New Quarter

I've got a shiny new quarter! We are on the quarter system, so this is kind of a pun.

So, what is on tap for the next few months? Class #1 is Complex Analysis, which is not just normal analysis but harder, but specifically analysis (think calculus) in the field of complex numbers. I don't know why but it seems like complex numbers are a thing that everybody who's had at least Algebra II should know about, but nobody seems to know by name. They're just numbers of the form a + bi, where a and b are real, and i is the familiar square root of -1. There are bunches of ways to think about them, but usually people think about the real numbers as being a line and the complex numbers being a plane where the real numbers are one axis and the imaginary numbers are the other axis. Other times people think of the complex numbers, with a "point at infinity" attached, as a sphere. Then the real numbers would form a great circle on the sphere, as would the imaginary numbers, or any other line, for that matter. The most important thing, though, is that this is the last of my "required" courses.

Class #2 is part two of Applied Functional Analysis, which I am taking just to fill out my credits and because the first quarter was almost laughably easy, so I'm assuming we'll continue on in that fashion for the next ten weeks.

Class #3, which I haven't gotten officially recognized as a class yet, but only because I haven't talked to the secretary over break, is a reading class with the professor who taught my Topology I & II courses over fall and winter terms. He asked me if I would want to finish (the relevant parts of) the book we had been using and then read some stuff about model categories and other nonsense, probably because I was the only one who liked all this abstract nonsense and diagram chasing and wasn't busy doing research about numbers in boxes or some such. So, I'm keeping busy reading and re-reading stuff until I can't think any more.

As far as teaching goes, I was expecting to be doing a recitation section or two for what I call "math for dummies," but the first years don't have any teaching experience yet, so they got all those, which means I am relegated to the tutoring center for nine hours/week. Until now, the most I've had in there is six hours/week, so I am trying to get my hours scheduled to days that aren't before large exams. Those are the worst days because all these people who have never gone to class and can't do basic algebra come in and ask you to basically teach them everything in a few hours, and it is frowned on to just tell them that that is stupid and they are stupid and should just drop their class because they will assuredly fail, even if it is true.

I should specify: there are a few types of students who come for help.

1. Students who come in periodically for help with specific problems/concepts that they don't understand - Most of these students are good and nice to work with because you can't do problems for them and then let them do similar problems, or just guide them through the problems step by step, or sometimes just casually watch over them as they work. Usually these students come from one of the many calculus classes or linear algebra, which is understandable, since linear is the first time that most people have to think abstractly at all, and sometimes calculus problems just involve seeing some clever transformation that isn't obvious the first time, but also sometimes they come from lower classes, trying to figure out when to apply what formula.

2. Hopeless students - These students are usually continuing education students or sometimes low-level people trying (often for the 2nd or 3rd time) to get the math requirements for their major done. These students are a huge pain to work with because while they try, they will never succeed, and you can't just dismiss them, because at least they are trying and aren't putting it off till the last minute. I don't know whether it speaks to the absolutely horrendous state of math education K-12 in the U.S., or if there is just a segment of the population that is completely incapable of reasoning or the use of symbols to further that reasoning. One of the biggest hurdles is people that, as I mentioned earlier, can't do even basic algebra. I really think that if you graduated high school without the ability to do everything in a typical Algebra I (I am talking h.s. or junior high level) class without any real thought or effort, you are basically on the same level as someone who is still sounding words out when they read and your school completely failed you.

The reason I can't figure out whether it is the fault of the educational system or of just some people is that people seem to lack a fundamental understanding of what they're doing, rather than the specifics of how to do it. For example, you are probably familiar with the quadratic formula, and most people that come in have at least that formula drilled into their heads. But if I give you the equation of a parabola and tell you to find x-intercepts, would you know to use that formula? People don't make the connection between an equation and a graph, that the graph is a set of points with coordinates (x,y) that satisfy a given equation. They don't seem to even understand what that means, i.e., that an equation is a statement which is either true or not. They can't distinguish between an equation and an expression, and instead just memorize how to solve equations and are baffled by what an equation, or any mathematical expression, for that matter, is saying. I don't know whether this is something that most people can be taught or if it is just something you have to work out for yourself.

3. Awful students - These are the previously mentioned students who don't care enough to do anything but care about getting passing grades. Another thing they love to do is try to make you do their take home quizzes/homework for them. Some of them realize that we won't just do their problems for them, but will do similar problems, so they "cleverly" copy their problems to another sheet of paper and then ask us to do those problems. I don't know why people think this will work. It is insultingly dumb and really annoying.

So that does it for my rant, and since this is already a basically impenetrable wall of text, I'll just end the update here.

Not Much Happening

As the title would leave you to believe, there's not much happening. The term is almost over, which is nice because I won't have to deal with all the work that teaching (teaching assisting?) Data Analysis entails. It's not a horrible class or anything, but the grading takes forever since it's all explanations, not math. I often have to read through paragraph+ length answers despite the fact that all the questions can be answered properly in AT MOST two sentences. The problem is that the class is for bio major seniors who have spent the last four years learning to regurgitate terminology mindlessly, and who have taken to heart the lesson that the less you know, the more you should write. So, there are all these practically novella length completely wrong answers, which I have to read through. The kids that are good at it have started to learn how to be concise and right answers are generally easy to grade, anyway, because I can just comb through them for the right combination of words, but wrong answers take forever because I have to try to figure out what they are thinking (which is usually nothing) in order to give partial credit, and then I have to write something about how they are wrong or what they should have said.

They just had an exam this last week, which means the grading was even worse because I have to be more careful and the length of their answers has been stepped up War and Peace levels. One girl, who is almost surely the worst student of all time, actually wrote a whole page of nonsense that didn't even answer the question, and then a request for at least partial credit for addressing something tangential to what was asked. I wrote "this does not approach an answer" and gave her 0 points for that. They lose points with me for wasting my time with stuff like that. Other things I have written on exams include "don't write me a novel," "answer the question," "do this [with an arrow pointing to the directions]," and "baffling." That last one actually made some other TAs laugh. They are generally sympathetic, but not sympathetic enough to help grade.

Most of the classes have many sections, taught by a few instructors/TAs, and there are usually one or two TAs who just grade, which means when the low level classes have an exam, there's an all day grading session in the common area/break room/whatever it is called where the mailboxes and coffee are. I have been part of a couple for a class last spring, and it's almost nice because it's social and there is pizza. Of course, the social aspect can be not so great if you don't like the people you are working with, but sometimes I get jealous, since when I am grading exams, it is just me and the one professor who teaches the class in our separate offices for nine hours at a time and there is no pizza :(

On the learning side of things, there's not much happening, either. I have three courses this term, as I am trying to finish up all my course requirements this year, which means three courses per term. They're not bad, though.

One of them is applied functional analysis, which is probably the easiest course I have had here. It's like baby analysis, definitely a step down in difficulty from the analysis course I had here last year, which was intended to be preparation for the qualifier, so it was fairly rigorous. This course is for people who have already had analysis, but the book we use is for someone who never has, so a lot of the exercises are nothing new if you have a solid grasp of things like vector spaces and metric spaces. Since this post is already rambling and fairly low-content, I guess I will explain.

A vector space is just a set of vectors that meets certain basic algebraic conditions. This may sound daunting, but it isn't. It just means that you can add and subtract vectors and stay in the space, as well as multiply them by scalars (which basically just means real numbers or complex (!) numbers). The addition and multiplication have to behave nicely, too. They have to be associative and commutative and whatnot.

A metric space is just a set that has a distance function on it. I think I've talked about them before, and it's not super hard to figure out what is meant by a distance function, anyway. It just needs to tell you the distance between two points in a way that a distance does. For example, d(x,y) = |x - y| is a metric for any Euclidean space [n-tuples of numbers].

In this class, we are specifically interested in normed vector spaces, which is basically a vector space that is also a metric space, and the metric behaves like an absolute value, and even more specifically in inner product spaces, which are normed spaces where the norm (absolute value) comes from something that acts like a dot product, if that means anything to you. So, it's not highly interesting from my point of view, but it's alright.

My second course is abstract algebra, which is concerned with abstract algebraic (duh!) spaces, such as rings and groups and whatnot. I had a class in it in undergrad, and I haven't picked up a whole lot here, but it's nice to get the review and to learn Sylow's theorems, which are super useful.

My third class is my most interesting, so I arbitrarily numbered it third. It's algebraic topology, which I can only define in an abstruse way. Topology is generally the study of topological invariants, and algebraic topology is just a branch where these invariants are algebraic structures like groups. There's more to it than that, but I can't really explain everything. It's the field of study that primarily lead to category theory, which is probably the most abstract form of math. It's also very difficult to understand in my experience. A category in the mathematical sense is just a collection (not even just sets!!!) of objects with morphisms, which are basically just arrows between the objects that you can compose if the head of one points to the tail of the other. This makes everything basically into a category, and they can usually be thought of in tons of different ways if you just change what you want to be the morphisms, and ultimately you sort of don't care about the objects at all. It's very bizarre. Here's an example:

A group is an algebraic concept. It's just a set where there's a "multiplication," which is associative [ (ab)c = a(bc) ], and there's an identity element ,e, which acts like a 1. That is, ea = ae = a, and every element has an inverse, which basically means you can divide. Anyway, we can think of a group as a category with only one object, which is the group itself, and all the morphisms are multiplication by one of the elements of the group, so that all the arrows in the category point from the only object back to itself, so it's a very simple category.

For a totally different example, you can think of the collection of all sets being the objects of a category, and the morphisms just being functions from one set to the other. Since if f:X -> Y and g:Y -> Z, then we can compose the functions like g(f(x)), this makes sense as a category. Technically, we also need identity morphisms, but for each set X, the function i(x) = x suits works fine. So, this is a gigantic category. In fact, it is so big that the objects don't form a set at all, but rather a proper class. IT'S CRAZY.