Hi, I'm Jeff VanderKam. Years ago, I did math contests galore around North Carolina (some people, including my wife, still remember me as the guy that threw a fit on stage at state MathCounts in '86--I'd like to think I've matured a bit since then). After high school I went to Duke, and these days I'm a fourth-year math graduate student at Princeton University. In May I should get my PhD, with a thesis of analytic number theory research done with my advisor, Peter Sarnak. Those of you who watched the "NOVA" episode about the proof of Fermat's Last Theorem may recognize the name--he was the one who spilled the beans about Wiles' proof before Wiles announced it. Incidentally, if you didn't see the "NOVA" episode, shame on you, it's fairly short on math but long on human interest stories. Plus it's a lot of fun to see really smart people get kind of goofily excited about something.
So what do I do these days? Math research. What's that? Good question. Usually it's looking for patterns in numbers/shapes/formulas, then trying to explain them (it's the explaining that we usually get excited about, mathematicians more than perhaps anybody else really want to know "why?"). Who cares? Another good question. The amazing thing is, this stuff matters a whole lot more than it ought to (people occasionally refer to this as the "unreasonable effectiveness of mathematics"). Mathematicians see it as their job to keep finding patterns and relationships, and they figure that some will be more useful than others, but it's never clear whether a given result will be useful to others, at the time or in the future.
A few examples of things, both of which bear some relation to mathematicians around my neck of the woods. Ever notice how many graphical images on web pages first transmit as "chunky" versions, then gradually accumulate more smoothness and detail? They're using something called "wavelets", created by mathematicians (including Ingrid Daubechies, here at Princeton) about twenty years ago. Turns out that they're the fastest way to transmit data of the sort that goes into pictures, although that wasn't their original purpose. While we're talking about web pages, ever notice that the Netscape start-up page talks about "RSA security"?
The RSA code methods are about as purely number-theoretic as you can get. At their heart is a relationship about divisibility that was first found by Fermat three hundred years ago (and, unlike his "Last Theorem", he actually proved this one). And he certainly wasn't working on the problem to find a code that computers couldn't break, he was just messing around with patterns.
What kind of math am I doing? Analytic number theory involves using techniques from calculus to answer questions about whole numbers. The classic problem, and the one that gave birth to the field 200 years ago (but still isn't completely solved) is this: How frequent are prime numbers? You know, the numbers (2,3,5,7,11,...) only divisible by themselves and one. Early on it looks like there's a lot of them, but if you keep counting, they get rarer and rarer. Gauss, who loved doing this kind of stuff, kept counting them, and discovered something rather startling: the number of primes less than a large number X is really, really close to X divided by the natural logarithm of X. So primes account for about 1/ln X of the numbers less than X. Like a good mathematician (or in his case, a great mathematician) he asked "Why?" After all, the natural log (the logarithm with base e = 2.71828...) only comes up in calculus, what can that possibly have to do with prime numbers? It took fifty years for Riemann to give an intuitive reason, another fifty for people to finish filling in the details, and we still don't know everything about prime frequency (in fact, Riemann's guess at "how far off is X/ln X from the actual number of primes?" is nowadays called the Riemann Hypothesis, and its various spinoffs are now probably the most important unsolved problems in mathematics today). So there it is again: Gauss was just messing around with patterns and numbers, and found something that has led to some of the most amazing and beautiful (and yes, applicable) mathematics of our time.
So how does one "do" mathematics? Despite what the general public thinks, math isn't just about being able to multiply numbers in your head quickly, or memorizing thirty digits of pi (I know good mathematicians who struggle to calculate 15% tips in restaurants). We've got computers and calculators for that. Mathematics is about finding structures and truths in the world of patterns, and explaining why they're there. So fiddle with numbers, draw patterns, tie knots, examine shapes--maybe you'll spot something we've missed!
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