Title: FI-algebras

Abstract: An FI-algebra encodes a family of algebras with symmetric group actions, and an ideal of an FI-algebra represents an infinite family of ideals with symmetry.  I will give an overview of some results about when such ideals are finitely generated, and how to compute with them.  Then I will explain how to compute Hilbert series of these ideals.  Along the way we will see some surprising connections to combinatorics, such as well-partial orders and regular languages.