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Please Don't Tell Me to do the Math(s)

USA | Tuesday, 9 June 2015 | Views [236]

As I was updating my Googele Calendar with the schedule that I got from the program, my main thought was “we sure are spending a lot of time in Reserach Seminars.” This was followed by “well, duh. It is a summer research program.” But really, it is a lot of time. 10:30-12, and then again 1:30-3 MWF, and 1:15-3:15 TR. This in addition to a short course in Analysis (beginning 8:30 Monday morning, though it's 3:30-5 on Wednesdays and Fridays) and Algebra (Tuesday/Thursdays 9:45-12). And some Tuesday or Thursday afternoons, we'll aslo have colloquia.

It's a pretty busy schedule, at least during the day, for the first 3 weeks. We do get a break of over an hour each day for lunch, though, so it shouldn't be too bad. After 3 weeks, the short courses end every waking moment (really, every moment, because why would we need sleep?) is devoted to research or getting into grad school. At least that's what I'll assume happens, because don't have a schedule, just a list of times for GRE prep and a grad school panel. So either we're constantly doing work, or we just loll about all day dreaming about our futures. I'll find out which in a couple of weeks.

In the mean time, I get to find out more about everything that's on the schedule for this week, or at least this day.

The morning began with us figuring out how to use our meal cards. (It's like it is at my home university, only instead of handing them my Carthage ID, I hand them my Maimi key card. And then they swipe it and the cash register displays how many points I have left in the day. They don't give me a receipt with that information, though, so that's a little disapointing.

My breakfast was black coffeee, a chocolate chip muffin, and awe at how big their student center was. There were so many different places I could eat on campus with my meal points. Even a store where I could use up any of the $25 remaining at the end of the day. We'll see how long it takes me to get bored with the options, but for right now, I'm just amazed by the size of the building. And then it was time to go to class.

The professor began by describing the general structure of the class. It is, by it's nature, a short course, designed to give us just a taste of real analysis. The first half of the course would be based off an undergraduate curriculum, and the last half would get into more graduate level material.

The next thing he did was hand out notes. He had the same theory that my Real Analysis professor at college had. Basically, if students spend their entire time writing down everything on the borard, they aren't paying as much attention to what he's saying. The notes for this short course were all handwritten, which made me realize how much the typeset notes/textbook I'd had during the semester-long course had spoiled me.

Having taken a semester of Real Analysis before, the material we covered today was all review. And not the vague review of “I know what real numbers are.” “Oh, yeah, seuqeunces. Those came up in Calc 2.” Actual review of material that had been covered more rigorously the semester before. Regardless, the professor was good, I like the general subject, and I look forward to the rest of the course, when we move into material I haven't studied and tutored already.

After Analysis ended, we had a half hour break. I checked out the reading room next to our classroom. So many books. And almost all o fthem were about math. (Some were physics or engineering.) It was so beauitufl. Unfortunately, none of the books can be taken out of the room.

After that came the research seminar. He began by asking how many of us had taken an abstract algebra class (5 of the 8) and then how many had taken topology (3 out of the 8). He then proceeded to give a scattered explanation of group theory, with a lot of examples and non-examples. Including several examples from topology.

I had a very good abstract algebra course, and I was struggling a bit to appreciate his explanation of concepts I understood pretty well. So when he started discussing topology, it became a struggle just to not get lost.

He talked a bit about the braid group on n strands, which was not a group I'd encountered before. Specifically, we were looking at the braid group on 4 strands, and spent a bit of time discussing the generators and the relators for that set. (It would be another couple of hours before he'd give anything approaching an explanation of the representation of a group.) Essentially, you take strands in three dimensional space, and you pass them in front of or behind other strands. (That's a terrible description. Here, have a picture.)

Braid group, description and sketches

It seems like an interesting group, and I woulnd't mind having more time to play with it and experiment with some of its properties.

Another examples of a group that the instructor wanted to give us was the Fundamental Group of X at x. (Side note: I have enough problems writing lower-case letters. X compared to x is really not making my notes easier to read.) This turned out to require so much background topological knowledge that not only did it take the remaining twenty minutes of the morning class, but also the 90 minutes of the afternoon session.

The path we took was nearly as convuluted as some of the examples drawn on the board. (Side note: at present, all the boards at Miami University are chalk boards. This is good for the people who like the chalk dust and everything related to that. I am not one of those people, and would be happier with white boards.) Right before lunch, we had a mini lesson on Poincaré and the history of his namesake conjecture. Then more about in a bit more detail so we could unpack what Poincaré had said in his original statement (which he'd proved) and then how he'd discovered he was wrong, and used fundamental groups to prove it.

Now we were ready to compute fundamental groups. Er... actually we weren't. Let's back up a bit and go over the definition of a metric. And when you definie that, it's best to give an example or two about a metric on a familiar space. So he defined several different distances on R^2, specifically the regular distance function and the taxicab metric.

With those in place, he could define epsilon balls (memories or epsilon neighborhoods in real analysis made the concept slightly more familiar), open and closed subsets. (Despite the name, open and closed are not opposites. A set can be open, closed, both, or neither. That or is exclusive.) And with that in mind, he could define a topolgy determined by a metric. Hmm... maybe he had best back up and define a topology?

OK, with that out of the way we could go back to our three distance functions. That's right, there are now three of them. Ooh, proably means we need another one to be defined.

Now, back to the generals about topolgoies and why they're useful. The can be used to definie continuous functions, convergence seuquences, conectivety of spaces, and compactness. It was somewhere in there than Fernando asked a question about a topology being fine. So the professor kind of explained what course and fine were, then warned us that some people would take the exact opposite definition of those two words. (How? Why?)

And finally, we were ready to learn about computing fundamental groups. Unfortunately, by this point, we only had time to do one example. And fundamental groups are not so intuitive that you only need one example to understand them. Especially when, upon reaching the end of the discussion, it becomes apparent that generators and relators are still a topic of confusion. I'd had previous exposure to them (it was relevant for a paper I wrote in Abstract) but I'm pretty sure I was the only one in the class who had. So, with the last few minutes we had left, the instructor tried to give us a working definition. And then it was over.

Over the course of the day, he dropped a couple of hints about material would be on the research problem, but that was it. So, at the end of the day, I was left with an incredibly shaky introduction to topology (apparently it never makes sense on the first exposure anyway, so at least I'm getting it out of the way) and no more idea of what the research will be on than I had before. Guess it's good I have hours more of research seminar this week And next week. And the week after.

Oh, but the day wasn't over yet. Nope. As we were packing up, Hannah, the graduate assistant for the program, told us there would be a review session at 7:00 in the Stoddard Lounge. So we had a couple of hours to relax, unwind, and get dinner. And then we gathered together for more math.

She went into a lot more examples, beginning with general groups, then moving on to free groups, representation of groups (the generators and relators) and then to fundamental groups. Lots more examples.

One thing I learned about myself today was how much I prefer rigorous proofs to examples and handwaving. I think that's one of the things I liked so much about Real Analysis when I took it. It required you to go through and prove everything, even the steps that seemed like they should be obvious. Intuitive is not the same thing as trivial, and things that are easy to understand are not always easy to prove. (I think that's what a lot of people hate about that class.)

Especially during the topological discussions, I felt lost without the formal definitions. Examplesa re nice and necessry, but you can't prove anything with “well, you said it worked in that other example, and this looks a bit like that, so it probably works here too.” I need definitions and theorems and proofs to know how to use these examples. I got a lot of that in the Real Analysis class, and a bit of that after lunch in the research seminar, but not as much as I would have liked.

I wonder what the algebra short course will be like. Or what the research seminar tomorrow will be like, since I never know where the instructor's going to go next. Hopefully he'll have the time to introduce the problem. It would be nice to know what, specifically, I'm doing with the rest of my summer.

Tags: algebra, analysis, math, topology

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