Taras Panov (Manchester) : Combinatorial aspects of torus actions ABSTRACT We describe new relationships between torus actions on manifolds orsome more general spaces and combinatorial objects such as polytopes, simplicial complexes, cubical complexes, and subspace arrangements. The case of our particular concern is simplicial and cubical subdivisions of manifolds and, especially, spheres. New constructions which allow to study such combinatorial objects by means of commutative and homological algebra are described. This approach unifies commutative algebra methods applied in the combinatorics by Stanley and topological approach to torus actions by Davis and Januszkiewicz and gives rise to the theory of moment-angle complexes, currently being developed by V.M.Buchstaber and the author [1]. The theory centres around the construction that assigns to each simplicial complex $K$ with $m$ vertices a $T^m$-space $\cal{Z}_k$ with a special bigraded cellular decomposition. In the framework of this theory, the well-known non-singular toric varieties arise as orbit spaces of maximal free actions of subtori on moment-angle complexes corresponding to simplicial spheres. Different combinatorial invariants of simplicial complexes and related combinatorial-geometrical objects acquire a nice and surprisingly regular interpretation in terms of the bigraded cohomology rings of the corresponding moment-angle complexes. [1] V.M.Buchstaber and T.E.Panov. Torus actions, combinatorial topology and homological algebra, Russian Math. Surveys 55 (2000), no. 5. Available at http://arXiv.org/abs/math.AT/0010073.