A polynomial $p$ is bigger than $q$ if $p(x)> q(x)$ for all $x$ and a classical area of mathematics (called semialgebraic geometry) focuses on providing algebraic ways to test if such inequalities hold. Semialgebraic geometry can be generalized to polynomials with matrix variables which is fortunate, since many problems in systems engineering have matrix unknowns. The difficulty here is that the polynomials are in variables which do not commute. Particularly important (for numerical solutions) is convexity and there has been a recent quest to classify convex noncommutative polynomials. The conclusion turns out to be simple. The talk will be on the mathematics and the motivation.