  The starting point of this paper is a duality for sequences of natural
numbers which, under mild hypotheses, interchanges subadditive and
superadditive sequences and inverts their asymptotic growth constants.
  We are motivated to explore this sequence duality since it arises naturally
in at least two important algebraic-geometric contexts. The first context is
Macaulay-Matlis duality, where the sequence of initial degrees of the family of
symbolic powers of a radical ideal is dual to the sequence of
Castelnuovo-Mumford regularity values of a quotient by ideals generated by
powers of linear forms. This philosophy is drawn from an influential paper of
Emsalem and Iarrobino. We generalize this duality to differentially closed
graded filtrations of ideals.
  In a different direction, we establish a duality between the sequence of
Castelnuovo-Mumford regularity values of the symbolic powers of certain ideals
and a geometrically inspired sequence we term the jet separation sequence. We
show that this duality underpins the reciprocity between two important
geometric invariants: the multipoint Seshadri constant and the asymptotic
regularity of a set of points in projective space.

Commutative Algebra;Algebraic Geometry