Mathematical tools.

Madison.

I've been thinking about mathematics a fair bit¹ lately. This is due, for the most part, to the fact that the New York Times noted the Fields Medal awards today. This is, in case you weren't up on your high-level scientific honors, the mathematical equivalent of a Nobel Prize. Though a number of prominent mathematicians have been awarded Nobel Prizes, it's always been for the application of their research to other fields, like Economics or Physics. The Nobel - with the exception of Literature - seems like awards for practical results, after a lifetime of work; the Fields Medal² only goes to young researchers (under 40), and doesn't need to apply to anything particularly useful.

Useful to you and me, the average person, that is. Fields Medalist-level research is big, big stuff in the world of mathematics. Potentially physics, chemistry and other related fields, too.

The big deal this time around - Fields Medals only come once every four years - is that Grigory Perelman has officially solved the Poincaré conjecture. (Actually, he did it about three years ago, but the verification of his work has taken quite some time.) Without getting into the details³ of it, he's managed to solve a central, nagging problem in topology. Henri Poincaré posited it in 1904; a century later - using a novel combination of mathematical tools unavailable to Poincaré - a proof exists.

What caught my eye in all this was that the Poincaré conjecture is⁴ one of the Millennium Prize Problems, a group of seven unsolved classical mathematical problems. The Clay Mathematics Institute, at the turn of the millennium, set out seven one-million dollar prizes for published, peer-reviewed solutions to these problems. One example is the Navier-Stokes equations, which is used to describe fluid flow. I studied them in college. The equations have been around since the 1800s, and have been useful in all sorts of applications. The difficulty lies in the fact that no one really understands them, or can really use them without making a lot of assumptions⁵ along the way.

So, math is much more intense than most of us give it credit for. This probably applies to most fields of study, I suppose, but math gets a pretty bad rap all around. I think that much of this is due to the fact that the average brush with mathematics is boiled down to simple process. From grade school arithmetic through multivariable calculus and differential equations, nearly every math course⁶ I took in my years of school was based on solving problems, in search of an answer. Usually something numerical.

At the dentist's office⁷ yesterday, the hygienist commented that she really liked math. It was straightforward, she said, and you could always work backwards to check your answer. Even if I didn't have a mouthful of dental implements, I wouldn't have the heart to correct her. I used to think about it that way, which, when you're studying engineering, is a sufficient sort of understanding. Not exaclty enlightened, but practical enough. Since then, though, I've started to realize the complexity of the whole field.

I'm beginning to think of mathematics as a tool. A powerful one, like a good hammer. Everyone has some conception of a hammer; nearly everybody's used one at some point. It's something you'd expect to find in any toolbox.

For those who don't use one regularly, a hammer's a basic brunt-force type of object. You tap a nail in lightly, then try to whack it in with all your might, ideally avoiding any sort of thumb-crushing. Don't pay attention to how you're using it, and you'll end up with a sore arm and wrist. Every so often, a glancing blow bends the nail over, and you've got to start again. This is math for most of us. We'll use it when we have to, but we're no good at it, and it's probably a crappy hammer, anyway.

Really, though, a hammer takes some skill. Some practice. As simple a tool as it may be, it demands your respect. A skilled carpenter can sink nails much faster, much easier and with greater accuracy than most could with a pneumatic nail gun. The tool, more like an extension of the arm, can do a phenomenal amount of work with very little effort. Sure, building a house with hand tools takes a long time, but so does solving the Poincaré conjecture. Not many of us could really do either in our lives.

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¹Despite the fact that I'm married to a mathematician, and that a good number of my friends are mathematicians, I don't spend much of my day thinking about it. In fact, I tend to glaze over when the subject comes up.

²You could also think of it like the MacArthur Genius Grants. We like what you're doing; here's a pile of cash, no strings attached.

³Since I'm a bit fuzzy on the problem, in general. The details are far beyond me. "Coffee cups = donuts" is about as far into topology as I get.

⁴Was?

⁵The first case, if I recall correctly, was to assume that the fluid wasn't moving, in order to predict pressure changes at varying depths of a fluid. When nothing's moving, all sorts of terms become zero and drop out, which makes calculations much easier.

⁶9th grade geometry had some very basic proof work in it, but even that was directed at a single right answer.

⁷Always remember to brush and floss regularly. Your dental hygienist will thank you.