The celebrated Brascamp-Lieb (BL) inequalities, and their reverse form of Barthe, is a powerful framework which unifies and generalizes many important inequalities in analysis,
convex geometry and information theory.

I will exemplify BL inequalities, building to the general set-up. I will describe the structural theory that characterizes existence and optimality of these inequalities in terms of their description (called BL-data). But can one efficiently compute existence and optimality from given BL-data?

I will describe a recent polynomial time algorithm for these problems. The analysis of this algorithm uses tools from several areas, including invariant theory, quantum information theory, and non-commutative algebra, and resolves some algorithmic problems in these areas as well.
I hope to explain some of these connections, and some extensions of this work in the talk as well.

Most of this presentation is based on the paper https://arxiv.org/abs/1607.06711