An international group of mathematicians, including two researchers at the University of Vermont, has released a new kind of mathematical tool: an online atlas that provides detailed maps of previously uncharted mathematical terrain.
The "L-functions and Modular Forms Database," or LMFDB, exposes deep relationships and provides a guide to new mathematical landscapes that underlie current research in several branches of science, including quantum physics, computer science, and cryptography.
“It's a massive collaborative effort involving over 100 mathematicians from around the world,” said Christelle Vincent, an assistant professor of mathematics at UVM who has been working on the new atlas over many months. “It’s both beautiful and functional, shining light on surprising and profound relationships in the abstract universe of mathematics.”
Best of the collection
A staggering amount of computational effort went into creating the LMFDB: about a thousand years of computer time spent on calculations by multiple teams of researchers. 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.
For Vincent, who studies a kind of mathematical object called an elliptic curve, the new LMFDB database is “like a museum with all of our best specimens,” she said. “You can find rare and hard-to-produce items there—that can let a researcher or student study something they didn’t know existed or that would be impossible to reproduce on their own.”
The new project is also a bit like “the first periodic table of elements,” Vincent notes. The team, supported by the National Science Foundation and others, has been able to find enough of the building blocks that “we can begin to see tantalizing structures and find surprising and intriguing relationships,” she said.
Just like the elements in the periodic table, the fundamental objects in mathematics fall into categories. Those categories have names like L-function, elliptic curve, and modular form. These broad categories naturally divide into smaller subcategories, each with its own personality. Every object has connections to objects in other categories, which Vincent and other project members refer to as its “friends.” One of the major goals of the new project is to determine the friend network, and to understand how the quirky behavior of a particular object can influence its friends. “Our new database begins to show a clear picture of these many mathematical relationships,” Vincent notes.
“The LMFDB is really the only place where these interconnections are given in such clear, explicit, and navigable terms,” said Holly Swisher, a project member from Oregon State University. “Before our project it was difficult to find more than a handful of examples, and now we have millions.”
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. “Their equations can be weird,” UVM’s Vincent says, but they are tremendously powerful and useful, she notes. “They’re like Goldilocks; often they’re just right.” Because of their qualities, elliptic curves 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.
“We focused a lot of effort on the usability and accessibility of the new database,” Vincent says. “We believe it will help lead to the discovery of new mathematics.”
Toward unified theory
The LMFDB only reaches a small fraction of the mathematical universe. But the worlds being explored are ones of particular interest: they cross a multitude of areas, guided by a network of conjectures at the forefront of mathematical research.
For example, the connection between elliptic curves and modular forms is just a small part of the Langlands Program, a vast web of conjectures proposed by Robert Langlands, at the Institute for Advanced Study, in the late 1960s. The Langlands Program is enormous in scope, but also vague in some of its details: it serves as a framework for the types of relationships provided in the LMFDB database, but it does not describe the specific connections. The explicit nature of these connections is the subject of a great deal of current research, and in many cases they are now cataloged in the new database.
Another example arises from prime numbers, which have fascinated mathematicians for millennia. The distribution of primes appears to be random, but mathematicians have yet to prove this conclusively. Under the Riemann hypothesis—perhaps the greatest unsolved problem in mathematics—the distribution of primes is intimately related to something called 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. Tantalizingly, patterns found in the study of these L-functions also arise in complex quantum systems, and there is a conjectured to be direct connection of this area of mathematics to quantum physics.
Christelle Vincent and Taylor Dupuy, both members of the Department of Mathematics and Statistics at UVM, have been deeply focused on this new database effort over the last year. “As an avid user of the LMFDB in my own research I felt that it was incumbent on me to help out in whatever way I could to make the database accessible to others,” Vincent said. “I really see this project as being a great example of the community spirit of mathematicians. We love to help each other, give back, and share what we have with others.”