WHERE MATHEMATICS MEETS THE INTERNET

Introduction The Internet has experienced a fascinating evolution in the recent past, especially since the early days of the Web, a fact well documented not only in the trade journals but also in the popular press. Unprecedented in its growth, unparalleled in its heterogeneity, and unpredictable or even chaotic in the behavior of its traffic, “the Internet is its own revolution,” as Anthony-Michael Rutkowski, former executive director of the Internet Society, likes to put it. At the same time, folklore has it that mathematics lies at the heart of Internet operation. After all, the argument goes, mathematics is the language of computers, and the Internet is currently connecting tens of millions of them and still doubling every year [Lo98]. Yet the Internet is a new world, one where engineering reality wins over tradition-conscious mathematics and requires “paradigm shifts” that favor a combination of mathematical “beauty” and high potential for contributing to pragmatic Internet engineering. In this article we take a look at how the Internet differs in fundamental ways from the conventional voice networks, how the (r)evolution of the Internet is impacting the world of mathematics in the small as well as in the large—both on how mathematics is done and, for understanding the network itself, on what sort of mathematics is done—and why this, in turn, makes Internet engineering a gold mine for new, exciting, and challenging research opportunities in the mathematical sciences.1