Thursday was the algebra short course. As with analysis, we were moving on to more advanced material that we might encounter in more detail in a graduate course.
It began with a description of Zorn's Lemma, and an application to prove that every proper ideal is contained in a maximal ideal. Dr. Azhar said that it was one of those useful things that wouldn't come up in an undegradaute, which is a lie. Two weeks into my Abstract 2 class, my partner and I were presenting a problem where we needed that fact as a part of the proof. Using Zorn's lemma would have been a pain and make the presentation take even longer, so we tried to gloss over it quickly and hope the professor wouldn't notice. (He did.)
And then we started learning about modules. Which was very exciting.
A few months ago, my other abstract algebra professor was warning us that when we reached grad school, there would be so many things that people would say “surely you know about *,” and then we'd think “I've never heard of that before,” and run off to look it up. And after that hpapens 5 or 6 times, you start wondering if your undergrad professors taught you anything.
When she'd attended Carthage, her professors had at least had the excuse of not being algebraists. So they hadn't entirely known what was really important to know for grad school algebra courses and future life in algebra. By contrast, Dr. Jensen, having just finished her PhD had a very good idea of what sorts of things would make our lives easier if we learned them as undergrads. Modules was one of those subjects. She didn't get around to talking about them because she thought that Galois was cooler. She acknowledged that it was an importannt field and gave us (mplied) permission to write her angry e-mails from grad school, but she didn't find modules terribly interesting, so she wasn't going to do anything with them.
The short course doesn't give us the time to give more than a general picture of what modules are, how they behave, and why they're cool. Hopefully that general picture will be enough to stop me from writing an angry e-mail in a couple of years.
The research seminar began with the a discussion of our work from last night, and the increasing base of examples where what Dr. Farley was hoping we could prove did not hold true. So, following the scientific method, it's time to re-evaluate and form new hypotheses. Or just quietly abandon that project and move on...
Diagram groups and binary trees. Seems promising. It contains several of my favorite words in mathematics.
Diagram groups begin with a semigroup presentation. We'd spent a lot of time talking about group presentations.(A long time. Two or three days in class, plus an evening session or two.) The discussion on group presentations was thorough, formal, and involved. The discsusion of semigroup presentations boiled down to “we can basically extend this to semigroups. You all have experience with semigroups, right? No? Eh, it's not that much of a problem.”
Even without knowing how on earth anyone woul deven think to investigate this, diagram groups rae really cool. You take a word and write it along the top of a line. Then, you use the relators to make loops showing the ways you can change the word into an equivalent one step by step. (It's one of those things that's kind of hard to explain, but really easy to demonstrate.) All of the diagrams for a given presentation that end with the exact same (not merely equivalent) word on the top and bottom form a group nder the operation of combining diagrams. It reminded me a lot of grade school art projects. (Technically only one art project, but I liked it a lot, so I can still remember it. Which is more than I can say about most of the things that happened during those years.)
This is a group, though we kind of need to take it on faith. Dr. Farley sketched out some of the ideas, like how you form an inverse of an element, and and why it showed be associative. But mostly we were believing him when he said that it was a group. We also trusted him blindly when he the diagram group of the semigroup presentation with one element and the relation x^2=x was isomorphic to Thompson Group F.
Thompson Group F can be thought of as a group of self-homeomorphisms of the Cantor set, if that's a way you like thinking about things. (Although if that's true, much of your life is probably disapointing because there's very litle that can be thought of as a self-homeomorphism of the Cantor set.) It can also be thought of as a group where the elements are pairs of finite binary trees where every tree has the same number of leaves. (Guess which way I like thinking about this group?) He showed us how to add elements, but we were also taking it mostly on faith that this was also a group. For all we knew Dr. Farley could be lying to us and neither collection is actually a group, let alone anything like the other.
Towards the end of class, we went back to what we'd been discussing the day befor, with CAT(0) space and K(G,1) complexes. I feel like it's starting to make a little more sense.
After class, I dropped my stuff off in my room (Thursdays are so much shorter when we don't have a colloquium) and set off for the King Library.
One of the comments I'd made and everyone had seconded during the discussion about what we'd improve about SUMSRI was access to the library. Sam had mentioned that he'd talked a woman into checking a book out of the library for him. (Dr. Phillips: “That's the math student version of getting someone to buy them beer.”) Being able to check out books would make things so much better. (And by “better” I mean “more comforting.” Probably other people had reasons other than just liking being surrounded by books.)
I was thinking about that, and it occurred to me that the point of a library isn't to be super restrictive. In fact, they generally want to encourage people to use it. During lunch, I'd searched for some information about non-Miami University access to borrowing privileges. Once I found the right website (Why do they have a Cuban heritage collection? I mean, that's cool, but kind of random. Ohhhh. Wrong Miami) I found two things that looked promising. One of them was the argument that I was affialted with the university, and the other was that for borrowing privileges, I would be wiling to pay $25 to become an Affilated Patron of the Libraries. So after class, I set off.
It turned out to be easier than I'd been preparing myself for. It helped that the woman at the circulation desk was clearly pretty experienced. (She knew enough about SUMSRI's existence to know that in previous years they'd had all of the cards set up before the students showed up.) So I gave her my key card, told her some of the information about myself, and a couple minutes later she handed it back to me and told me that it should work, and if the other library had any problems, I should get them to call King library.
Awesome.
So of course I immediately headed over to BEST to check out some of the books I'd been looking at longingly and scanning the pages of before. I left the library skipping and could feel myself smiling without seeming to be able to stop. I could check out books! Apparently that's all it takes to make me happy.
