My group decided to meet daily at 10 am for the first week, which gives me plenty of time each morning to do some research work (that’s when my brain is at its freshest). I started at 7:30 at The Gov Cup where I made some progress on merging several drafts of The Paper. Some ideas for streamlining some of the mathematics also occurred to me. Why use cases if you can help it, right? When I reached the point where I would need a new a hard copy of The Paper, I turned to outlining a different paper. (This one is work from the past year with my senior research fellow.) It was hard to shift gears that quickly between projects, though, and I was disappointed with my work on the outline.
During the meeting with my REU students, we talked about how the research reading went. This discussion led us to create a list of terms that we should define. We started with one they all knew as a baseline (“ring”) and ramped up to a special kind of ring (“standard graded k-algebra”, where k is any field). The classic example of a standard graded k-algebra is a polynomial ring over k where each variable has degree 1. We also defined “module,” “graded module,” “Hilbert function,” and we did some examples. The research project possibilities all include working with a quotient of a polynomial ring by a homogeneous regular sequence (also known as a “complete intersection”). Thus, I focused my examples on quotients of the polynomial ring by homogeneous ideals. We started to talk a bit about free resolutions and Betti tables, too. Based on our morning session, I assigned some homework.
By afternoon, my triple latte had worn off, so I spent some time on the general bureaucratic professor stuff I alluded to yesterday. I’m about to start my third year at Hamilton College, and that means I have to start putting together my reappointment materials. Instead of creating any of these materials, I made a list. Then I made a more precise list. Then I added due dates. Then I decided that it’s summer, so that was enough for one day. Back to the fun stuff!
To round out the work day, I read a bit ahead of where my students were in their research papers with an eye toward what we might talk about next time (free resolutions, Betti diagrams), and I prepared a few examples to clarify some definitions (I hope!).