Yikes. These past two days have not been particularly productive, research-wise — I think I made negative progress. Indeed, on Wednesday, I met all sorts of challenges that made everything harder than it needed to be. For example, I realized that I forgot to pack my cable for my external hard drive, rendering my collection of digital math textbooks inaccessible for now. This oversight is less of an issue for my research and more of an issue for assigning well-crafted problems to help my REU students learn the necessary background material. I did create some problems, and I worked them through, and they’ll do for now. But, still, arrgh!
To change up the format, which has so far ended up with my lecturing, I got to our classroom early on Wednesday and wrote three warm-up problems on the board. They ended with the note, “Come find me in half an hour or when you’re done.” I was purposefully vague there; I wanted to see if they would give up at 10:30 or keep working. At 10:50, they came and found me, and they’d worked through the problems together. They each presented one of the problems, and it was clear that they had been working together. After today, I have no doubt that they will form a good team.
Alas, I did end up lecturing a bit, but at least it was less. We (okay, I) defined “complete intersection,” “graded module homomorphism,” “free resolution,” “Betti diagram,” “pure diagram,” and “degree sequence.” Examples followed, and I asked them to work through, by hand, examples Betti diagrams of complete intersections and to use the Boij-Soederberg decomposition algorithm by hand for next time.
On Thursday, we spent a lot of time working with the algorithm and talking about what a rational cone is. I managed to lecture less. I finally showed them how to access Macaulay2 online and load the Boij-Soederberg package, which does all kinds of Betti table calculations.
I was quite fond of the following pair of problems for these last two days:
Assigned Wednesday: Find an example of a complete intersection with Betti table__________ 1 1 - - - B = - 3 6 3 - - - - 1 1
or explain why such a complete intersection doesn’t exist.
Assigned Thursday: Use the Boij-Soederberg decomposition algorithm to justify the claim that there is no Cohen-Macaulay R-module with the Betti table B from yesterday’s assignment.
Here’s hoping tomorrow is a bit more productive for me!