1. CJM 2009 (vol 61 pp. 888)
 Novik, Isabella; Swartz, Ed

Face Ring Multiplicity via CMConnectivity Sequences
The multiplicity conjecture of Herzog, Huneke, and Srinivasan
is verified for the face rings of the following classes of
simplicial complexes: matroid complexes, complexes of dimension
one and two,
and Gorenstein complexes of dimension at most four.
The lower bound part of this conjecture is also established for the
face rings of all doubly CohenMacaulay complexes whose 1skeleton's
connectivity does not exceed the codimension plus one as well as for
all $(d1)$dimensional $d$CohenMacaulay complexes.
The main ingredient of the proofs is a new interpretation
of the minimal shifts in the resolution of the face ring
$\field[\Delta]$ via the CohenMacaulay connectivity of the
skeletons of $\Delta$.
Categories:13F55, 52B05;, 13H15;, 13D02;, 05B35 

2. CJM 2007 (vol 59 pp. 1008)
 Kaczynski, Tomasz; Mrozek, Marian; Trahan, Anik

Ideas from Zariski Topology in the Study of Cubical Homology
Cubical sets and their homology have been
used in dynamical systems as well as in digital imaging. We take a
fresh look at this topic, following Zariski ideas from
algebraic geometry. The cubical topology is defined to be a
topology in $\R^d$ in which a set is closed if and only if it is
cubical. This concept is a convenient frame for describing a
variety of important features of cubical sets. Separation axioms
which, in general, are not satisfied here, characterize exactly
those pairs of points which we want to distinguish. The noetherian
property guarantees the correctness of the algorithms. Moreover, maps
between cubical sets which are continuous and closed with respect
to the cubical topology are precisely those for whom the homology
map can be defined and computed without grid subdivisions. A
combinatorial version of the VietorisBegle theorem is derived. This theorem
plays the central role in an algorithm computing homology
of maps which are continuous
with respect to the Euclidean topology.
Categories:5504, 52B05, 54C60, 68W05, 68W30, 68U10 

3. CJM 2005 (vol 57 pp. 844)
 Williams, Gordon

Petrie Schemes
Petrie polygons, especially as they arise in the study of regular
polytopes and Coxeter groups, have been studied by geometers and group
theorists since the early part of the twentieth century. An open
question is the determination of which polyhedra possess Petrie
polygons that are simple closed curves. The current work explores
combinatorial structures in abstract polytopes, called Petrie schemes,
that generalize the notion of a Petrie polygon. It is established
that all of the regular convex polytopes and honeycombs in Euclidean
spaces, as well as all of the Gr\"unbaumDress polyhedra, possess
Petrie schemes that are not selfintersecting and thus have Petrie
polygons that are simple closed curves. Partial results are obtained
for several other classes of less symmetric polytopes.
Keywords:Petrie polygon, polyhedron, polytope, abstract polytope, incidence complex, regular polytope, Coxeter group Categories:52B15, 52B05 
