BackManjul Bhargava and the magic of mathematics
Everything about Manjul Bhargava suggests love for mathematics. To me, it is fundamental to why he was awarded the Fields Medal
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For a few years, I was the resident magician at my son’s birthday parties. That stopped when the kids realized I was repeating the same three tricks. (Funny how you can’t pull wool over the average four year-old’s eyes).
Still, I always thought one of those three was a pretty good trick. You ask someone to memorize a card out of a pack, then without ever looking at the faces of the cards, you deal them into three piles, thrice. After the third time, still without looking, you pick out the memorized card. Except with jaded four year olds, it always sets off delighted gasps.
What makes it work? No sleight of hand, but just simple mathematics. And that’s why I’m thrilled to find that one of the winners of the 2014 Fields Medal—the big prize in mathematics—teaches a variant of this trick in an entry-level mathematics course. In fact, it’s apparently a whole semester worth of magic: the course is called the Mathematics of Magic Tricks and Games. What a route to learning and loving mathematics.
Manjul Bhargava is the professor I mean, a number theorist at Princeton University. His purpose with that course is to show his students that mathematics is not at all the clinical and dull by-rote subject so many of us come to see it as in school. Instead, it is fun. It is fabulously creative. It is, believe it or not, an art. If more of us start seeing it that way, less of us will be frightened of the subject. There’s zero reason to be scared of numbers, but every reason in the world to grow to love them.
Everything about Bhargava suggests this love for his subject. To me, it is fundamental to why he was awarded the Fields Medal.
Bhargava works with exotic mathematical objects called hyperelliptic curves. One particular breed of these curves are elliptic curves, and these turned out to be crucial to solving Fermat’s famous Last Theorem (see my column Margins of a theorem, 19 August, 2011). In fact, the man who solved the Last Theorem in 1993, Andrew Wiles, was Bhargava’s PhD thesis advisor.
Now it’s a good bet that you used elliptics today. They are useful in cryptographics, the science (or in the spirit of Bhargava’s work, call it an art) of making codes. Every time you do a credit card transaction online, your card number is encoded so it cannot be stolen—and that process uses elliptic curves. Bhargava once had this to say about these curves: “Intellectual stimulation, beautiful structure, applications—elliptic curves have it all."
Take note of that word beautiful.
One important problem with hyperelliptic curves has to do with rational numbers. These are numbers that can be written as a fraction, such as 2/3, or 5/1, or 365/413.
In contrast, irrational numbers, such as pi or the square root of 2, cannot be reduced to a fraction. The problem is this: given the mathematical equation that describes a hyperelliptic curve—a hyperelliptic equation—how many rational solutions does it have? In other words, if you draw that curve on a graph paper, how many points will it pass through that have rational coordinates? None? One? Two? Four? An infinity of them?
For example, you could draw a circle to pass through a particular rational point. A little thought will persuade you that the point diametrically opposite it will also be rational. Are there more such? How many?
Turns out these are hard questions to answer for hyperelliptic curves. And that difficulty is the reason they are useful for online security. For in effect, an aspiring cyber thief will have to answer that hard question.
But Bhargava has found ways to learn about the relationship between hyperelliptic curves and their rational points. (Famously, one method involved a Rubik’s Cube, which I’ll leave you to read about).
For example, he has shown that as the curve gets more complex, or as the degree of its equation increases, the chance of it passing through rational points decreases dramatically.
This is already too arcane for this column, I realize. But the remarkable thing about Bhargava, for me, is not so much that he has made real advances in areas of number theory that have defeated mathematicians for decades. Instead, it is in the ingenuity and passion he is known for that he brings to his work.
Awarding him the Fields Medal, the International Mathematical Union put it eloquently: “A mathematician of extraordinary creativity, he has a taste for simple problems of timeless beauty, which he has solved by developing elegant and powerful new methods that offer deep insight…He surely will bring more delights and surprises to mathematics in the years to come."
Honestly, how many of us expect to see words like creativity, timeless, elegant, insight and delights in relation to mathematics and its practitioners?
Yet delights and surprises are exactly what we should expect from a man who calls elliptic curves—those purely mathematical animals—beautiful.
Me, I want to enrol in that course about magic tricks.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences.
Comments are welcome at dilip@livemint.com. To read Dilip D’Souza’s previous columns, go to www.livemint.com/dilipdsouza
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