New bridges between metric geometry and algebra

A manifold submetry is a map from a Riemannian manifold to a metric space, with smooth and equidistant fibers. Such objects appear in the literature in relation to rigidity phenomena, especially in the context of non-negative sectional curvature. In the past few years, it was shown that there is a one-to-one correspondence between manifold submetries from spheres and certain algebras of polynomials called Laplacian algebras, and through this correspondence a dictionary has been created between geometric properties of manifold submetries and algebraic properties of the corresponding algebra. In this talk we will survey these results, and discuss recent generalizations of these results to compact symmetric spaces and normal homogeneous spaces.