HANOVER, N.H. - An international group of mathematicians at Dartmouth College and other institutions have released a new online resource that provides detailed maps of previously uncharted mathematical terrain.

Images and an animated graphic are available at http://aimath.org/aimnews/lmfdb/images/

The "L-functions and Modular Forms Database," or LMFDB, is an intricate catalog of mathematical objects that maps out the connections between them. Both beautiful and functional like an atlas, the LMFDB reveals deep relationships in the abstract universe and provides a guide to previously uncharted territory that underlies current research in several branches of physics, computer science, and mathematics. This coordinated effort is part of a massive collaboration of researchers around the globe, and includes developing new algorithms and performing large calculations on an extensive network of computers.

An international group of mathematicians at Dartmouth College and other institutions have released a new online resource that provides detailed maps of previously uncharted mathematical terrain. Credit: American Institute of Mathematics

The scale of the computational effort that went into creating the LMFDB is staggering: a total of nearly 1,000 years of computer time spent on calculations by multiple teams of researchers.

"Our project is akin to the first periodic table of elements," says LMFDB project member John Voight, an associate professor of mathematics at Dartmouth. "We have found enough of the building blocks that we can see the overall structure and begin to glimpse the underlying relationships. The LMFDB provides a coherent picture of a vast web of interconnections in clear, explicit and navigable terms. The worlds being explored are ones of particular interest: they cross a wide variety of domains, guided by a network of conjectures at the cutting edge of mathematical research."

One of the great triumphs in mathematics of the late 20th century was achieved by British number theorist Andrew Wiles in his proof of Fermat's Last Theorem, a famous proposition by Pierre de Fermat that went unproven for more than 300 years despite the efforts of generations of mathematicians. The proof has been the subject of several documentaries and it earned Wiles the Abel Prize earlier this year. The essence of Wiles' proof was establishing a long conjectured relationship between two mathematical worlds: elliptic curves and modular forms. Elliptic curves arise naturally in many parts of mathematics and can be described by a simple cubic equation; they also form the basis of cryptographic protocols used by most of the major internet companies, including Google, Facebook and Amazon. Modular forms are more mysterious objects: complex functions with an almost unbelievable degree of symmetry. Elliptic curves and modular forms are connected via their L-functions. The remarkable relationship between elliptic curves and modular forms established by Wiles is made fully explicit in the LMFDB, where one can travel from one world to another with the click of a mouse and view the L-functions that connect the two worlds.

The LMFDB tabulates data which has been produced over many decades, and which is now available in one place in a unified format. The scale of the computational effort involved in the LMFDB is staggering: hundreds of years of computing time were involved in compiling the database along with thousands of hours of human effort. 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 to discover uncharted mathematical worlds.

"Experiments are at the heart of mathematical research. In the face of extraordinary complexity, underlying structure often emerges only after studying thousands of examples," says Edgar Costa, a LMFDB contributor and postdoctoral fellow at Dartmouth. "The LMFDB provides a common location for all of this data to be shared, allowing us to discover connections between our observations."

Many of these calculations are so intricate that only a handful of experts can do them, and some computations are so big that it makes sense to only do them once. The LMFDB provides a sophisticated web interface that allows both experts and amateurs to easily navigate its contents. Each object has a "home page" and links to related objects, or "friends." The LMFDB also includes an integrated knowledge database that explains its contents and the mathematics behind it.

"We are mapping the mathematics of the 21st century," says LMFDB project member Brian Conrey, director of the American Institute of Mathematics. "The LMFDB is both an educational resource and a research tool which will become indispensable for future exploration."

source: **Dartmouth College**