We introduce the notion of rigid embedding in a grid surface, a new kind of plane drawing for simple triconnected planar graphs. Rigid embeddings provide methods to (1) find well-structuredâ€¦ (More)

Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions encoded by simplicial complexes. There are numerous equivalent waysâ€¦ (More)

This note constructs the flat toric degeneration of the manifold Fln of flags in Cn from [GL96] as an explicit GIT quotient of the GrÃ¶bner degeneration in [KM03]. This implies that Schubert varietiesâ€¦ (More)

Alexander duality has, in the past, made its way into commutative algebra through Stanley-Reisner rings of simplicial complexes. This has the disadvantage that one is limited to squarefree monomialâ€¦ (More)

Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated N-graded module. The functors associated with Alexander duality provide a duality onâ€¦ (More)

New representations of tree-structured data objects, using ideas from topological data analysis, enable improved statistical analyses of a population of brain artery trees. A number ofâ€¦ (More)

For a radical monomial ideal I in a normal semigroup ring k[Q], there is a unique minimal irreducible resolution 0 â†’ k[Q]/I â†’ W 0 â†’ W 1 â†’ Â· Â· Â· by modules W i of the form âŠ• j k[Fij ], where the Fijâ€¦ (More)

Let Q be an affine semigroup generating Z, and fix a finitely generated Z -graded module M over the semigroup algebra k[Q] for a field k. We provide an algorithm to compute a minimal Z-gradedâ€¦ (More)