Stanley-Reisner rings and torus actions

Abstract

We review a class of problems on the borders of topology of torus actions, commutative homological algebra and combinatorial geometry, which is currently being investigated by Victor Buchstaber and the author. The text builds on the lectures delivered on the transformation group courses in Osaka City University and Universitat Autonoma de Barcelona. We start with discussing several well-known results and problems on combinatorial geometry of polytopes and simplicial complexes, and then move gradually towards investigating the combinatorial structures associated with spaces acted on by the torus. Parallelly, we set up the required commutative algebra apparatus, including Cohen-Macaulay/Gorenstein rings and Stanley-Reisner face rings of simplicial complexes.