# Combinatorial Commutative Algebra

### Irena Peeva

Cornell University, Ithaca, USA### Volkmar Welker

Philipps-Universität, Marburg, Germany

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## Abstract

There has been very fruitful interaction between the fields of Combinatorics and Commutative Algebra since the 70's \cite{BrHe93}, \cite{Eis95}, \cite{Sta76}, \cite{Sta96}. Notably, there has been a surge in interest and research during the last 8 years: a great variety of new ideas and techniques were introduced, and substantial progress was made. Projects in this direction have been undertaken by both established mathematicians and graduate students or postodcs. Currently, the main centers for such research are Germany, Italy, Japan, and USA. The toolset from Commutative Algebra that helps to solve combinatorial problems ranges from Hilbert-series to local cohomology. On the other hand, Combinatorics enriches Commutative Algebra by supplying questions, methods and results that ask for a more general setting, a setting which in many cases has a ring theoretic framework. The Oberwolfach workshop on ``Combinatorial Commutative Algebra'' was organized as an attempt to gather researchers from Combinatorics and Commutative Algebra in order to announce the latest developments, spread new problems, and spark further interaction. For that purpose only very few and only longer talks were scheduled, and they all reported on exciting recent developments. The talks covered a wide spectrum of topics ranging over f-vector theory, algebraic shifting, simplicial complexes, polytopes, Gr\"obner basis, free resolution, powers of ideals, Hilbert-Kunz functions, and related questions in Classical Algebraic Geometry. The talks were confined to the morning session and the afternoons were kept free for research. Existing teams continued their collaboration and new teams were formed; conjectures announced during a lecture before lunch did not exist anymore at dinner time. Mathematics was on the move. Gil Kalai envisaged in his talk yet another round of progress through the interaction of Commutative Algebra and Combinatorics. We hope that this conference has made a contribution for this vision to become true. The success of a conference is determined by many factors, one of them is the atmosphere at the conference location. The Oberwolfach staff created the perfect atmosphere and we are very grateful for their hospitality. \begin{thebibliography}{aaaaa} \bibitem{BrHe93} Bruns, Winfried; Herzog, J\"urgen: Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics. Vol. 39. Cambridge: Cambridge University Press. (1993). \bibitem{Eis95} Eisenbud, David: Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics. Vol. 150. Berlin: Springer. (1995). \bibitem{Sta76} Stanley, Richard P.: Magic labelings of graphs, symmetric magic squares, systems of parameters, and Cohen-Macaulay rings. Duke Math. J. 43, 511-531 (1976). \bibitem{Sta96} Stanley, Richard P.: Combinatorics and commutative algebra. 2nd ed. Progress in Mathematics (Boston, Mass.). Vol. 41. Basel: Birkh\"auser. (1996). \end{thebibliography}