Monday, January 31, 2011

Types of Proofs

When you're first taking a course in math proofs, you learn about things like direct proof, proof by contradiction, proof by induction, and so on. Now that I have more experience, these are the types of proofs that I've experienced:

Proof by Algebra
These are the types of proofs most often encountered in a math class that isn't very proof-oriented, like statistics or (sometimes) linear algebra. The problem will say something like
A Palanquin Duo is a pair of numbers, x and y, such that x+ y = a for some real number a. Then the product xy is called a Palanquin Product. Show that the minimum Palanquin Product for a given a occurs when x = y = a/2.
[Yes, I made this up completely.] You see this and you go "shit, that's a bunch of weird crap" but when you go to do it, it just requires some algebra (or calculus, or whatever - basically some calculations/math that are sort of obvious in context). Often these are pretty easy because it's sort of obvious what the next step is, and you just have to have faith that if you keep going, the proof will work out.

Proof by Definition
These proofs usually ask you to verify that something specified in the proof is, in fact, an Thing, where the Thing is something that is defined in the course. For example,
Let A1 and A2 be two topologies on a space, X. Show that A1 intersect A2 is also a topology on X.
When you first, in your math career, encounter this type of proof, you often think something like, "My gosh, why would that be true?" But if you look at the definition of a topology (or whatever), you can easily verify that all of the conditions are easily met using the assumptions that you're given.

Proof by Construction
I may not be using the word "construction" here as it is usually used in math. But sometimes you are asked to show that something exists and the easiest way is to actually show how it can be made. For example,
Show that every Lebesque-measurable function may be approximated by a step function such that [blah blah conditions].
I see these in analysis, mostly, where you end up doing a lot of stuff with epsilons and deltas and building up a giant edifice step by step. These types of proofs can be very hard.

Proof by Induction
The canonical induction proofs are when you want to prove something for every natural number, for instance,
For $n>1$, show that $2 + 2^n + 2^3 + ... + 2^n = 2^(n+1) - 2$
To do this, you prove that the statement is true for n=1 and then that, if it is true for some n, it is true for n+1. But there are much more complicated and interesting types of induction proofs out there as well. Induction is fun - when you can get it to work, it often feels like cheating.

Proof by a Trick You'd Never Have Thought Of
I assume this is self-explanatory. Usually the professor will show these proofs in class rather than expecting you to do them as homework. There is a reason some theorems are named after the mathematician who first proved or formulated them!

Saturday, January 15, 2011

Terrifying Product

Every time I go to Kroger, I pass this product, and am gobsmacked by how horrifying it is:You may not be able to read the jar in this image, but it says 'Walden Farms Calorie Free Alfredo Sauce.'

Good luck sleeping tonight.

Thursday, January 13, 2011

My Spring Schedule

This is my schedule for Spring 2011.

Mondays & Wednesdays
10:00-10:50AM Logic & Set Theory
2:00-3:20PM Statistics

Tuesdays & Thursdays
11:00AM-12:20PM Intro to Topology
2:00-3:20PM Real Analysis

10:00-10:50AM Logic & Set Theory

So I have basically a pretty consistent schedule every day except that on Fridays I have no afternoon class.

Wednesday, January 12, 2011

Doomed. Doomed, I Tell You!

I've been facing a dilemma (two conflicting lemmas!) for the past few weeks, and I have finally resolved it, though I have resolved it in the direction of ultimate doom. Pray for my sanity, friends.

The dilemma is this. Last semester, I took the first semester of a two-semester logic & set theory course. I sort of hated it a lot in the middle, but I look back at it very fondly, and the stuff to be covered this coming semester is stuff that seems to come up a whole lot and that I really want to understand. I'm extremely tempted to take the second semester, and normally one does take both semesters of such a thing.

However! I took four courses last semester, and, though it was originally planned for me to take four this semester as well, the graduate advisor asked both me and the other fellowship recipient to only take three courses this semester. One of the courses I need to take is an Intro to Topology class. I think that class will be relatively easy ("relatively" being the key word here), and I'm not that excited about it, but I have never taken a topology class and I am planning to take the topology core sequence next year, and that just won't work. So: topology ho!

But I still want to take logic, and I don't know when the course will even be offered again. Also, though this may sound silly, all of my favorite people from my cohort are in that class. They are the brightest and (especially) the hardest-working of us, and being around them is really great for me.

I had decided in favor of topology - we must be practical, after all. But then Ed let me know that the logic professor is worried the second half of the class won't even make. So I changed my mind and have now signed up for both courses. This means I'll have four classes again, except that, unlike last semester, none of my courses will be a total blow-off like the pedagogy one was.

There are two reasons this should be possible, and one fairly compelling reason that it shouldn't be. First, other students routinely take three classes and have a TAship that is supposed to be half time. Many don't work 20 hours a week at their TAships, but some do work many hours. Second, I did not, objectively speaking, work all that many actual hours last semester. There is theoretically enough room for the extra hours topology will add to my life, without compromising time to eat, sleep, relax, and so on.

However, last semester there were at least a few weeks (perhaps the middle half of the semester) when I was really and truly stressed, basically walking around like a zombie and fantasizing about going back to my old job and having an easier life. The stress subsided near the end, partly (perhaps) because the analysis prof stopped assigning homework, but also partly (I feel) because I figured out how to chill out a little bit. I think maybe--maybe--I can make that skill carry over to this new semester.

The following things are clear:
  • It's possible for me to drop a class if I need to, but I would feel bad about how it would look (not on my transcript, but in terms of how the professor teaching the course would view me).
  • I'll need better time management; I need to get better at working early and often.
  • If I succeed at this, I'll feel really good about it.
  • If I succeed at this, I'll have learned some extra math that I'll be really happy and excited about.
  • If I succeed at this, it will have made me stronger.
So that's that.

Wednesday, January 05, 2011

A Second Chance

A couple of weeks from today, I start my second semester of grad school. The first semester went great, in retrospect - I ended up getting A's in all four classes (woot), and I remember it all extremely fondly. If I didn't remember having complained about how stressed out I was, I'd think I enjoyed the entire thing start to finish. I guess, as usual, I enjoy it retroactively. This probably bodes well for my real-time enjoyment of future semesters.

This whole grad school thing is really amazing. One of the most awesome things about it is the other people in my cohort (and in other years, potentially). There is a large social group I feel very comfortable in, and I am really close to two people in particular. I'm not used to having so many acquaintances and friends to enjoy, close at hand like this.

I went to a New Year's Eve party at Drew's parents' house in Missouri this year, and stayed for a couple of days. Jared and I drove up together. (These are the two I am close to.) I actually saw midnight on New Year's Eve for the first time in a few years, and I stayed up very late drinking somewhat heavily (though not so much that I became ill, passed out, or had a hangover the next day). I actually felt socially likeable and accepted. I felt like a fun person whom other people would naturally like.

Counting Drew, there were five people from our program there, but the majority of the attendees were Drew's other friends, most from her undergraduate school (a SLAC, I think). I liked many of them very well indeed, and there was nobody I felt I didn't like.

It is a little strange to think about. I am about halfway between Drew's friends and her parents, age-wise. Is it creepy to be 36 and hanging out with a bunch of people who (on average) just finished college? They seemed to accept and like me. I didn't feel out of place. I don't feel, moment-to-moment, like I am not one of them, despite the ~13 years of work experience I had before starting grad school.

What all of this feels like is a great big giant miraculous do-over for me. I feel like this is the life I should have had, but couldn't grasp when I was these people's age, but here I am now, getting to do it. I'm pretty immensely grateful for it, too.