A Minimal Poset Resolution of Stable Ideals

We use the theory of poset resolutions to construct the minimal free resolution of an arbitrary stable monomial ideal in the polynomial ring whose coefficients are from a field. This resolution is recovered by utilizing a poset of Eliahou-Kervaire admissible symbols associated to a stable ideal. The structure of the poset under consideration is quite rich and in related analysis, we exhibit a regular CW complex which supports a minimal cellular resolution of a stable monomial ideal.Comment: 25 pages, 2 figure

Distribution of Crown of Thorns (Acanthaster planci) Outbreaks along the North Shore of TutuilaDistribution of Crown of Thorns (Acanthaster planci) Outbreaks along the North Shore of Tutuila

Outbreak levels of Crown of Thorns starfish (Acanthaster planci, COTs) have been occurring around Tutuila since 2011. These outbreaks are thought to be caused by increased nutrients due to terrestrial runoff. Outbreaks were confined to the south side of Tutuila in 2011 and 2012, but spread to the north side in 2013. A possible reason for the observed outbreaks on the south side could be that most of Tutuila’s population live on the south side of the island. Thus, we hypothesize that due to the small population of people on the north side of Tutuila, there should be limited COTs outbreaks on the north shore. Our study was conducted by using the tow boarding method along the north side of Tutuila Island. Tow boarding is conducted by towing two snorkelers behind a boat over a reef slope. While being towed, the reef is scanned and GPS coordinates are taken for any outbreaks that are seen. After taking the coordinates, the average depth, time, and intensity of each outbreak are noted. Each site is recorded as light, moderate, or heavy. During our tow board surveys, we came along a stretch of moderate and heavy outbreaks along Fagasa Bay to Fatuelo point, as well as between Fagamalo and Aloau Bay. Contrary to our hypothesis, the data collected showed that there were large outbreaks on the north shore, and that the outbreaks were very heavy. With this data, divers have eradicated 1115 COTs to date in the area that was surveyed

The Betti poset in monomial resolutions

Let $P$ be a finite partially ordered set with unique minimal element $\hat{0}$. We study the Betti poset of $P$, created by deleting elements $q\in P$ for which the open interval $(\hat{0}, q)$ is acyclic. Using basic simplicial topology, we demonstrate an isomorphism in homology between open intervals of the form $(\hat{0},p)\subset P$ and corresponding open intervals in the Betti poset. Our motivating application is that the Betti poset of a monomial ideal's lcm-lattice encodes both its $\mathbb{Z}^{d}$-graded Betti numbers and the structure of its minimal free resolution. In the case of rigid monomial ideals, we use the data of the Betti poset to explicitly construct the minimal free resolution. Subsequently, we introduce the notion of rigid deformation, a generalization of Bayer, Peeva, and Sturmfels' generic deformation