We’re pretty close to Theorem 1.
When I started designing my REU project, I worried that I was either too narrow or too ambitious. I thought the students might finish in week 1…or not get anywhere at all. After all, my experience mentoring student research has been with the amazing and talented Robert Huben. On some days, I felt like I should be happy with less from Robert, and on other days, I thought I should push for more.
Mike Orrison, of Harvey Mudd, visited the REU on Wednesday and talked about the generalized Condorcet criterion and the underlying linear algebra used to prove a surprising result.
The Condorcet criterion is said to hold for a voting system if it guarantees that, if a candidate would win each head-to-head election, then that candidate would win the election.
In our usual voting system, we only ask voters to vote for their first choices.
The students in my research group are superlative! They’re funny, intelligent, hard-working; basically, they kick serious ass. It’s easy to get frustrated during the research process. The project I designed is no different, and we’re facing our first hurdle.
When you read someone else’s proof, often you’ll think, “Well, yeah. Duh. That’s easy enough.” But here’s the thing: every problem is hard (until it’s easy). At least half of the work is the process of framing the problem the right way so that you can use the right tools.
When I was but a wee undergraduate at the Colorado College, one of my professors encouraged me to work on a research project with him one summer. Over the next couple years, we wrote two papers together. That was Josh Laison, and he is now one of my friends. I don’t think I realized until partway through grad school that Josh is only six years older than me. He was a young visiting professor, but I was also a couple years older than my classmates.
The title pretty much sums it up for the day. We’re working toward a strong foundational understanding of the Koszul complex and how to build it from copies of 0 -> R -> R -> 0. In each such complex, the interesting map, R->R, is multiplication by a (homogeneous) form.
The tensor product differential is a bit beastly. The modules in the complex are just new free modules (because the tensor product of free modules is another free module).
The remote video experiment was a general success, I think. I was gone just long enough for my group to work on the project just to the point of feeling frustrated and hamstrung (which is, sadly, the primary feeling of doing math research). I was expecting them to feel that a little sooner in the program. I tried to set the project up so they’d have plenty of opportunities to feel confused, overwhelmed, and unsure of what to do next.
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).
Color me impressed. Our REU students are working on some interesting stuff. From uniquely pancyclic matroids (a matroid generalizes a matrix) to algebraic voting theory (measuring fairness through invariance under group actions) to decompositions of Betti tables (understanding the numerics of free resolutions), we’ve got an excellent crop of projects. On my end, it was challenging to score aspects of the presentations while also paying close attention!
When I was a graduate student, I would go to 3-4 hours of classes a day, teach a couple classes, and even go to an hour or two of seminars.
