Abstract: There is a complete characterization of the zeros sets of the Hardy space Hp. However, at the present, we have just some partial characterizations for most of the relatives of Hp, e.g. the Bergman space and the Dirichlet space. For the latter space, we will discuss Carleson’s condition and its generalization by Shapiro-Shields. Then we give some new families of zero sets which are not covered by the preceding classical results.

Abstract: The plan of the talk is to look at some explicit recipes for constructing minimal free resolutions of certain 'nice' classes of monomial and binomial ideals in a polynomial ring.

The general idea is to introduce appropriate simplicial or polytopal complexes which encode the information of a free resolution and read off the resolution from the complex.