Monday, December 15, 2008

Research vs Homework

i don't know how these two things differ in other fields, but it seems there's a key difference between research and homework. Sure, they both involve solving problems, generally using methods previously discovered. But homework is normally structured in a much nicer format than research. Generally homework questions come at the end of a chapter which just introduced a new theoretical framework for solving problems. So these homework problems will generally just test a student's ability to put this theoretical framework into action (even if its the theoretical action of a proof). Sometimes these problems will ask you to go back a few chapters and use other theoretical work as well, but thats the most they ask.

Research is a different beast in itself. The research problems are generally just concepts that somebody notices or thinks up, so something that can seem obvious may be really hard to prove (see Goldbach Conjecture). Further, there's no general recipe for all the type of things you'll need in research. And if it happens that you need something that you haven't yet learned, then I guess you've got some studying to do.

I'm thinking about this right now because I just finished working on a research problem that was pretty annoying. When I first heard the problem proposed, I knew very little about integer programming or linear programming. I had related the problem to the Maximum Independent Set (MIS) problem and felt pretty annoyed that because the MIS problem is NP-Complete (lets just say really hard), I wouldn't be able to get further. Back then, I simply wrote computer programs to try to enumerate all the feasible solutions - but stopped when I was able to prove that the programs would take a long LONG time to finish. I was able to solve some smaller instances of the problem by hand and left the larger instances alone.

Well this weekend, I went back to the larger instances. Initially I started with some of those same thoughts, and the lazy programmer in me just wanted to write a nested for loop and call it a day. But again, being able to see how long that program would take to run, I kept doing things to increase the running time of the program and was able to finish up that larger case.

The key improvement I made this weekend was formulating the problem as an integer program and solving the LP relaxation (with a little bit of branch and bound). This seems like such an obvious way to approach the problem to my mind today, but I received so much resistance from my co-authors when I initially proposed this as a way of tackling this problem that I just gave in. But after they were unable to move past a few different enumeration techniques, this idea came back into my head with positive results.

I'm trying to wrap my head around this whole "research" concept. I know that I fell in love with it in undergrad, but each problem seems to possess its own uniqueness. I remember when I was taking proof based math classes, I would systematically attack homework problems:
- What am I trying to prove?
- What are the key terms in the assumption?
- What are the key terms in the conclusion?
- Are there any theorems that relate these key terms?

Normally by the time I've gotten to the fourth question, I can get to a proof. Its not so simple in research. First of all, I don't generally know what I'm trying to prove, except that its unproven. Sometimes, its helpful to make a claim like "I'm trying to prove a matrix is totally unimodular", but if I'm unable then it doesn't mean that the matirxis NOT totally unimodular, it means that I haven't proven that it is totally unimodular.

Second, I'm normally asking a question that can be formulated in many different ways. So the idea of key terms depends a whole lot on which format I choose to work with. Sometimes it doesn't matter, but other times after the research has been completed I'll see why I couldn't have come to the same result with a different format. But things are always easier after the fact.

And third, unlike homework where we're working out of a standard text book, research is just building on whats already been done. There may be a theorem that was discovered last month (and hence not in any text book) that helps me to proceed with a given format. But I won't know this by simply looking at a text book. So another part of it is looking back and forth at research journals and publications. Or as was the case with this weekend, I may need to look at other areas. This may provide me with a different way of looking at the problem that makes the question being asked more concrete.

But I do think that the way I learned to break down homework problems can be an effective way of doing research, but its a much more iterative process (hence the "re" in research).

Saturday, December 6, 2008

Blinded by Love

I was blinnnnnnd
dead by love
of the friendship we had built.
I had never
thought of you in that way
I had never
until that day
that we became more than friends.
It seemed so right
it seemed just like
it was meant to be
til that day when we
talked about dating seriously
I don't know why we disagreed
maybe it was something about me
or maybe you never liked me
but that breakup really hurt me.

I was blinnnnnd
dead by love
or was it love
or was it just us
was it just us
laying there in the bed
til the condom slipped off
and I lost my head
wondering if we should wed
til the condom slipped off
and you came and said
that your friend
missed a month.

I was blind
dead by love
by images of you and me
living happily
that made me
lose sight
of reality
my immagination ran away with me
you played my heart
had me write you
even after you had told me
it wasn't meant to be
I blamed me
wanted to do things differently
to go back in time
and make you love me.

Monday, December 1, 2008

Me and Math

I've just gotta say that I love math. I really don't hear that enough in my everyday life. It seems like many people around me, even in academia, see math as merely a means to an end, or as high school hurdle that they once stumbled over and thankfully never have to use again. Well, this is not a blog to convince anybody else to like math. Its just an expression of how much I like math.

I feel like I was introduced to math kinda by mistake. What I mean by 'math' here is the math I took after set theory. See, prior to this stuff, I pretty much hated math. I especially hated that we weren't allowed to use programming calculators on tests. See, I had somewhat of an affair with computer programming back in high school. This affair lasted until about my sophomore or junior year of college.

I always thought that math was stupid. My theory was who wants to sit and add up numbers all day? What would be the point of just using formulas that have already been discovered? I especially grew irritated when I would learn something like the Law of Sines, Law of Cosines, how to invert a matrix, or how to use row reduction to solve a system of equations. I would learn these things first by programming things into a calculator. Then I was told I couldn't use these programs on tests, which only frustrated my relationship with math. What's the point of using 20th century teaching methods to teach math to a 21st century classroom?

I remember entering college with a major in computer science. I was a pretty good programmer and this was the middle of the dot com era, when everybody was making money on the Internet. It just so happened that a computer science major at my undergrad consists of so many math classes that all I needed to have a math minor were a few classes - among these classes was set theory.

Set theory was a class that changed my life.

This class was like philosophy and debate for mathematicians, complete with paradoxes and all. But this was more structured than my debate classes in high school. In high school, we had topics like "Capital Punishment is Justified" and had to argue either the affirmative or the negative. The person who won was the person who made the better arguments or spoke more eloquently - not the one who spoke the truth (because truth here is ambiguous).

In math, its all about the truth! We assume only the basic necessities (which are agreed upon) and prove EVERYTHING else. There's no room for ambiguity and no room for errors. That's just beautiful to me.

I remember when I first proved the irrationality sqrt(2). All through middle school, I can remember being told that certain numbers are irrational, but never really understood what that meant or why it was true. When I first saw that proof, it was just an amazing sight. And the most beautiful thing is that each step followed logically from the one before it, and step one was just a statement of the assumption. There was no citing a survey, or claim that "correlation implies causation".

It was just pure beauty. And what made it more beautiful was its simplicity.

I don't often get to talk with people who share this same love of math, but when I do get such an occurrence I sit and smile.

Today, I'm smiling!
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