Analysis has grown on me every time I’ve taken it. The first time I took it, I liked it, but it was basically just a math class. A math class that involved more rigor and writing than I was used to, but nothing super special.
The next analysis course I had was over the summer. It was kind of a bridge course between the Real Analysis course I’d already taken and the Integration and Theory of Measure course I was about to take. A rough summary of my days: nothing makes sense. Nothing makes sense. What am I doing? Why am I studying math? This should make sense, because it made sense the last time I learned it, but nope. I have no idea what he’s saying. And topology? Literally no clue what’s going on. Nothing makes sen… oh, analysis. Hey, I understand what’s going on! And… we’re moving on to new material, and I STILL understand what he’s talking about. These proofs are complete and well-explained. And nothing is used without being defined. This is wonderful!
It’s not that extreme this semester. (My History of Math class also made sense.) But, as it got closer to Tuesday, I found myself really looking forward to Intégration. It helped that this week, the course would only be two hours long. (I cannot believe that I’m describing a two-hour class like it’s a short course. Because, this semester, it kind of is.) And this was why I’d come to Paris. This is what people had missed when they’d made comments like “you’re only taking math classes? Why?” I wanted the chance to test my French, my math, and mostly myself. And Intégration still felt like a very reasonable test.
The class began by finishing up countability, this time with Cantor’s proof of the uncountability of the real numbers. Once again a student tried to ask me a question in a whisper, once again I had no idea what he was saying. (Fortunately, the person in front of me did, and was able to answer his question.) The proof concluded, and some students raised their hands to ask questions for clarification.
I really wanted to. Not out of confusion, but because the proof, as he wrote it up, wasn’t quite right. I’d seen the proof, or slight variants on it, several times before. I knew what the problem was, and I knew why you could easily show it wasn’t a problem. (Proof sketch: Create a new number by choosing the kth digit as any digit that isn’t the same as kth digit of (k-1)th number in your list. Problem: .29999… is the same as .30000… Basically, nines are problematic. Solution: Avoid them! This still gives you 8 choices for each digit of your new number, which is 7 more than, strictly speaking, you need.) Unfortunately, I wasn’t entirely sure how to express that in French. I played around with phasing while the other students asked their questions, but didn’t get to the point where I was comfortable raising my hand in front of a lecture hall full of students who (mostly) all spoke French fluently to ask a question that I already knew the answer to. So I stayed quiet. Even when the professor asked if there were any other questions.
He concluded by giving a brief description of why the proof wasn’t complete, since the decimal representation is not unique. Because nines are a problem. It might have just been me projecting, but I think he sounded a little disappointed none of us had asked about that. Too late for it now, and it was time to move on.
So we started a new chapter, and new definitions. It started on relatively familiar ground, with proofs that were purely manipulations of basic set theoretic operations. Then things got a little more complicated, and I got a little more lost, but nothing that some time with my notes wouldn’t be able to fix.
One of the things that I really like about the professor is the way he is able to ground abstract concepts. So, when he was talking about lim sup and lim infs of series of sets, he gave us an example, let us play around with it for a while, and then gave us a different way of thinking about the lim sup and lim inf. And with that way, the examples became a lot clearer.
Same with today’s lecture. He talked about the “tribu engendrée par un sous ensemble,” and then gave a list of other mathematical concepts that used the exact same idea. And then a little bit later, he gave us a very specific example to work with, (σ({A, B}), if anyone cares) asking us to find how many elements were in the set. People made guesses, and then he went up, with input from the class, wrote up all of the unique elements of the set. Once we had all of them, he was able to use a Venn diagram to show us a much easier way to determine the number of elements, if not the elements themselves. (A few days later, I was sitting in topology trying to remember the best explanation I’d ever gotten in that course. That was the winner for a while, until I remembered it was analysis and not topology.)
The professors is still very quiet, but I moved a couple rows down, so it was less of a problem. Besides, he talks slowly enough that I can write down every proof in its entirety, which for me is far more important. I’m liking the class a lot, and wondering what the TD will be like.
The class let out late, and I was slightly pressed for time, so I ended up practically needing to run to my first harp lesson. However, this was nothing new, since last semester my harp lessons began at the exact same time French got out. (Theoretically, it also ended at the same time Calc 3 started, but fortunately she’d usually let me go early, so I had time to make it to class.)
The harp lesson took place at the Madame Luce’s apartment. She grew up in Minnesota, so she obviously knows English, but the lesson took place in French for reasons involving “good practice” and “learning more than just math vocabulary” and “listening to Erin.” I reserve the right to request the lessons take place in English if it turns out to be too much.
It began with me talking a little bit about myself and my previous experience, and she gave me some information about renting a harp. And then I was told to go into the small room, (closet would probably be the more accurate term) sit down at the pedal harp, and play something. I tried to remember any of the pieces I’d learned for juries (which I always have memorized by the end of the semester) but I couldn’t. Eventually I had to go with arpeggios and try and ignore the nagging feeling in the back of my head that I’d been able to play a much more interesting etude at the end of last semester.
After that, Madame Luce found an exercise for me to play, slowly, and paying a lot of attention to my fourth finger and making sure it was more rounded, and also that I kept a gap between my thumb and my index finger. The directions were familiar, because my instructor at Carthage had mentioned similar things, though not with the same focus we went over it here. Between every measure, when I placed my hand, I would pause. At the beginning, she needed to reach over and correct my placement, but that was less frequent near the end, which presumably meant I was learning.
Then we switched places and she played through several pieces and told me to choose my favorite. I did, and then it was my turn to play again. I played through each of the notes in the first half, slightly startled by how easy it seemed. It was pretty repetitive, and the left hand only did two things, both of them very basic. When I stop to think about it, it’s not really surprising that this piece would feel easy: it probably was. The piece was interesting, and it’s not like I could just sit down and play it hands together without any thought, but it didn’t have as much going on in it as I was used to.
I’d started playing the harp at Carthage, so I had the luxury of an instructor who knew exactly what I was capable of. She’d occasionally give me a piece just to work on to have a break from what I was preparing for the jury, but generally, the music she gave me was chosen carefully to stretch my current abilities. Madame Luce did not yet have a solid grasp on what I was and wasn’t capable of. That was part of the point of giving me a song to work on. I wanted to see where the rest of the semester would bring me, and I really wanted to have more time with a harp just to play for myself. That, however, would have to wait until I was able to get a rental.
We arranged the next meeting, and I left to wander around. I had two hours before I needed to be at Stephanie’s for dinner. If I’d realized that both lines from the metro stop near my apartment stopped near her apartment, I probably would have gone back home. Instead, I walked around for a while looking into shops. I bought Stephanie a box of chocolates to thank her for hosting (see? I do pay attention during the meetings we have) and found an intriguing shop that sold yarn and clothing. Not knit clothing, just your average (not that I really know what average looks like) boutique clothes. And yarn. I’m not sure why they did that, but I liked the theory: this way, if I got bored looking at clothes, I could go stare at yarn until I was re-energized. Maybe it works in reverse for friends of knitters?
After 90 minutes, I was ready to stop walking and head directly to Stephanie’s. But when I pulled out Google maps, it asked “are you sure you don’t just want to walk? It would be faster.” So, sighing at another victory of Google over the Paris Metro, I listened.
I ran into Clara outside of the building, so I let her read and follow the instructions in the e-mail. (I’d already needed to follow Madame Luce’s instructions to get to the lesson. There’s a limit to how many unfamiliar apartments I’m comfortable finding on any given day. It’s more than two, but whatever.) There, we met up with the others and talked about our weekends, our courses, and pretty much anything, actually.
Dinner was in four courses, but, contrary to the trend I’d half come to expect at French houses, there were only two kinds of alcohol served. Kire, served with the appetizers and continued for the main course, and red wine for the cheese. There was a regional theme going on, most notable with the kire and the caramel. (I don’t know food well enough to be able to distinguish between regional specialities.)
Appetizers were something that had “chou” in the name: a fluffy scone with a cheesy taste, carrots, baguettes, and several different kinds of sauces. The main course was escargots and several kinds of quiche-like dishes. I’m not quite sure how many, because it’s hard to tell which of them were the same and which were different. Then there was a cheese course, with eight different varieties. Several of them were rather strong, including one that I found difficult to swallow. And the only thing I had to wash down the taste was red wine, which wasn’t much of an improvement. This was especially problematic since I was talking to Stephanie at the time, but I managed to grab the last piece of bread (next person in that situation could fend for themselves) and focus on the taste of something a little more banal. Other than that one cheese, I found most of them very good, and it’s always a nice experience to try a cheese course in a French house. Mainly because a cheese plate is an authentic French experience, and, unlike a restaurant, a French host won’t make you choose between cheese and dessert. Dessert was, as I’d mentioned, caramel-themed. Two or three kinds of caramel cake, plus more bread with a caramel spread.
There was a lot of food, and a lot of catching up to do. Christine was there, as was Danielle, the other tutor. (Christine’s humanities, Danielle’s social science, and if I want a tutor I need to talk with Stephanie to arrange someone special.) There was also a Brown graduate student who several people had had as a professor, Erin, and Stephanie’s sister. So basically just a lot of people. There wasn’t quite room for everyone to sit comfortably, but if a few people were fine on the ground or perched on sofa arms, we could and did manage.
It was a nice couple of hours, but hardly a relaxing end to a day. Between that, the continued presence of classes in my life, and the fact that I was still carrying my backpack and hadn’t seen my apartment in 10 hours, I declined karaoke and caught the nearest metro back. And, as it occurred to me that I was closer to being able to settle into a routine, I added my classes and the cultural activities to my calendar.
Mondays were wonderful. Tuesdays were good. Wednesdays were not the best, but fine. Thursdays were a little strange, but I didn’t start too early and could do basically anything I wanted with a five hour break. Fridays were making me wonder why I hated myself. It was going to be a fun week.
