Tensor Products of Complexes

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.

Back in Salem

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!


WMC REU – Day 8

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.

The first mini symposium

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. Then I’d go home and do my homework and class prep. That’s the kind of stamina I wish I still had.  After a mere 90 minutes of listening intently to math, my brain is still swimming at the end of the day, and I have too many unconnected ideas, questions, and intuitions to do anything useful.

Bring on the printing!

Hooray, hooray! I have printing privileges! This development will seriously help with the paper writing. I envy all of the digital natives out there who don’t have to print a document to proofread it. Alas. That’s the way I learned, and my efforts to edit on-screen yield pretty poor results. Upshot: this is a game-changer.

My REU students and I talked about one of my favorite mathematical topics (today, in the context of rings and modules): how do you tell how big something is? They didn’t realize that they were asking this question — they were, more innocently, asking what a Cohen-Macaulay ring is. We toured the ideas Krull dimension and depth. We discussed in what ways each was a measure of the size of the ring (or module). We looked at the Emmy ring, CC[x,y]/(xy,x^2), and showed that depth need not equal Krull dimension.

It was a good time.

On Tuesday next week, we’ll have our first REU mini-symposium. Each research group has 20-30 minutes to present. We spent some time today outlining possible presentations. As those of you who know me realize, I’m a very competitive person.* So I suggested that we have a little low-stakes competition. Hold up, NSF, I didn’t suggest financial gains or anything like that! Just a round of beverages for the winning group paid for by their mentor; the mentor of the winning group earns a beer on the other mentors). I am confident that my group is capable of winning this competition, but I’m trying to keep the competitive edge out of our meetings. (Deep breath, Gibbons!)

At noon, we headed off to a picnic for all of the scientific research groups on campus. I liked meeting some of the other Willamette faculty. My co-mentors Erin McNicholas (voting theory) and Colin Starr (unipancyclic matroids) are great, and this seems to be true of every new faculty member I meet. In the afternoon, around 3 pm, the REU broke for root beer floats and games. Finally — a chance to crush someone at SET! (Remember what I said about being competitive?)

Now, as I finally wrap up work for the first week of the REU, I’m hoping for a fun weekend in Salem.

*This is why, if I like you, I won’t challenge you to a game of Scrabble. I want to stay friends after.

WMC REU Days 3-4

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!

WMC REU, day 2

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!).