In Integration, we learned a few more theorems about integrals and sequences of functions (such as Borel-Canelli Lemma and Fatou’s lemma) and defined negligible sets (having a measure of zero) and properties that are true almost everywhere (true everywhere except on a negligible set.) And the professor got a dig in at number theorists because they tend to work with the integers, which are negligible. (Note: it is not recommended that you find a number theorist and tell them this.)
Finally, we could move on to defining the Lebesgue Integral in general. Because really, what’s the point of an integral that only works on positive functions? Fortunately, extending it to work on not-necessarily positive functions is really easy: take a function, take the integral over the positive part of the function and subtract the integral over negative the negative part of the function. (If that makes sense, then you’ve probably worked with integrals without using the Fundamental Theorem of Calculus somewhat recently. If that didn’t, you now understand what my history of math class is like.) And finally, we got another theorem about integration (convergence dominé) and a look at how the Lebesgue and Riemann Integrals compare. (Lebesgue is better!)
My Harp instructor was in the Netherlands, so my lesson was moved to Thursday. Basically, not having a vacation when everyone else does sucks.)
For Topology, we’d actually been given homework. In this case, “homework,” meant “do it at home, so that the first thing we do in class is go over the answers.” Unlike the usual “spend ten to twenty minutes working on it quietly in class before we go over the answers.” Nothing was actually collected. Which was good, because although the first of the problems had been relatively straightforward, I had not gotten the second one, despite working on it for a number of hours. (The second problem had come from a final, so that was something to look forward to. It turned out to use theorems about compactness, even though it was in the section on completeness, a chapter before compactness was introduced. Yay.)
Once those were gone over, we could resume our normal classwork and have a while to work on problems in class and then go over them. We jumped straight from hard questions on metric spaces to straightfoward ones about compactness, whch left me second-guessing myself. But no, for the most part my answers turned out to be right. They really were that easy. I found myself working ahead, so when he assigned us two more homework questions, I already had one of them done. (As I would find out, only the easy one. The second, multi-part question was pretty brutal.)
During Russian, I took an unhealthy amount of amusement and satisfaction out of watching native French speakers struggle with numbers. It’s amusing because, as anyone who’s studied French can tell you, their numbers make no sense. Up through the sixties it’s fine, but then 70 is sixty-ten, meaning all of those irregular teens you need to memorize come up again. Eighty suddenly becomes four twenties and then, in the best of all possible worlds, ninety is four twenties and ten. Four twenties and eleven. Etc.
Maybe it works both ways and, when the French study other languages, they don’t understand why other countries (including, it should be noted, Belgium and Switzerland) think seventy and ninety are important numbers. Whenever the Russian professor tells us a page or exercise number in French and then Russian, I wait for her to repeat it in Russian, because it’s easier for me to make sense of it than the French. (Ironically, my skill with numbers was at its peak when I was taking a history class. I don’t think I’ve used a number larger than ten in any class I’ve taken this semester.) My French classmates don’t seem to agree with me.
Although the numbers themselves are simple, the same can’t be said about using the numbers. It’s not as bad as all of the counters in Japanese, (is the object flat or is long? Because those are completely different words, and you’d best memorize them) but you do need to pay attention to whether the noun should be in singular nominative (or accusative, or whatever other case it would naturally be in) because there’s one of them, genitive because there are *2,*3,*4 of them, or plural. It takes a bit of practice, but we got a lot in class as well.
I was doing pretty well until she started talking about countable and uncountable objects and my brain tried to connect that to analysis and failed. So throughout her explanation I was trying to find a way to mathematically redefine what she was saying, which was really that some quantities were discrete (three apples. Two loaves of bread.) and others weren’t (one liter of water. Two cups of flour.) And, if you can forget all of the mathematical meaning “dénombrable” is now loaded with, it makes sense. You can count how many watermelons you have, because that’s a discrete quantity. You can’t count grapes, because they’re continuous. (Wait a second…)
Meanwhile during the Topology TD, we finished the chapter on compactness and moved on to connectedness. He began by defining paths and path-connectedness, which is one of the few things in topology that I think makes intuitive sense. Imagine you have a space. If you can find a path between any two points in the space, the space is path-connected. Path takes on a specific mathematical meaning, of course, but if you strip it of that and just try and picture it. So two parks are connected if there is a walking path between them. Two beaches are connected if you can swim between them. Two roads are connected if you can drive between them. You can use that as a working definition in the real world, which is not true of most things in topology. (Next time someone asks if something’s open, respond “it’s definitely not closed” and see how many glares you get when they come back and it turns out not to be open either.)
While the professor was explaining a proof, the student behind me leaned forward to ask me what “CPA” (connexe-par-arcs, or path-connected) stood for. And I was able to answer him. I finished writing down the proof, aware that I was smiling. Two months ago, I might have asked the student to repeat himself, stared at him blankly, and then shook my head with an apology because I couldn’t respond to him, and I needed to focus on the professor or I would miss something major. But now, a student had just asked me a question in the middle of class, and I’d responded. Correctly. I really was learning and adapting to being here.
During the harp lesson, I got information about when and where the “audition” (meaning recital. It’s one of those French faux-amis you need to be aware of) for Madame Luce’s students would take place. No idea what piece I’ll be playing, how many students will be there, or generally what’s going to be expected of me, but I know where and when I should show up, so that’s a start.
I also got started on a new piece, “Greensleeves.” This arrangement was special for two reasons. First, it was written to be played by two harps. The first part can stand alone pretty well, but the second seems to bear no relation to what I'm used to the piece sounding like.
The other difference is that you're going along merrily in key of B flat when suddenly, there's an F sharp. If you're more used to playing another instrument than you are to the harp, you might not get why that's a problem. Say you're on piano. Sure, you need to remember that the F should be sharp, but other than that, it's basically as easy to play an F sharp as it is to play an F natural.
On harp, anything that deviates from the initial key signature is a pain. It requires physically changing the tuning of the string. On pedal harp, it means your feet now have something to do. On lever harp, it means your left hand (which, it should be remembered, has notes of its own to play) needs to reach up and change the lever. And, if you try and change the note while the string is still vibrating, it sounds horrible.
So the new piece was certainly a challenge.
Integration continues to lag behind the class a bit, so we were still doing exercises simply related to measure and not moving on to, you know, actual integration. But the partiel that we were going to be taking the next week was only going to work on measure, so it was OK.
We also got back our controls. Recall, we got the answer keys immediately after finishing the test. Which was seen as sufficient correction that they didn’t write anything on the papers they gave back. Instead, we got another sheet of paper that had on it our scores and those of everyone else in the class. So not only could I see what problems I’d gotten wrong, I could see what problems the people I sat next to had gotten wrong too. Joy. (Turned out that one of the questions that I’d considered pretty simple was not. That would have been better if I hadn’t missed other things that were, objectively, pretty simple.)
History of Math resumed the normally scheduled four hour blocks. So I got to re-remember how terrible that was. He did not have the midterms graded or, from the sounds of it, even looked at, but he asked us how we thought they went. General consensus of the class was that it was hard and the parts written in 18th century French sucked. It’s always comforting to realize that my French classmates struggle with some of the same things that I do. Turns out some things, like a struggle with essay questions and certain math problems, transcend cultural differences.
During the CM we were up to the Renaissance, though during the TD we were still on ancient Greeks. Non-Euclid Greeks, which wasn’t necessarily an improvement. There were still stylistic difficulties that continue for several centuries. But other mathematicians were occasionally willing to use ordinal numbers and occasionally actually measure things, so that at least was different.
And, finally, when I just wanted the week to be over, (another hour of History of Math, followed by) French class. Not much of interest there, though we were talking about nominalisation so, after class, I went up to ask him about something that had been bothering me for a bit- nominalisation of an adjective by adding “un” in front of it. Because I’d never really seen that before this semester, but suddenly it had been all over my math classes. So instead of saying things like “Let O be an open set” they’d say “Let O be an open.” “Let A be a Borelian.” “Let f be a lipschitzienne.” (side note: how come Lipschitz becomes an adjective, but a Cauchy sequence is always “de Cauchy” instead of “cauchienne?” The more I gave examples (especially with names) the better the professor understood what was going on. Because he’d never seen that before, so basically, he told me that French mathematicians were completely making things up.
Super comforting to know.
