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And then there's topology

FRANCE | Sunday, 13 September 2015 | Views [225]

I didn't realise until I got to campus that my next class (by process of elimination, that one had to be topology) that this class did not take place at the same lecture hall my previous one had. In fact, this one didn't take place at a numbered lecture hall. It took place at "amphitheater F1," wherever that was.

I pulled out a campus map and located building F. (I've never felt like a freshman as much as I have at UPMC.) Great. I headed there and entered the first door. There were signs about chemistry, and the fire map that I found did not have any labeled classrooms. I left that building and looked at other doors. They all appeared to be unmarked at the top of a double-helix staircase. It became increasingly obvious I was not going to find the classroom myself.

I located a group of students who looked friendly and like they knew what they were doing. (By which I mean they were standing around talking and not running frantically around. My standards were pretty low.)

The directions I received essentially boiled down to "go up the nearest staircase and find the classroom." Eh, it was a step in the right direction. (No pun intended.) So I followed the first half of her directions. I even found an amphitheater, but it was labeled "F76", which seemed pretty far off. I chose a direction and kept going until I found someone else who looked like they knew what was going on. (In retrospect, I should have paid attention to whether the numbers were going up or down. At the time, I didn't even think about it because I didn't expect the numbers of the classrooms to behave normally.)

At the end of the building, I found a large group of people standing outside a classroom. I asked one of them where amphitheater F1 was, hoping she'd say "oh, right here."

"Tout droit. Tout droit." ("Straight ahead, straight ahead." Not to be confused with fordroite," "on the right." In fact, it was straight ahead and on the left.) So I had to run back down the hallway. Finally, I saw my first "helpful" sign. The sign told me the lecture hall was another floor up. There was another person right in front of me, so I entered on his heels and tried to pretend the door slamming loudly behind me had nothing to do with me.

We were a little over five minutes late, so I missed the description of the class and landed in the middle of the historical context. Maybe it was because I missed the beginning, but his explanation made no sense. Something about physics and geometry and calculus and points in space... I don't know. I couldn't see the connections between what he was saying at all. It wasn't until I saw that the person in the row below me had written "Topologie et calcul différentiel" that I was able to relax with the thought that at least I was in the right course.

Things got better when he finished whatever context he was trying to give and went into actual material. For the approximately three minutes it took the professor to define distance, I understood exactly what he was talking about. Then he started giving examples, and I was quickly lost.

You don't need to be a mathematician to come up with an easy example of a distance function. In fact, of you're kind of creative and drive in a city where streets are all at right angles or parallel, you might be able to come up with another comprehensible example. With a solid grounding in math and the full definition, you can come up with a lot of interesting, but still graspable, examples. The professor gave us none of those.

His first example was the PageRank algorithm. (I'm pretty sure that's what it was. Otherwise it was a similarly named algorithm that dealt with likes on social networks.) Besides being a sophisticated and proprietary algorithm, it's not even an actual example of a distance function since it's not necessarily symmetric.

The next example was the mechanics of incompressible fluids. The set in question was the set of diffeomorphisms that preserve volume, and, since that is a set that everyone is intimately familiar with, you can tell it's going to be a simple example, right? He went on to define "historiques," trajectories, and I believe velocity (several math/physics majors have told me French physics is really easy to understand because the words are almost the same. That, of course, relies on you knowing the right physicsy words.) Finally, he gave a theorem that probably let him use those definitions in a distance function. I was left with so many questions, primarily "will this be on the test?" and "is anyone else as confused as I am? Is everyone?"

Finally, he gave an "easy" example: Hamming distance. This measures the difference between two strings of bits by counting the number of places in which the value of the bits are different. It's not that difficult a concept, and, if necessary, I might even be able to show it's a distance function. It's still not necessarily intuitive.

Number of times I’ve received a mathematical definition of distance in the past six months: 4. Number of times I’ve thought it made sense: 1. (It’s always the same definition, but without good examples, it’s really hard to understand why that’s the definition.) So at least this class was in the majority?

He went on to give more definitions and theorems. These were semi-familiar from the summer, and he stayed away from examples, so they made more sense.

During the class, I had the new experience of not being able to write fast enough to keep up with the proofs. This was due in large part to the professor using abbreviations. And, if I wasn’t paying careful attention to what he’d said (or even if I was but his symbols didn’t make sense) I needed to stop to puzzle over what that stood for, and then write the entire word. Sometimes I had to give up, write “q.c.q.” and hope that was 1) right 2) enough for me to figure it out later. (He did not have the neatest handwriting.) Eventually, I realized that by trying to finish every proof, I was playing a losing game of catch-up and missing everything he was saying. So I skipped a few lines and moved on to what he was currently showing. I had to do that one more time before the end of class.

My favorite moment came when he started talking about a “application lipshitzienne” at the very end. Besides being on more familiar ground, (Lipschitz functions were something covered in Real Analysis, so I felt way more confident about my knowledge  of them then I felt about the patchwork intro to topology I’d gotten over the summer) Lipschitz is a fun word to say or hear. The fact that French tries to pass this obviously German name off like it’s a real French adjective amused me a lot. (I needed some source of amusement, and 1 hour 53 minutes of topology taught in French doesn’t have a lot of alternatives.)

With the conclusion of the course, it was time to head back to the Brown in Paris office for our weekend homestay information. (Thursday was kind of the inverse of Tuesday, with classes in the morning and a meeting in the afternoon.) When I was most of the way to the office, I stopped to grab lunch at the nearby bakery. And then on to the office.

It was scheduled as a two hour meeting, with the promise that we’d get our information about where we were going this weekend (read: tomorrow) and who we were staying with. The first hour was dedicated to a discussion about the proper way to make cultural observations. Erin mentioned that, when making those kinds of observations, it’s best to always ask why instead of leaving something with “C’était bizarre.” Which made me feel guilty, because I had just used that phrase with Stephanie when describing how my class had gone. So I tried to think of other ways to express the differences between the way my professor had explained the distance function and the way that I thought it should have been explained. I was drawing a blank, and sorely missing the presence of other people who knew enough math to be able to understand why the professor’s examples of distance were rather unhelpful.

We went on to the “find the Francophone town you’ll all be staying at” portion of the afternoon. Not, this was a plural you. None of us knew where we would be, but we did now know the name of the cities, towns, and villages we would collectively be.

Now, on to practical information about trains, tickets, and what to expect from our hosts. The tension in the room had grown since the beginning of the meeting, and it was impossible to ignore it by the end of that discussion. So, at last, they passed out the information. I would be staying with Laure, a woman in Dijon. And, with that information, I could go home and prepare for the weekend.

Tags: campus, dijon, math, topology

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