# WMC-REU-2015

## 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!).

## Greetings from the Willamette Mathematics Consortium

Today marks the first day of the WMC REU. I never went to a formal REU as an undergraduate, so I’m a first-timer just like many of the students. My first task today finding a suitable coffee shop for my summer caffeinated home base. (I settled on The Governor’s Cup, which sports an exposed brick wall, generously-sized table tops, and coffee that’s roasted in-house).

Although I’d corresponded with my three research students via email, it was a treat to meet them in person. All nine of the students seem like good people, in fact, and so do the other research mentors. It’s rare to find a group where there isn’t a single unlikeable person (which leads me to wonder if, in reality, I’m the unlikable one? Nah…).

Besides having found a coffee shop that will inevitably see many hours of work on my part, I have an office in the mathematics “hearth” at Willamette. I haven’t had much experience visiting places for long enough to have an office; I find myself distracted by trying to calculate the number of math books I have in common with the usual owner. In my professional opinion, it’s a big number. We also have several climbing guides in common. What is it about math and climbers? (One of the REU students is a climber, too.)

In my first meeting with my group, we talked about the basics — scheduling, their goals for the summer, my goals for the summer, project possibilities, and, of course, homework. Since the homework required starting to read research papers, we talked about how to do that.

They had plenty to do for the day, so we broke and I spent the afternoon working on revising a paper that I’m writing with my graduate advisors. If I can get the paper out of my hands by next week, I’ll be a jubilant mathematician. After a productive session, I outlined some of my other summer goals. I’ve got other research work to do, and some general administrative professor “stuff” that I have been, well, lazy about getting done. While I feel well-prepared for my role as an REU mentor, I would like to wrangle some resources together in case they’re helpful.

It’s been a fun day!