After a brief introduction to operator algebras, I will present new inequalities concerning entropy and energy; in particular, my recent relative entropy/energy ratio local bound in Quantum Field Theory. This bound is model-independent and rigorous; it follows solely from first principles in the framework of translation-covariant, local Quantum Field Theory on the Minkowski spacetime. 

In this talk, I am going to review recent progress in the mathematical understanding of the low-energy properties of dilute quantum systems. In particular, I am going to present a rigorous version of Bogoliubov theory and I am going to show how it can be used to obtain precise estimates for the ground state energy and the low-energy excitation spectrum and also to approximate the time evolution of Bose gases in the so-called Gross-Pitaevskii regime.