Pure and Applied Mathematics
One of the world's leading number theorists, working on Hilbert's 12th problem.
"Somewhere out there is a theory that would explain my empirical observations, and this theory has yet to be discovered. Mathematics thrives on such mysteries."
Henri Darmon is a mathematician specializing in number theory. Many scientists feel that mathematics is the purest of all the sciences, and most mathematicians will agree that number theory is the purest form of mathematics. Unlike other sciences, mathematics requires no laboratory experiments or observations of the natural world. It rests on its own internal consistency and logic, requiring nothing more than the intellect of the human mind. As one mathematician observed, it’s the only job for which you can lie down on a sofa, close your eyes and get to work.
What are number theorists doing when they close their eyes? They imagine how numbers relate to each other. They see patterns. They invent formulas and equations that capture these patterns and amplify them. Have you ever wondered about the curious features of the nine-times table (9, 18, 27, 36, 45, 54, 63, 72, 81)? Notice that the digits for any of these numbers always add up to 9 (1 + 8 = 9, 2 + 7 = 9, et cetera). Note the symmetry of the series: After you hit 45, the numbers become mirror images of the ones that came before (45/54, 36/63, etc.). This is a very simple example of what number theorists think about, but they go much deeper than this. They want to know why the nine-times table behaves like this. (It has to do with our counting system having 10 digits, including zero.) They like to figure out what these patterns and relationships mean.
While they rarely look for practical applications, number theorists have provided many useful tools for all the other sciences. For instance, by picking a counting system that uses two digits (0 and 1) instead of 10, mathematicians developed a way for machines to do mathematics. It’s called binary arithmetic and is the basis of all computer language. Usually, number theorists are not looking for ways to use numbers for practical things. They are just fascinated by the amazing natural relationships that can be discovered in the world of numbers.
To understand, Darmon suggests the simple exercise of creating a series of numbers adding up the reciprocals of all the squares. So the first one would be 1/12 which is just 1 and the second term would be 1/22 which is 1/4 to get: 1 + 1/4 + 1/9 + 1/16 + 1/25 ... and so on.
Each added fraction increases the total, but it’s smaller than the one before, so the sum approaches but never quite reaches a limiting number. This number turns out to be π2/6 (pi squared, divided by 6, or 1.64493406685…). How did pi arise from something completely unrelated to anything circular? And how can you prove this relationship? Questions like these fascinate number theorists.
Like adventurous explorers seeking new terrain, mathematicians are always looking for newer and more fascinating number patterns. Darmon thinks of the world of numbers as a natural wilderness, like the Arctic tundra or the Amazon rainforest. Just as these places on Earth have abundant plant and animal relationships, the number world is a place loaded with connections, patterns, structure and even animals of a sort. But you don’t explore it in a boat or a plane. You go there in your mind.
In the 1960s a couple of English mathematicians named B. Birch and H. Swinnerton-Dyer formulated a conjecture based on many complicated counting exercises they had done. Their conjecture, if it were ever proved, would provide a systematic mathematical method (an algorithm) to calculate the set of solutions to elliptic curve equations that describe Abelian fields. In May 2000, to set the stage for a century of mathematical problem solving, the Clay Mathematics Institute in Cambridge, Massachusetts, offered a US$1-million prize for the solution to each of seven big problems, including the Birch and Swinnerton-Dyer conjecture.
The Clay Institute was echoing an event that occurred in May 1900, when one of the greatest mathematicians of all time, David Hilbert of Germany, gave a famous lecture in Paris in which he laid out 23 extremely challenging problems for mathematicians of the 20th century. The 12th one was a question about generating Abelian fields. More than 100 years later, Hilbert’s 12th problem remains unsolved, but Henri Darmon is making progress toward a solution.
Put simply, Hilbert asked, “How do you get an effective and efficient method for constructing all the Abelian extensions containing a given field?” Years later, commenting on the problem, Hilbert said, “The theory of complex multiplication (of elliptic modular functions), which forms a powerful link between number theory and analysis, is not only the most beautiful part of mathematics but also of all science.”
So Henri Darmon’s pursuit is noble indeed. If he can find what he’s looking for in the mathematical wilderness, according to Hilbert he will affect all of science. He might also become a millionaire. A few years ago Darmon proved a connection between Hilbert’s 12th problem and the Birch and Swinnerton-Dyer conjecture. And it all hinges on clever manipulation of elliptic curves.
On the practical side, patterns generated by elliptic curves provide a very efficient way to encode and decode information. Such encryptions are small and fast to calculate, so they are ideal for quick transactions with credit cards, ATMs and online shopping.
The number system starts with the natural numbers: 1, 2, 3, 4 and so on. Next, add zero and the negative numbers to get the integers. Between each integer are fractions. Mathematicians call these the rational numbers. Each bigger group includes the smaller ones. After fractions come numbers that cannot be expressed simply as one number divided by another, like the square root of 2. This includes imaginary numbers, required to explain numbers whose squares are negative. Collectively they are called irrational or algebraic numbers. Beyond these are more exotic numbers like pi (3.14159 … ) or e (2.17828 … ), numbers with simple logical relationships — for example, pi is the ratio of a circle’s radius to its circumference — but they cannot be expressed as ratios of whole numbers. They go beyond ordinary counting, so mathematicians call them transcendental numbers.
Symmetry: Just as biologists catalogue plants, mathematicians categorize numbers. A very simple example would be all the even numbers, but a better tool is the number field, a collection of numbers that is preserved under addition, subtraction, multiplication and division and obeys rules involving symmetries like the multiplicative inverse (a x b is the same as b x a).
Number theorists measure the complexity of a number field by the amount of symmetry it possesses. Consider a square and a rectangle. You can rotate a square by any multiple of 90 degrees and flip it around any of its four axes of symmetry for a total of eight possible transformations. But a rectangle, because of its unequal sides, has only four such ways of flipping and rotating. The figure depicts the number of symmetrics below each shape.
Darmon works with a kind of number field that obeys a special kind of symmetry. When two symmetries are applied to an Abelian field, they can be applied in any order and the outcome will be the same; like flipping, then rotating, the square above. In a sense, Abelian fields are the simplest class of number systems because they have this “symmetry of symmetries.” Mathematicians would like to find equations that describe all the possible Abelian fields and it turns out that elliptic curves can help.
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Solutions to many mathematics problems are waiting to be discovered. Hilbert’s 12th problem, the Birch and Swinnerton-Dyer conjecture and other millennium problems from the Clay Institute remain unsolved.
Amir Aczel, Fermat’s Last Theorem
, Delta, 1997. An easy to understand history of all the math behind Wiles's proof of Fermat's last theorem.
To learn more about elliptic curves, try Ed Eikenberg’s website at the University of Maryland.
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