We demonstrate how primary decomposition of commutative monoid congruences
fails to capture the essence of primary decomposition in commutative rings by
exhibiting a more sensitive theory of mesoprimary decomposition of congruences,
complete with witnesses, associated prime objects, and an analogue of
irreducible decomposition called coprincipal decomposition. We lift the
combinatorial theory of mesoprimary decomposition to binomial ideals in monoid
algebras.

This survey of methods surrounding lattice point methods for binomial ideals
begins with a leisurely treatment of the geometric combinatorics of binomial
primary decomposition. It then proceeds to three independent applications whose
motivations come from outside of commutative algebra: hypergeometric systems,
combinatorial game theory, and chemical dynamics. The exposition is aimed at
students and researchers in algebra; it includes many examples, open problems,
and elementary introductions to the motivations and background from outside of
algebra.

The face ring of a simplicial complex modulo m generic linear forms is shown
to have finite local cohomology if and only if the link of every face of
dimension m or more is `nonsingular', i.e., has the homology of a wedge of
spheres of the expected dimension. This is derived from an enumerative result
for local cohomology of face rings modulo generic linear forms, as compared
with local cohomology of the face ring itself. The enumerative result is
generalized in slightly weaker form to squarefree modules.