A team of mathematicians from 12 countries has begun charting the terrain of rich, new mathematical worlds. The mathematical universe is filled with both familiar and exotic items. The “L-functions and Modular Forms Database,” abbreviated LMFDB—a sophisticated web interface that allows both experts and amateurs to easily navigate its contents.
According to Benedict Gross, an emeritus professor of mathematics at Harvard University, “Number theory is a subject that is as old as written history itself. Throughout its development, numerical computations have proved critical to discoveries, including the prime number theorem, and more recently, the conjecture of Birch and Swinnerton-Dyer on elliptic curves. The LMFDB pulls together all of the amazing computations that have been done with these objects.
Prime numbers have fascinated mathematicians throughout the ages. The distribution of primes is believed to be random, but proving this remains beyond the grasp of mathematicians to date. Under the Riemann hypothesis, the distribution of primes is intimately related to the Riemann zeta function, which is the simplest example of an L-function. The LMFDB contains more than twenty million L-functions, each of which has an analogous Riemann hypothesis that is believed to govern the distribution of wide range of more exotic mathematical objects. Patterns found in the study of these L-functions also arise in complex quantum systems, and there is a conjectured to be direct connection to quantum physics.
A recent contribution by Andrew Sutherland at MIT used 72,000 cores of Google’s Compute Engine to complete in one weekend a tabulation that would have taken more than a century on a single computer. The application of large-scale cloud computing to research in pure mathematics is just one of the ways in which the project is pushing forward the frontier of mathematics.