Antonio Lerario: The Wasserstein geometry of algebraic hypersurfaces

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 space of hyperusrfaces, which has the meaning of an inner Wasserstein distance. For instance, this distance is of Weil-Petersson type. Time permitting, I will connect to the condition number of polynomial system solving.

This is joint work with P. Antonini and F. Cavalletti.