http://www-math.mit.edu/~rstan/pubs/pubfiles/30.pdf

Hilbert functions of graded algebras

Let Δ be a triangulation of a (d − 1)-dimensional sphere with n vertices. The Upper Bound Conjecture states that the number of i-dimensional faces of Δ is less than or equal to a certain explicit number ci(n, d). A proof is given of a more general result. The proof uses the result, proved by G. Reisner, that a certain commutative ring associated with Δ is a Cohen-Macaulay ring.

The Upper Bound Conjecture and Cohen‐Macaulay Rings

/pdf/balanced-cohen-macaulay-complexes-4bgigtziph.pdf

Balanced Cohen-Macaulay complexes

Maximum betti numbers of homogeneous ideals with a given hilbert function

Toric topology emerged in the end of the 1990s on the borders of equivariant topology, algebraic and symplectic geometry, combinatorics and commutative algebra It has quickly grown up into a very active area with many interdisciplinary links and applications, and continues to attract experts from different fields The key players in toric topology are moment-angle manifolds, a family of manifolds with torus actions defined in combinatorial terms Their construction links to combinatorial geometry and algebraic geometry of toric varieties via the related notion of a quasitoric manifold Discovery of remarkable geometric structures on moment-angle manifolds led to seminal connections with the classical and modern areas of symplectic, Lagrangian and non-Kaehler complex geometry A related categorical construction of moment-angle complexes and their generalisations, polyhedral products, provides a universal framework for many fundamental constructions of homotopical topology The study of polyhedral products is now evolving into a separate area of homotopy theory, with strong links to other areas of toric topology A new perspective on torus action has also contributed to the development of classical areas of algebraic topology, such as complex cobordism The book contains lots of open problems and is addressed to experts interested in new ideas linking all the subjects involved, as well as to graduate students and young researchers ready to enter into a beautiful new area

G.F. Clements

Papers

A Generalization of a Combinatorial Theorem of Macaulay

The minimal number of basic elements in a multiset antichain

A generalization of the Kruskal-Katona theorem

Another generalization of the Kruskal-Katona theorem

On a min-max problem of leo moser