Optimal transport is the general problem of moving one distribution of mass to another one as efficiently as possible, typically keeping track of the ambient geometry. In this seminar I will present recent results on the optimal transport problem between algebraic hypersurfaces of the same degree in complex projective space. I will explain how this problem, which is defined through a constrained dynamical formulation, is equivalent to a Riemannian geodesic problem away from the discriminant. I will discuss the main properties of the distance obtained in this way on the whole space of hypersurfaces, which has the meaning of an inner Wasserstein distance. For instance, this distance is of Weil-Petersson type and has some nice convexity properties. These properties are particularly interesting in the context of Smale 17th problem (on polynomial system solving) and the problem of regularity of roots of polynomials. This is joint work with P. Antonini and F. Cavalletti