This is the first of two volumes of a state-of-the-art survey article collection which
originates from three commutative algebra sessions at the 2009 Fall Southeastern American
Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse
areas of commutative algebra and build a bridge between Noetherian and non-Noetherian
commutative algebra. These volumes present current trends in two of the most active areas of
commutative algebra: non-noetherian rings (factorization ideal theory integrality) and
noetherian rings (the local theory graded situation and interactions with combinatorics and
geometry).This volume contains combinatorial and homological surveys. The combinatorial papers
document some of the increasing focus in commutative algebra recently on the interaction
between algebra and combinatorics. Specifically one can use combinatorial techniques to
investigate resolutions and other algebraic structures as with the papers of Fløystad on
Boij-Söderburg theory of Geramita Harbourne and Migliore and of Cooper on Hilbert functions
of Clark on minimal poset resolutions and of Mermin on simplicial resolutions. One can also
utilize algebraic invariants to understand combinatorial structures like graphs hypergraphs
and simplicial complexes such as in the paper of Morey and Villarreal on edge
ideals.Homological techniques have become indispensable tools for the study of noetherian
rings. These ideas have yielded amazing levels of interaction with other fields like algebraic
topology (via differential graded techniques as well as the foundations of homological algebra)
analysis (via the study of D-modules) and combinatorics (as described in the previous
paragraph). The homological articles the editors have included in this volume relate mostly to
how homological techniques help us better understand rings and singularities both noetherian
and non-noetherian such as in the papers by Roberts Yao Hummel and Leuschke.
