A set theoretic ideal over an underlying set X is, intuitively, a collection of "measure zero" subsets of X. Two standard examples are 1) the ideal of Lebesgue null subsets of the reals, and 2) the ideal generated by complements of closed unbounded subsets of the least uncountable ordinal; the latter is called the nonstationary ideal, and will be the main focus of the talk. If I is an ideal on X then there is a natural quotient, denoted B_I, which forms a Boolean Algebra. If I is the nonstationary ideal, the algebraic behavior and antichain structure of B_I is highly independent of the standard axioms of mathematics. For example, it is independent whether or not the canonical Boolean embeddings of B_I into its liftings are regular embeddings; moreover this is closely tied to so-called large cardinals. I will discuss recent solutions to several open problems in the area. This is joint work with Martin Zeman.