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.