Working with Betti diagrams can be challenging. To get the most bang for your buck, you should embed them in a rational vector space. But then you want to cut down the dimension of the ambient space by finding equations that Betti diagrams satisfy (but that random tables of rational numbers need not satisfy). My group is grappling with this by trying to understand how Boij and Soederberg did this in their paper (using the Herzog-Kuhl equations). Once they get the general gist of the technique, their task is to come up with similar equations for their particular project.
For my part, I’ve got a few pages of reductions and notes to this end, but I keep reminding myself that it’s not my research project (or at least, not entirely mine) and that the best pedagogical stance I can take is one of “benevolent neglect.” That is, spend a little time with the group each day, and then step back and let them work and learn and discover. Repeat the next day. I’m sure that as the weeks wear on, I’ll vary my level of involvement. But for now, they’re doing great work on their own!
I’ll be traveling for a few days to take care of some personal business, so I’ll have plenty of opportunities to practice benevolent neglect. I’ll Skype with my group at least once a day while I’m gone, but they’ll be on their own to formalize their problem and start investigating it.
Tonight, my group will bond by heading up to Portland for dinner (my treat, although they don’t know that part yet). Then they’ll drop me off at the airport. It’s hard to know from reading applications whether a group of three students will work well together, but my group exceeds expectations. They are a very strong working unit. They’re also really great human beings.