Abstract: The classical Nagata conjecture can be viewed as
a polynomial interpolation problem in projective plane. It has
been reinterpreted in terms of Seshadri constants some ten
years ago.In the present  talk I will recall another kind of
asymptotic invariants attached to a homogeneous ideal
(or ideal sheaf), the Waldschmidt constants, and explain how they come into the picture when considering the classical Nagata conjecture. Then I will show that the obtained setting can be vastly generalized leading to a tower of conjectures governed by a family of polynomials which have been overlooked so far.

This will be a report on a joint work with Brian Harbourne,
Marcin Dumnicki and Halszka Tutaj-Gasinska, available at: