A generalization of Cohen-Macaulay simplicial complexes

The Upper Bound Conjecture (UBC) states that the number of i-dimensional faces of a simplicial complex is less than or equal to a certain explicit number. The UBC was proposed for simplicial polytopes by Motzkin in 1957 and proved by McMullen in 1970. In 1964 Victor klee suggested that the same statement should hold for all simplicial spheres and this was indeed established in 1975 by Richard Stanley using a characterization of Cohen-Macaulay simplicial complexes, that a certain commutative ring associated with simplicial complex is a Cohen-Macaulay ring, due to Reisner (1974). In this talk we will study a generalization of Cohen- Macaulay simplicial complexes which contains Buchsbaum simplicial complexes.