Linear matrix inequalities (LMIs) play a role in many areas of applications and the set of solutions to one is called a spectrahedron. LMIs in (dimension-free) matrix variables model most problems in linear systems engineering, and their solution sets are called free spectrahedra. These are exactly the semialgebraic matrix convex sets. This talk will discuss analytic maps between free spectrahedra and, under certain irreducibility assumptions, classify all those that are bianalytic. The foundation of such maps turns out to be a very small distinguished class of birational maps we call convexotonic. The results depend on new tools in noncommutative analysis, such as a Positivstellensatz for analytic functions whose real part is positive on a free spectrahedron, and fine detail, geometric in nature locally and algebraic in nature globally, about the boundary of free spectrahedra.