Working on proving the theorem today.
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. I’m sure it was baffling at times, working with Dr. Gibbons and Ms. Hyde.
Let me return to my point. Based on research as an undergraduate and now my sample size of one research student, I wasn’t abundantly confident in my ability to design a challenging, satisfying, and fun project. I’m overstating my anxiety a little bit; if I thought I couldn’t pull it off, I wouldn’t have agreed to mentor. Nevertheless, I was anxious even into the first week. (Luckily, in grad school, I learned to manage and even transform anxiety into a weird calm competence.)
Well, folks, even if my project had some flaws, it’s turned out to be a good project, and my group is working like a smoothly oiled machine. Double induction? Ha! They eat double induction for breakfast. Three cases with several subcases? No big deal. I’ve taken a slightly more active role in the project now, trying to highlight some opportunities to make progress. I think we’ll have Theorem 1 nailed down by the middle of next week. There’s a second theorem I’d like them to prove, and I think that will be a bit more straightforward (knock on wood). From these two theorems, we get some nice structure results.
I think part of our progress comes from a change of scenery. We got locked out of our classroom, so we’ve been working in a common space with a couple of comfortable chairs.
Alas, my team didn’t win the first round of research symposium talks, but they have a chance to redeem themselves on Wednesday next week. Here’s the silver lining: I’d feel guilty if they won the research symposium talks every week. They’re already exceeding expectations!
As for team-building, we had fun at the weekly picnic, we laughed our way through a session on “Do’s and Don’ts” for presentations, and we enjoyed a rousing game of “Rabbit Rabbit Moose Moose” at Friday Floats this week. The fun will continue into the weekend: some of the students and some of the faculty are going to karaoke this weekend. Watch out, Bowie.
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. But you could imagine having voters list their preferences. For instance, if I had to rank my cats, I’d rank them M > T > S (Moo’s my favorite, then Tipper, and lastly, there’s Sophie).
Now, I could ask my friends to rank my cats, and maybe I’d come up with:
- M > T > S – ten ballots
- T > M > S – four ballots
- T > M > S – five ballots
- S > T > M – two ballots
We can take away a few things from these rankings:
- No one likes Sophie (poor Sophie!).
- Moo wins the election!
- I have a lot of friends (okay, okay: it’s a fictional example).
- Tipper beats Moo (11 to 10); Tipper beats Sophie (19 to 2); Moo also beats Sophie (19 to 2).
Item 4 means that Tipper is the Condorcet winner. However, if we use our usual “most first place votes” election, we’d see that Moo is the winner.*
If you have more than three candidates, the results get weirder. You can have someone who wins every head-to-head competition (call the 2-winner Bob), but someone else could win in every three-way battle (call the 3-winner Alice).
The surprising and cool result is that once Bob’s 2-winningness (my term) is overshadowed by Alice’s 3-winningness, Bob can’t go on to be a 4-winner. He can’t be any n-winner for n > 3. And the proof, my friends, rests in linear algebra. You can read all about the work Mike did with his REU students in their paper: Generalized Condorcet winners.
*In reality, Moo would win by any metric because she’s fantastic.
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. We have a problem that we’re trying to solve, but there are a lot of competing pieces of information.
To try to help out with the framing process, I took the various moving parts and used them one at a time in a sequence of reductions (numbered 0 through 4, ish). After the reductions, we have a lot of powerful assumptions that we can use. At the end of our working day on Tuesday, I felt confident that we had set ourselves up for a quick slam-dunk to win the game.
…Ha. When I got home, I started playing around and found that I couldn’t get very far. As of this Wednesday morning, it turns out, no one got very far. Back to the whiteboard for us.
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. I think my first impression of Josh involved math puns and 80s pop songs. Inevitably, I became friends with Josh and his wife, Steph.
Josh is now a tenured professor at Willamette University, and he is one of the co-PIs for the WMC REU this year. Now that I’m Salem for the summer, it’s fun to spend time with Josh, Steph, and their hilarious 5-year old. One of Josh’s mathematical interests lies in games and gaming.* One of my mathematical interests lies in describing anything I can get my hands on with polynomials. We’ll see if the intersection of those interests leads to a project.
I’m enjoying getting a chance to talk about math with Josh, but it’s also fun to banter about the craft of teaching. He’s got myriad great ideas for classes, and he’s got a lot of experience trying new things. I’ve picked up some great ideas, and I feel a renewed sense of courage to go back to Hamilton and try new stuff. Sometimes even great educators take a promising idea into the classroom only to watch it fail. When they’re up-front about it, it gives the rest of us the guts to try innovative techniques. Sometimes they flop. Life goes on. Josh has also taught me a lot about working with undergraduates, both as my undergraduate research mentor and now as my colleague. I’m trying to soak up as many lessons as I can.
When I’m not hanging out with the REU folks or Josh and family, I have plenty of time to myself. In Salem, it’s easy to find things to do on the weekend to be out among people (but not necessarily with people). That means that an introvert can have a nice time out and about without feeling awkward. Farmers markets, coffee shops, bookstores, movies, reading in a park: none of these activities requires a car or other people. It’s hard to find that in upstate New York.
A weekend feels wasted if I don’t get any work done. So I scheduled some math and other stuff to get done on Sunday morning. Happily, I accomplished the modest goals I set out for myself. Tonight I’ll work a little more, and then I’ll see about an impromptu Skype date with my partner Ryan and my cats.
*Josh’s daughter is already adept at strategy games, and this can be embarrassing for me. Luckily, she’s a (mostly) gracious winner.
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). However, the differential is a Frankensteinian mix of differentials from the two complexes you tensor together. Let’s extend the gruesome metaphor for a minute: if you want to stitch together a bunch of human parts to get another human, you have to be careful about how you stitch them together. So too with the differentials; here, there are a bunch of (-1)^e, where e depends on… stuff. (I have to remind myself that I’m writing a blog, not a textbook!)
TL;DR: These signs that occur so that the resulting module homomorphisms square to zero.
The morning started with a crash course on tensor products of modules. I assigned a few standard homework exercises that focus on using the universal property to exhibit the basic isomorphisms that we need for the tensor product of complexes. Then we started tensoring complexes together like mad. We’ll pick up next week with an equivalent characterization of the differential of the Koszul complex. This alternative description is much more straightforward. For one, it’s easier to show that if you start with a regular sequence over a polynomial ring, the Koszul complex is acyclic. That is, it’s a minimal free resolution of the quotient of R by the ideal generated by the regular sequence.
We’re making some progress on the research project, too.
I’m optimistic for a fun weekend in Salem and looking forward to next week’s math.
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. That’s not because I’m naturally sadistic or uncaring; that’s because I want them to have a real research experience, and these feelings are part of the process. To become a researcher, you have to learn to live with doubt and uncertainty — and then to get to work. I also want to give them a taste of criticism and rejection, because that’s part of the process, too. But ultimately, I want them to learn a lot of math and discover something new. They’re on track to meet both of those objectives.
However. Having said all of that, I want to be supportive and helpful, too. I’ll prepare a few lectures on related algebra topics (tensors products of modules and complexes) and give them a little homework to do. Again, this mimics research: when you realize you have to learn something new to start working on a problem, it’s a relief. Working through the basics of a theory is like classwork, and by the time you’re doing research, you’re good at classwork.
I’ve also tried to rephrase the research problem a few different ways for the students, based on what they’ve described to me. The point here is to try to help them interpret the problem in a way that will lead to a specific course of action. I’m being purposefully vague about what they’re doing; as I keep telling my math friends, you’ll just have to wait for the paper for the full picture. Consider this blog one long teaser.
As for my work… Before I left last week on my personal trip, I finished editing my parts of a paper and sent it off to my coauthors for their next turns. Phew. One item off the to-do list. I also made some progress on writing a project description for a grant proposal. I struggle with writing grants; finding the appropriate level of assumed knowledge on the part of the reviewers trips me up.
Time to get my homework done for tomorrow!