Alessio Figalli, University of Texas, Austin

TITLE AND ABSTRACT:
Stability results for geometric and functional inequalities and applications


Geometric and functional inequalities play a crucial role 
in several problems arising in the calculus of variations, 
partial differential equations, geometry, etc. More recently, 
there has been a growing interest in studying the stability 
for such inequalities. The basic question one wants to address 
is the following: suppose we are given a functional inequality 
for which minimizers are known. Can we prove that if a function 
almost attains the equality then it is close (in some suitable sense) 
to one of the minimizers? The aim of these lectures is to 
describe different ways to attack this problem, together 
with some applications of this kind of results.