Given a category $\mathcal {C} $ of a combinatorial nature, we study the following
fundamental question: how do combinatorial properties of $\mathcal {C} $ affect algebraic …

The purpose of this paper is to prove that certain limits of polynomial rings are themselves
polynomial rings, and show how this observation can be used to deduce some interesting …

We introduce and study equivariant Hilbert series of ideals in polynomial rings in countably
many variables that are invariant under a suitable action of a symmetric group or the monoid …

We introduce FI-algebras over a commutative ring K and the category of FI-modules over an
FI-algebra. Such a module may be considered as a family of invariant modules over …

Convex sets arising in a variety of applications are well-defined for every relevant
dimension. Examples include the simplex and the spectraplex that correspond, respectively …

The resolutions of determinantal ideals exhibit a remarkable stability property: for fixed rank
but growing dimension, the terms of the resolution stabilize (in an appropriate sense). One …

We study the asymptotic behavior of the Castelnuovo–Mumford regularity along chains of
graded ideals in increasingly larger polynomial rings that are invariant under the action of …

Symmetric ideals in increasingly larger polynomial rings that form an ascending chain are
investigated. We focus on the asymptotic behavior of codimensions and projective …

We introduce a new class of monomial ideals which we call symmetric shifted ideals.
Symmetric shifted ideals are fixed by the natural action of the symmetric group and, within …

Recently, chains of increasing symmetric ideals have attracted considerable attention. In this
note, we summarize some results and open problems concerning the asymptotic behavior of …