## Daniel Robertz

An involutive basis is a special kind of (in general non-minimal) Groebner basis containing additional combinatorial information about the ideal or module it generates. In this talk I concentrate on particular involutive bases: Pommaret bases. It is shown that a Pommaret basis allows to construct almost immediately a free resolution for the module it generates. Moreover, the constructed generators for each syzygy module form a Pommaret basis again. If a homogeneous ideal has a (finite) Pommaret basis, then its Castelnuovo-Mumford regularity can easily be read off from the constructed free resolution.

## Charles Crissman

Many well-known theorems in commutative algebra are of the form: ``an ideal maximal with respect to not having property P is prime.'' Famous examples of P include: "intersecting a given multiplicatively closed set" and "being finitely generated." Despite the similarity of all of these results, their proofs have always been disparate, without any unifying pattern. I will discuss a new "Prime Ideal Principal" of Lam and Reyes that unifies the proof of these results. I'll explain the clever but very elementary proof of their result and talk about some of the beautiful new examples that the theorem covers.