On some fundamental inequalities of quantum mathematical physics

By the early 1970's two important conjectures, the Wigner-Yanasee-Dyson conjecture and the Strong Subadditivity of Quantum Entropy conjecture, had attracted the attention on many mathematicians and physicists. These conjectures were both proved in 1973 by Leib and by Lieb and Ruskai respectively. The methods introduced for their solution were powerful, and immediately found other applications, for example in Lindblad's 1975 proof of the Data Processing Inequality, now one of the cornerstones of quantum information theory. In this talk I will briefly explain the history before turning to modern developments and a modern perspective that has led to new inequalities and stronger versions of known inequalities. In this latter part, I will focus on recent work of myself, and myself in collaboration with Alexander Mueller-Hermes and with Haonan Zhang. I will also briefly discuss some recent applications.