In these talks I will describe a relation between gauge theory instantons and quantum many-body systems. Certain observables in supersymmetric gauge theories admit exact representations in terms of interacting quasi-particles. This correspondence leads to effective descriptions in terms of integrable many-body dynamics and provides new tools for studying non-perturbative phenomena in quantum field theory.

I will explain how these structures give rise to quantum limit shapes and stochastic processes closely related to random walks. This perspective reveals new connections between gauge theory, probability, and statistical mechanics, and leads to new results in many-body dynamics.

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.