New results on the Lieb-Thirring inequality

The Lieb-Thirring inequality is a generalization of the
Gagliardo-Nirenberg inequality, which plays a central role in the
analysis of large quantum systems. In this talk I will first introduce
the inequality and then focus the discussion on its best constant. After
reviewing what is believed, what is known and what is open, I will
present new results on the value of the best constant as well as
numerical simulations in 1D and 2D.
Collaboration with Rupert L. Frank (Caltech, USA) and David Gontier
(Paris-Dauphine, France).