Comparing Powers of Edge Ideals

Given a nontrivial homogeneous ideal $I\subseteq k[x_1,x_2,\ldots,x_d]$, a problem of great recent interest has been the comparison of the $r$th ordinary power of $I$ and the $m$th symbolic power $I^{(m)}$. This comparison has been undertaken directly via an exploration of which exponents $m$ and $r$ guarantee the subset containment $I^{(m)}\subseteq I^r$ and asymptotically via a computation of the resurgence $\rho(I)$, a number for which any $m/r > \rho(I)$ guarantees $I^{(m)}\subseteq I^r$. Recently, a third quantity, the symbolic defect, was introduced; as $I^t\subseteq I^{(t)}$, the symbolic defect is the minimal number of generators required to add to $I^t$ in order to get $I^{(t)}$. We consider these various means of comparison when $I$ is the edge ideal of certain graphs by describing an ideal $J$ for which $I^{(t)} = I^t + J$. When $I$ is the edge ideal of an odd cycle, our description of the structure of $I^{(t)}$ yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.Comment: Version 2: Revised based on referee suggestions. Lemma 5.12 was added to clarify the proof of Theorem 5.13. To appear in the Journal of Algebra and its Applications. Version 1: 20 pages. This project was supported by Dordt College's undergraduate research program in summer 201

For the Love of Mathematical Research: A Conversation with Undergraduate Research Students

One of my passions as a professor is creating opportunities for students to ask questions about mathematics. Posting about students\u27 perspectives on mathematics research from In All Things - an online journal for critical reflection on faith, culture, art, and every ordinary-yet-graced square inch of God’s creation. https://inallthings.org/for-the-love-of-mathematical-research-a-conversation-with-undergraduate-research-students

On the Fattening of Lines in P3

We follow the lead of Bocci and Chiantini and show how differences in the invariant alpha can be used to classify certain classes of subschemes of P^3. Specifically, we will seek to classify arithmetically Cohen-Macaulay codimension 2 subschemes of P^3 in the manner Bocci and Chiantini classified points in P^2. The first section will seek to motivate our consideration of the invariant alpha by relating it to the Hilbert function and gamma, following the work of Bocci and Chiantini, and Dumnicki, et. al. The second section will contain our results classifying arithmetically Cohen-Macaulay codimension 2 subschemes of P^3. This work is adapted from the author\u27s Ph.D. dissertation

A Guide to Digital Decluttering: A Review of Digital Minimalism

Digital minimalism is not Luddism, which rejects the technological innovations of the day. Instead, it rejects the way in which most people engage these innovations. Posting about ­­­­­­­­the book Digital Minimalism from In All Things - an online journal for critical reflection on faith, culture, art, and every ordinary-yet-graced square inch of God’s creation. https://inallthings.org/a-guide-to-digital-decluttering-a-review-of-digital-minimalism

Mathematical Beauty

With all the brokenness in the world, is the study of the mathematical aspects of Creation worthwhile? Posting about ­­­­­­­­the beauty of mathematics from In All Things - an online hub committed to the claim that the life, death, and resurrection of Jesus Christ has implications for the entire world. http://inallthings.org/mathematical-beauty

Burnt-Out Lightbulbs: A Review of Can\u27t Even

Given the ways in which personal technology and social media amplify burnout, millennials, who came of age as the Internet invaded all aspects of life, are perhaps especially vulnerable to it. Posting about ­­­­­­­­the book Can\u27t Even from In All Things - an online journal for critical reflection on faith, culture, art, and every ordinary-yet-graced square inch of God’s creation. https://inallthings.org/burnt-out-lightbulbs-a-review-of-cant-even

\u27Digital\u27 Fruits of the Spirit?: A Review of Analog Christian

Social media platforms are built to encourage addictive use, stimulating the same neurological connections as slot machines and drug addicts to encourage us to return to the well for further despair. Posting about the book Analog Christian from In All Things - an online journal for critical reflection on faith, culture, art, and every ordinary-yet-graced square inch of God’s creation. https://inallthings.org/digital-fruits-of-the-spirit-a-review-of-analog-christian