No one really wanted to get up for analysis on Monday morning. I woke up with my alarm at 7, decided it wasn't happeing, added half an hour to my alarm clock, and fell back asleep. For the first time all program, Dana did not wake up before her alarm clock. When we knocked at Maram's door a little after 8 to see if she wanted to come to breakfast with us, she told us she'd just meet us at class. It was pretty clear from her expression that she'd only woken up a few minutes before.
We had time to buy breakfast, but not really the time to eat it. So we headed to the classroom and blatantly ignored the “no food or drink” signs the janitors had put up. (Fun fact: they removed the garbage canss from the room, then had to put up those signs because suddenly, people started leaving trash on the ground and on the tables.)
At least it was analysis, and not the research seminar. I'm not sure I would have been able to get up for an 8:30 lecture where the teacher was rambling and the subject was unclear. But a class where the definitions are precise, the theorems are existent, and the teacher is organized? I can deal with that. Even if it does neeed to be at 8:30.
Thse notes were probably the least comprehensive set he'd given us. He apologized for that, but explained that when he'd been writing them a few years ago, he'd thought that the way he'd written them was logical. He'd since changed his mind but not rewritten his notes. Most analysis class days, we could follow along in the packet pretty closely, and only need to write down one theorem that he proved in class but not in the notes. Today, we had to skip two pages and write down about four additional proofs.
The good thing was that the material was new, but still accessible. The better news was that it made other things make slighly more sense.Only slightly more- it wasn't one of those moments where all the pieces suddenly fit together and life made sense. (That would be nice.) But I now understood half of a converstaion people had gotten into earlier about the definition of compactness in topology vs. in analysis. And there are som ereally cool theorems on compactness.
Unfortunately, writing them down proved slightly difficult in between cleaning up the coffee that I'd spilled. All over both my notebooks and my the page of professor-given notes. The page in my notebook that took the majority of the damage was the exact same page I'd spilled coffee on the week before. So apparently that's a new thing for me to do Monday mornings. Glad I didn't have Real Analysis at 8:30 Monday mornings until this summer. I don't think my notebooks could have survived that. (Also legitimately glad that I'm using pens that don't run when I spill a liquid on a page that I wrote on. The parts of the notebook that had coffee spilled on them are still about as legible as they normally are.)
After analysis came the research seminar. We didn't pick up where we left off on Friday. Nor did we take advantage of the work we'd done over the weekend to take a couple more steps forward. Instead, we dropped that subject and started on something new and fairly different.And, at least in my mind, slightly more sensical.
Favorite thing I learned today: “Abelianize” is a word. It means to make an infinite group finite, and therefore change it from a non-commutative group to an Abelian one. (After a week, it still disappoints me that most of my summer classmates wouldn't understand what I'm talking to. I have yet to come up with a good explanation that doesn't make my Abstract Two class sound crazy.) I'm stil not entirely sure why we wanted to abelianize groups, but I'm glad that it's a thing we can do.
We learned how, for certain kinds of presentations, we could use the single relator to draw a polygon associated with the graph. This polygon (or polytope. The paper that the ideas came from calls it a polytope, but Professor Farley switched to calling it a polygon because that makes more sense) is an invariant, and could theoretically help us identify if two different-looking presentations represented different groups. Some of the details were a little sketchy, bu the process itself was straightfowrward enough.
At the end, he posed three questions for us to think about. The last was a question about whether a marked polygon could be the polytope of a hyperbolic group. To do this, we needed to know what a hyperbolic group was, which required another twenty minutes of explanation. And then all we had was the definition, and not a good way to use it. But it was time for class to be over, so...
After dropping my stuff back at the dorm, I returned to the library. This time, I settled myself in front of the topology books. (after detours to gaze longingly at books on graph theory, number theory, and algebra, and imagine how nice it would be if I did't feel like I needed to play catch-up on a course even as I was being drowned in new, higher-level material.) After skimming through the table of contents and prefaces of probably a dozen books, I finally found two that fit my criteria.Which is to say, they had a section on fundamental groups and did not assume I was a grad sctudent who had already taken a basic course in topology. Both books began with a flow chart that suggested various paths through the book, so by following their guidance I could presumably get to the chapters that interested me with the base knowledge I'd need and not too much superfluous knowledge. Even with those flow charts that let me skip several chapters, it will probably take me too long to get to the relevant material of the books. (I'm also aware that trying to read two books at the same time will probably slow me down. But I need to, at least until I can decide which one I like better.) I'm just hoping that the sections I can get through will make the material make a bit more sense.
I returned to the room when the library started warning me it would be closing soon, and Dana and I went to dinner. Then it was time to grab our stuff and head to the review session. The review session went into a bit more detail on Bieri, Neumann, Stredel Invariant which had been introduced and glossed over during class. It took us a while to work through the example, and we ended by deciding to ignore the question about hyperbolic groups, because those didn't make sense to anyone. But overall, things had made more sense than they had a week ago. I'm hoping he's building up to a new project, and one that requires a bit less background knowledge than the last one.
Maybe we'll find out tomorrow. (By which I mean Thursday, because what are the odds he's going to be able to explain another project in under 8 hours.) At any rate, I don't habitually spill coffee on Thursdays, so it's a step in the right direction.
