Betti tables forcing failure of the Weak Lefschetz Property

We study the Artinian reduction $A$ of a configuration of points $X \subset {\mathbb P}^n$, and the relation of the geometry of $X$ to Lefschetz properties of $A$. Migliore initiated the study of this connection, with a particular focus on the Hilbert function of $A$, and further results appear in work of Migliore--Mir\'o-Roig--Nagel. Our specific focus is on Betti tables rather than Hilbert functions, and we prove that a certain type of Betti table forces the failure of the Weak Lefschetz Property (WLP). The corresponding Artinian algebras are typically not level, and the failure of WLP in these cases is not detected in terms of the Hilbert function

The MatrixSchubert package for Macaulay2

We introduce the MatrixSchubert package for the computer algebra system Macaulay2. This package has tools to construct and study matrix Schubert varieties and alternating sign matrix (ASM) varieties. The package also introduces tools for quickly computing homological invariants of such varieties, finding the components of an ASM variety, and checking if a union of matrix Schubert varieties is an ASM variety.Comment: Accompanying the package MatrixSchubert, to be included in version 1.22 of the software system Macaulay

Betti numbers for connected sums of graded Gorenstein artinian algebras

The connected sum construction, which takes as input Gorenstein rings and produces new Gorenstein rings, can be considered as an algebraic analogue for the topological construction having the same name. We determine the graded Betti numbers for connected sums of graded Artinian Gorenstein algebras. Along the way, we find the graded Betti numbers for fiber products of graded rings; an analogous result was obtained in the local case by Geller. We relate the connected sum construction to the doubling construction, which also produces Gorenstein rings. Specifically, we show that a connected sum of doublings is the doubling of a fiber product ring

A flow cytometry-based screen for synthetic riboswitches

Riboswitches regulate gene expression through direct, small molecule–mRNA interactions. The creation of new synthetic riboswitches from in vitro selected aptamers benefits from rapid, high-throughput methods for identifying switches capable of triggering dramatic changes in gene expression in the presence of a desired ligand. Here we present a flow cytometry-based screen for identifying synthetic riboswitches that induce robust increases in gene expression in the presence of theophylline. The performance characteristics of our newly identified riboswitches exceed those of previously described natural and synthetic riboswitches. Sequencing data and structure probing experiments reveal the ribosome binding site to be an important determinant of how well a switch performs and may provide insights into the design of new synthetic riboswitches