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The Next Four Weeks

USA | Wednesday, 29 July 2015 | Views [315]

At the beginning of the blog, I gave the schedule and admitted that I had no idea what the last four weeks would look like, but could only assume it was all research all the time. Then I got through the first three weeks and stopped updating the blog. A reasonable person might assume that was because my guess was right and I didn't have time to update it anymore.

That's not quite true. I still had free time, including more than enough time to update the blog. It was just a different kind of free time. One that didn't incline me to be writing during it.

Mathematical research is very different than the traditional conception of scientific research. In particular, “research” for a mathematician means “sitting in a room thinking about math. Maybe writing things down on a board occasionally.” Chemists tend to look at us funny because of that.

One advantage of this is that mathematical research can be done anywhere. Whiteboard helpful, but not required. (The first day that Fernando, Angelo, and I met together to work on the project, we used whiteboard markers on the windows. It was still there the last day of the program.) The disadvantage of not having a formal lab is that there's no need for you to ever stop working. If you can do just as well with napkins at a lunch table as a classroom surrounded on three sides by books and a whiteboard on the remainng wall, then you have no excuse to ever stop thinking about math.

This doesn't mean that I was always working on math. It does mean that I always felt like I should be. And by the time it was late evening ( my usual time to try writing up posts) I did not want to think about math anymore. And if I was going to think more about my project, I might as well be getting stuff done on my project. So the academic side of the blog was ignored.

There was also time to quiet the voice in my head saying “think of everything you could be doing right now” and just have fun. However, the voice in my head was not willing to stay silent for the time it would have taken to write it up. Even if it had, writing up the last four weeks in a way that hides any mention of work would have been very misleading. So the blog in general was ignored.

And now the program is over, so I have the time and energy to go back and add a quick summary.

Project-wise, things got messy pretty quickly. I spent a lot of time playing around with my discrete gradient vector field to make sure it satisfied all of the requirements. However, because the one about no closed non-stationary paths was a pain, I needed to create way more critical cells than anyone wanted to deal with by hand. So after a week, I needed to convert the pseudocode I'd come up with to describe the functions into real code.

A decent chunk of my Fourth of July weekend was spent relearning Python. The rest of the time was spent hanging out with the other program participants. Pretty much everything on campus was closed Friday and Saturday, so it was a constant struggle to not starve. We hadSam-cooked “family dinner” both nights, and for Saturday lunch, Griselda, Dana, and I found the one open meal hall and ate there. (Sam's food was much better than the food at the cafeteria.)

The next week featured more of me trying to remember idiosyncrhosies of Python I'd probably never learned. About five years ago, I'd done a fair amount of playing around, but not enough to get into details of functions and lists. Such as this one:

new_list = old_list

change new_list

print old_list, new_list

You've changed new_list, but not old_list, so the two should be different, right? Wrong! Turns out new_list is just another variable pointing to the exact same thing old_list was pointing at, so when you change the value of new_list, you're changing old_list as well. That was a fun bug to figure out.

Later that same week, I found a problem with the DGVF and closed non-stationary paths. A major one that I couldn't fix without assigning infinitely many two-cells to be critical (or collapsible, which I later found out was not possible.) Computing homology with infinitely many critical cells is not the sort of thing that sounds like it would be manageable, let alone easy or fun.

In the evening, I was talking with Angelo and Fernando (not deliberately. I needed to stop by to write something on the white board, and they happened to drop by while I wsas there, so it became a spontaneous group meeting.) when I realized it wasn't just my DGVF. Any discrete gradiant vector field you tried to put on Thompson Group T would have the problem of infinitely many critical two-cells. And I could prove it.

The proof ruined everything we'd been working on. It had two saving graces. The first was that I'd come up with it. (If anyone else had told me “here's why everything you've done for the past two or three weeks was doomed to failure from the beginning” I would have been so annoyed.) The second was that it was a really nice combinatorial proof. You didn't need to know any homology or topology or even algebra to be able to understand it. It was simple and concise, and if it weren't for that, I probably would have hated it.

As I was copying down the proof, Griselda came in and asked if we wanted to go for a walk. Having just lost all hope for our current solution, we had nothing to do until we met with Farley the next day, so we all agreed. Dana and Maram came too.

What was supposed to be a 10, maybe 15 minute walk changed when Angelo made a comment of “if we wanted to go to Walmart, we'd turn right here,” and Maram said “let's do it.” Over an hour later, we walked into the Walmart. Maram had by this point realized what a dumb idea that had been, and was complaining about it enough that people had forgottten she'd ever suggested it.

Once inside, Maram and I were very clear about wanting to sit down and have something to drink. (Dana and Grisleda were also clear about that, but much less effective, as they ended up following Fernando and Angelo to the sproting goods section.) Maram and I found furniture and sat down in the bean bag chairs. After meeting up with Dana and Griselda, we decided to ignore the significant step that was probably meant to deter people from sitting on the furniture and go for it. No store employee complained, but we had our excuses worked out in case they did. (“I'm sorry, I was just trying out the couch. I'm looking to buy one for a friend to use, and I need to make sure it's the kind of couch you can fall asleep in. Yep, it definitely is. I'll be back to buy it soon.”)

Testing out Walmart's futons

It was the kind of bonding expereince that only comes from terrible ideas that everyone decides to go along with. The next day,Dana, Griselda, and Maram were blaming Angelo and Fernando for coming up with it, and Fernando (who was at the gym pretty much every day of the program) was complaining that his legs hurts. We bought a ball to play with, though we almost forgot about that until it was too late.

The next day, Farley got to experience the same best/worst moment of “so that's the answer. But... that's not the answer we wanted.” He was able to give us a couple of theorems about connectedness and quotients and subcomplexes that he thought might help. It wasn't until my brain had time to shut down and recover during lunch that I realized why they were helpful. Essentially, rather than looking at the infinite space, we were able to look at a finite space instead. And then, although we might still have a lot of critical cells, we didn't have infinitely many, so we could still work with it. More importantly, the computer was able to work with it, and was able to perform these computations much faster than we could.

This was proven the next day (which, by the way, was a Saturday. No other group ever had to meet with Dr. Farley on a weekend, but we did. Twice. I'm not bitter about that at all...) with a brute-force way of dealing with the functions to ensure there were no closed nonstationary paths. Meanwhile, I was working on trying to prove that the DGVF I'd defined previously would work in this context. By splitting it up into multiple different cases, I was able to prove that there were none in dimensions two and, with another few days of work on it, in dimension 3 either. But case by case was painful, and I couldn't imagine how to extend it.

Until suddenly I could. It didn't even involve nested induction, which for a while I thought it would. (There was a brief horrifying moment where I thought it would involve a nested proof inside a nested proof inside a nested proof.) But by Wednesday, I was feling more secure in the DGVF, and went back to the program, hoping that I could train it to not just show me critical cells, but to se these critical cells to compute the homology. This required mattrices and a lot more work with Python, but alll of the pieces seemed to be fitting together.

On Friday, a friend of mine came to visit. With his car, which, after 6 weeks of being in Oxford without a way to get out, was a very welcome change. Friday night Maram, Griselda, Dana, Ben and I went to IHOP (Dana had never been before) and the next night we went out for Laser Tag and Mini Golf. When we got back to campus, we met up with everyone else to walk to get cookies, then Delaney, Maram, Dana, Ben, and I played Coup until 3 in the morning. (Things deterioated quickly at about the point we decided the game wasn't interesting enough and tried to add a simultaneous game of Never Have I Ever. None of us were up to the challenge of keeping track of two different games at 3 in the morning.)

On Sunday we met with Farley. The meeting was long and tiring, but it ended on a really positive note, with Dr. Farley saying he was mostly convinced about our proof, and the program printing out exactly what it should have. (Me: “I've never been so happy to have a computer print out 7 before.”) And then I went back to the lounge and tried to compute the homology in dimension 3. The program didn't like that as much. The night did not improve again until I forgot about the program and started playing games.

The game of the night was Drawful. It came in a set with Fibbage and You Don't Know Jack, among other games, though it was by far the best.There's one computer with the game on it hooked up to the TV, and each player needs a phone or tablet. Every player gets a phrase that they draw, and then the drawings are randomly sorted and displayed on the television. Now, each player except the drawer comes up with a phrase to describe the drawing. In the next phase, all of the sentences, including the original, are displayed, and everyone picks which they think the answer is, as well as awarding likes. You get points for picking the right answer, as well as points for each person who guesses your answer.

The Abyss

 We played a number of games of Drawful, then a few games of Fibbage (worked similarly, only with random facts instead of drawings. So not similarly at all.) and one round of You Don't Know Jack (what I learned about myself: I might not know anything about pop culture, but I can still identify the main character in Swan Lake) and then back to Drawful. Until, like, 5 in the morning. It doesn't seem like the sort of game that youc an keep playing for so long you completely lose track of time. It is.

 I spent most of Monday trying to debug my code. Including a number of times that I'd think I found a problem, only to realize no, my computer actually knew my function better than I did. Then in the afternoon, Dr. Farley showed up with a counterexample to my proof. Which meant I now had two problems to solve: the proof and the code. The two turned out to be related.

 Although my code really was working perfectly with the function I gave it, the function I'd given it was wrong. So I copied and pasted everything to a new file and started making major changes. This was happening in the guy's lounge Monday evening. I was vaguely aware of people trying to get a game organized, and of them throwing a giant plastic ball at me. But more importantly, my code. It was working! It was giving me the right homology up through dimension 4.

 With that settled, I could respond to the people asking me if I wanted to play Monkey in the Middle. We did, though we had to go inside when it got too mosquito-y for some people. (I did not find it too mosquito-y. Compared to Illinois, mosquitoes were practically nonexitent.) I corrected the proof, and then I decided to call it a night and sat around until other people reached the same conclusion and we started another game of Drawful, this time with more people. “We're not going to be up until 5 a.m. though,” Griselda said.  Around 3:30 she finally went to bed. The rest of us followed immediately after.

 The rest of the week was a rush of trying to fix the DGVF and the code and the proof simultaneously, because problems in one kept revealing problems in another. This at the same time that the paper and presentation needed to be written and rehearsed. It was a fun remainder of the week that led to us getting up to give the presentation and hoping to gloss over the fact that things didn't quite work. Turns out there are a lot of intricacies that come up in higher dimensions that can lead to closed non-stationary paths.

 On Thursday, after the presentations were given and the papers were as done as they were going to be, we had the final dinner, and then went back to the dorm to hang out together for the last time. It was bittersweet. Sweet because we were done with the projects, bitter because we might never see each other again.

 It would have been more bittersweet if either of the things I just wrote were true. The projects are not done. Not even close, to be honest. With a bit more work, we might be able to actualy prove (instead of the fake-proving that's worked so well for us so far) that we have a DGVF, and that could be publishable. The other two groups (Sam and Delaney are in a group, Maram, Griselda, and Dana are the last group) are in a similar position. And we're pretty sure we'll see each other again. Most people will see each other in January for the JMM, though Maram and I aren't sure that we'll be able to make it (or that SUMSRI will actually want to pay for our transcontinental transport.) But even without that, we'll probably see each other again. Maybe not as a full group, or in person, but we'll see each other again.

 Two months ago, I said goodbye to my friends from Carthage not knowing if or when I'd see any of them again. And then I talked one of into coming to visit me. The next year is going to be crazy in so many ways, but that doesn't mean it will be more unexpected than some of the other years of my life have been. I'm very unclear on what comes next, but I know that it doesn't need to be disjoint from this summer, or anything that came before now.

 SUMSRI is over. The same is not true of anything that I started there. I look forward to seeing what will become of the math and friendships I've developed over the past 7 weeks.

Tags: dgvf, endings, games, homology, math, programming

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