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The L-functions and Modular Forms Database, which Professor Nathan Ryan helped conceive nine years ago, and to which Professor Sally Koutsoliotas has contributed extensively, is a powerful tool for collaboration in mathematics.

*Image courtesy of the American Institute of Mathematics*

Nine years ago, Bucknell Professor Nathan Ryan, mathematics, was toying with an idea. That summer Ryan, who was then finishing a postdoctoral fellowship at UCLA before joining Bucknell in the fall, was with four other mathematicians organizing a workshop at the American Institute of Mathematics. Their plan was to lay the groundwork for a grand collaborative project: an encyclopedia of mathematical objects linking together disparate fields of mathematics. Not the least of the challenges standing in their way was the highly individual and frequently cloistered nature of their field.

"It was designed from the beginning to be beyond one person," Ryan said. "Because these different regions of the mathematical 'map' that we're trying to lay out are on the surface utterly different, you need people who can understand them."

Nine years later, the egalitarian collaborative group — which includes Ryan and Professor Sally Koutsoliotas, physics, and more than 70 other researchers in 12 countries — has unveiled its creation, a web database that moved out of beta development in May. The L-functions and Modular Forms Database (LMFDB) not only catalogues thousands of mathematical objects arising in number theory, but also the seemingly incongruous bonds between them.

"The hallmark of the LMFDB is connecting mathematical objects that are hard to connect," Ryan said. "It's not just linking web pages; the idea is that there is a philosophical underpinning, and figuring out where these mathematical objects fit into that philosophical structure is rigorous mathematical work."

The philosophical structure connecting these objects is the Langlands Program, a vast web of conjectures proposed by mathematician Robert Langlands in the 1960s that provides a framework for the types of relationships found in the LMFDB, called L-functions.

"Each object has a characteristic that happens to be an L-function, which goes along with and describes the object," Ryan explained. "But it turns out that an L-function can be shared by different objects, so there can be a connection between two once disparate objects through this underlying 'DNA.' "

Patterns and relationships are crucial to the study of mathematics, but the patterns and relationships between many mathematical structures are difficult to write down because of their sheer complexity. Providing mathematicians a database of those relationships makes it easier for them to make those observations, some of which would not have become apparent without the LMFDB.

"Each of these little pockets of study has a mathematical community of experts in it," added Koutsoliotas. "Using their own specific methods, groups perform calculations that can take hundreds of CPU hours. But their data are interesting to the broader research world, and the LMFDB brings all these pieces together."

Ryan gives the example of a graph plotting the location of zeroes in 10,000 different L-functions, created by LMFDB collaborator Michael Rubinstein of the University of Waterloo, Canada. A mathematician historically might have plotted a single row or column of the graph, and could not have observed the patterns that emerge in the larger field.

"There are fuzzy bands — I see clear striations, an accumulation of gray spaces," Ryan said. "There are clear gaps where different things happen. Looking at a row or column of this, you would never have seen the patterns that I see."

The scale of the computational effort that went into creating the LMFDB is staggering. Nearly a thousand years of computer time was spent on calculations, some of them at Bucknell's Linux Computing Cluster. One recent computation used more than 72,000 processing cores of Google's Compute Engine to calculate in one weekend what would have taken more than a century on a single computer.

But for Ryan, what's just as staggering is the collaborative effort that the project has involved, something that he said is still extremely unusual in mathematics. While scientific fields such as physics and the life sciences have embraced collaboration, math is still very much an individual pursuit.

"That idea has shifted in the last few years," Ryan said. "Having coauthors on a math paper is less scandalous than it was 15 years ago, but something of this scale — taking a modern crowd-sourced approach to building this giant tool — is still very unusual."

"In other collaborations, there is often a hierarchy," Koutsoliotas added. "There are spokespeople for the experiment and senior scientists. With the LMFDB collaboration, nothing gets approved unless by consensus. And the thing is, math trumps everything."

The LMFDB also aims to promote a broader understanding of the objects and connections it catalogues. While it's most useful for mathematics researchers, the LMFDB also offers in-text explanatory links, called knowls, to help the less-familiar identify concepts they do understand, and build out their knowledge.

"It's still intellectually honest and rigorous, but if you dig down deep enough we hope many people will end up finding things that they do understand," Ryan said.

For that reason, Ryan and Koutsoliotas said they believe the database will prove to be useful as an instructional tool. More than 100 academic papers have been written in conjunction with the LMFDB, and both professors have helped undergraduates at Bucknell use it for research.

JT Ferrara '17, a five-year math and electrical engineering major who just completed his fourth year, worked with Koutsoliotas on an empirical attempt to answer a long-standing conjecture in number theory known as the excess rank of elliptic curves.

"The LMFDB was crucial to this project because it provided us all the necessary data to develop these models, and as a student working on them, I also found the LMFDB to be a great resource for learning about the theory lying behind our calculations," Ferrara said. "It was a great experience and exciting to work on a research project that was part of such a large-scale, collaborative effort."

The LMFDB project is funded by the U.S. National Science Foundation, the U.K. Engineering and Physical Sciences Research Council, the American Institute of Mathematics and the EU 2020 Horizon Open DreamKit Project.

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