ABSTRACT:
In this talk I will review recent result about how the use of linear and nonlinear flows has been key to prove functional inequalities and qualitative properties for their extremal functions. I will also explain how from these inequalities and their best constants, optimal spectral estimates can be obtained for Schrödinger operators.
This is a topic which is at the crossroads of nonlinear analysis and probability, with implications in differential geometry and potential applications in modelling in physics and biology.