formulating this problem as a functional inequality
in the conformal factor, we show that if a metric almost
minimizes, then it is almost the standard metric (up to Möbius
transformations). Moreover, we obtain a refined inequality
with optimal exponents for two different notions of distance
to the set of minimizers. 
The talk is based on joint work with Rupert Frank.

We discuss stability results in Gagliardo-Nirenberg-Sobolev inequalities,
a joint project with  J. Dolbeault, B. Nazaret and N. Simonov. 
We have developped a new quantitative and costructive "flow method", 
based on entropy methods and sharp regularity estimates for solutions to
the fast diffusion equation (FDE). This allows to study refined versions of
the Gagliardo-Nirenberg-Sobolev inequalities that are nothing but explicit
stability estimates. Using the quantitative regularity estimates, we go
beyond the variational results and provide fully constructive estimates, to
the price of a small restriction of the functional space which is inherent to
the method.

We consider a system of N points in singular interaction of
Coulomb or Riesz type, evolving by gradient flow or
conservative flow (such as the point vortex system in 2D) with
or without noise. We describe the convergence to the mean-field
limit by a modulated energy method, that relies  on a functional
inequality of commutator estimate type. We also discuss the
question of obtaining global-in-time convergence and its
connection with modulated log-Sobolev inequalities.