A team of more than 80 mathematicians from 12 countries is charting the terrain of rich, new mathematical worlds, and sharing their discoveries on the Web. The mathematical universe is filled with both familiar and exotic items, many of which are being made available for the first time. The “L-functions and Modular Forms Database” abbreviated LMFDB, is an intricate catalog of mathematical objects and the connections between them. Making those relationships visible has been made possible largely by the coordinated efforts of a group of researchers developing new algorithms and performing calculations on an extensive network of computers. The project provides a new tool for several branches of mathematics, physics, and computer science.
Stephan Ehlen, a CRM-ISM Postdoctoral Fellow at McGill University working on Number Theory, is one of the more than 80 mathematicians contributing to the project in various ways. His research is closely related to modular forms. These are special functions that have a lot of symmetries (see example) and often count important quantities in Number Theory (like the number of ways to represent an integer as a sum of 4 squares).
Stephan has computed data about (holomorphic) modular forms together with Fredrik Strömberg from the University of Nottingham (UK) and they co-wrote the majority of the code that runs that part of the website. Stephan is one of the two organizers of a workshop that happens this week at the American Institute of Mathematics where around 15 researchers are already working hard on expanding the “Universe” of the LMFDB to include many more objects that Number Theorists are excited about.
Just like explorers, mathematicians seek to discover paths between apparently unrelated areas. Such discoveries can lead to breakthroughs when the connections are made explicit. For example, in the 17th century René Descartes forged a revolutionary connection between geometry and algebra. What was once innovative is now routine: every student learns formulas for lines and circles. The mathematical frontier has steadily expanded since the time of Descartes, with many diverging paths as researchers follow different sets of clues. Some of those trails may be leading toward the same destination, but unless they communicate with each other, the explorers on those paths may not realize how close they are.
The LMFDB team includes mathematicians from more than a dozen research areas, all of whom are building connections between their seemingly separate specialties. “Most of us are aware of these relationships in an abstract way, but it takes real work to actually figure out all the details,” said Andrew Booker from the University of Bristol in the UK. These details are made available on the LMFDB website, for everyone to explore and perhaps discover something new.
A “periodic table” of mathematical objects
Project member John Voight, from Dartmouth College, observed that “our project is akin to the first periodic table of the elements. We have found enough of the building blocks that we can see the overall structure and begin to glimpse the underlying relationships.” Similar to 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. The L-functions play a special role, acting like ‘DNA’ which characterizes the other objects. More than 20 million objects have been catalogued, each with its L-function that serves as a link between related items. Just as the value of genome sequencing is greatly increased when many members of a population have been sequenced, the comprehensive material in the LMFDB will be an indispensible tool for new discoveries.
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.” Holly Swisher, a project member from Oregon State University, commented that the friends links are one of the most valuable aspects of the project: “The LMFDB is really the only place where these interconnections are given in such clear, explicit, and navigable terms. Before our project it was difficult to find more than a handful of examples, and now we have millions.”
A “Grand Unified Theory” of mathematics
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.
One of the great triumphs in mathematics of the late 20th century was achieved by Sir Andrew Wiles in his proof of Fermat’s Last Theorem, a famous proposition by Pierre de Fermat that went unproved 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 this year. The essence of Wiles’s 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 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, 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 LMFDB.
Experimental mathematics for the 21st century
Mathematics has always been an experimental science: conjectures are formulated and tested based on evidence of all kinds. As we enter a new century of research, large-scale computer experiments now take the place of hand calculations, with the effect of accelerating the process of testing and discovery. John Jones, from Arizona State University, described this motivation for the project: “Many of us have made extensive computations, and we wanted to make this data available to other researchers and to link these projects together to aid mathematical progress. By joining forces, we now have a site for one-stop shopping of big data.” Many of these calculations are so intricate that only a handful of experts can do them, and some computations are so big and take so long that it makes sense to only do them once. The LMFDB also includes an integrated knowledge database that explains its contents and the mathematics behind it. Project member Brian Conrey, Director of the American Institute of Mathematics, thinks that the LMFDB approach is the wave of the future: “We are mapping the mathematics of the 21st century. The LMFDB is both an educational resource and a research tool which will become indispensable for future exploration.”
Benedict Gross, an emeritus professor of mathematics at Harvard University, had this to say: “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. During the past fifty years, modular forms and their L-functions have taken center stage at the forefront of number theory. The LMFDB pulls together all of the amazing computations that have been done with these objects. Having this material accessible in a single place will provide an invaluable resource for all of us working in the field.”
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.
The scale of the computational effort involved in the LMFDB is staggering: hundreds of CPU years were involved in compiling the databases, requiring thousands of hours of human effort.
A recent computation by Andrew Sutherland at MIT used 72,000 cores of Google’s Compute Engine to complete in hours a tabulation that would have taken more than a century on a single computer. As noted by Sutherland, “computations in number theory are often amenable to parallelization, and this makes it easy to scale them to the cloud.” 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.
A team effort
Mathematics has traditionally been viewed as a solitary activity, with most research papers in the 20th century written by a single author. The LMFDB collaboration involves more than 70 mathematicians from around the world. Activity on the project generally takes place during week-long workshops where the participants cycle between individual work, working in small groups, and large group discussions. The group has no leader — all decisions are made by consensus at workshop meetings. Project member John Cremona, at the University of Warwick, UK, describes the approach: “We have a principle that you can’t argue with mathematical facts, so many discussions end when someone explains the relevant mathematics.” Such explanations are necessary because the project covers a wide range of specialty areas, and each person only understands a small part of the overall project. Alina Bucur at UCSD notes, “This kind of team activity is also a lot more fun! The efforts of individuals do more than add together: the result is much more than the sum of its parts.”
A landscape of landscapes
The LMFDB charts a world populated with mathematical objects, and some of those objects have landscapes unto themselves. The objects called L-functions occupy a special place in the LMFDB because most of the inhabitants of the LMFDB world have a friend who is an L-function. Each L-function has its own separate terrain which can be viewed as a 2-dimensional map. An L-function landscape contains infinitely many valleys but no peaks: if you start wandering in some random direction the ground goes up and down but you will never find an isolated peak.
The greatest unsolved problem in mathematics, the Riemann Hypothesis, is a statement about the valleys in the terrain of an L-function. The Riemann Hypothesis is the conjecture that all the valleys in an L-function landscape lie along one longitude line. Most experts suspect that the Riemann Hypothesis is true, although the collective efforts of mathematicians over the past 150 years has hardly made a dent in the problem. The Riemann Hypothesis has profound implications, and there are hundreds of mathematical results of the form “If the Riemann Hypothesis is true, then the following important consequence is also true…”.
Since there are infinitely many valleys of infinitely many L-functions, the Riemann Hypothesis could never be checked by a computer. However, LMFDB participants have charted billions of valleys on millions of L-function landscapes, and so far all of them are on the line predicted by Riemann. The coordinates of those valleys are called “zeros,” because the height of the L-function is zero at that point. If an L-function was covered with altitude markers, there would be a “0″ at the bottom of each valley. A huge database of zeros is available on the LMFDB website.
The most basic L-function is known as the Riemann zeta function, named after the German mathematician Bernhard Riemann who described its properties in 1859. Riemann found the first 4 zeros of his zeta function by a pen-and-paper calculation, and in 1953 Alan Turing used one of the earliest electronic computers to locate more than 1100 of them. Much recent research has focused on the pattern of the zeros: are they randomly spaced along their line, or is there some underlying music? The data provided by the LMFDB is invaluable for researchers studying these questions because it isn’t necessary to go to the time and expense of locating the zeros: researchers can just focus on making new explorations.
The LMFDB team has discovered many new L-functions with surprising properties. The Riemann zeta function has an unusually large gap between its first two zeros: they are separated by a distance of 28.26 units. It had been proven that for a large class of L-functions, the type of L-functions which have been studied most carefully, none could have a bigger gap. Therefore it came as a surprise when team member Stefan Lemurell, from Gothenburg University in Sweden, found an L-function which had a gap of 28.99 units. That might not sound much bigger, but breaking a psychological barrier can alter one’s view of the world. Lemurell’s example was greeted by skepticism, particularly since he was dealing with an exotic type of L-function and few people understood the techniques necessary to investigate his result. Additional checking by other researchers confirmed his calculation, and now mathematicians realize there is a greater variety in the possible arrangements of the valleys in an L-function landscape. Lemurell’s example is the only known instance with a bigger gap than the Riemann zeta function, but researchers suspect there are more examples out there to be discovered.
The shadow of zeta
Interesting L-functions also come from high rank elliptic curves. One of most studied properties of an elliptic curve is its “rank,” a measure of how many solutions it has. When team members looked at the L-function of a high rank elliptic curve, they found huge ridges in the landscape. Those ridges were surprising, because the terrain of an L-function tends to be gently undulating. On looking closer, they realized that the ridges were located near to the valleys of the Riemann zeta function. It was as if the two L-functions were trying to be the opposite of each other. This happened in numerous cases: all the L-functions of high rank elliptic curves were trying to have ridges where the zeta function had valleys. The underlying cause of this phenomenon was found by team member Michael Rubinstein, from the University of Waterloo in Canada, who published a research paper providing an explanation. This was another instance where researchers had to adjust their views on the zeros of L-functions. It is now understood that Riemann’s original zeta function casts its shadow over the landscape of every L-function.
One of the most remarkable discoveries of recent years concerns patterns in the distribution of the valleys of the Riemann zeta function and of the other L-functions. It turns out that amongst all of the L-functions computed, one finds only three underlying valley patterns. This is highly surprising. These universal patterns are believed to be related to symmetries of the L-functions that are still not fully understood. Furthermore, the three patterns that appear are precisely those found in complex quantum systems, such as electrons moving in micro-circuits and disordered superconductors. This connection between L-functions and quantum mechanics is deeply mysterious. One achievement of the LMFDB project has been to determine what these valley-patterns tell us about fluctuations in the distribution of the primes, and about the likelihood of finding extremely high ridges in the L-function landscapes.
In addition to being a tool for high-level research, the LMFDB is also a resource for advanced students. This is accomplished by the “knowl.” The knowl, pronounced “nohl” was invented by team member Harald Schilly, a graduate student in mathematics at the University of Vienna, Austria.
Unlike a hyperlink or a pop-up, a knowl inserts new material inside the page you are viewing. It is ideal for providing a small amount of supplementary information, such as a definition or a brief biography. Schilly coined the name to represent “a small piece of knowledge.” Many online resources, particularly online textbooks and Wikipedia, would be vastly improved if they replaced most of their hyperlinks with knowls. Knowls are easy to add to any website, and the software is available for free.
History of the LMFDB
The LMFDB tabulates data which has been produced over many decades, and which is now available in one place in a unified format. The idea of pooling the computational results of researchers in several areas of mathematics was started at a workshop at the American Institute of Mathematics in 2007. The actual work on the LMFDB was started at a workshop supported by the National Science Foundation (NSF) in 2010. The majority of the work on the LMFDB was done at subsequent workshops supported by the NSF and by the Engineering and Physical Sciences Research Council (EPSRC) in the UK through a Programme Grant awarded jointly to Warwick and Bristol Universities, as well as at long-term programs at the Mathematical Sciences Research Institute (MSRI) in Berkeley, CA, and the Institute for Computational and Experimental Research in Mathematics (ICERM) in Providence, RI.
The official release of the LMFDB will take place on May 10 at the following events:
- A workshop at the American Institute of Mathematics, in San Jose, California
- A public talk and reception at Dartmouth College, in Hanover, New Hampshire, including a webcast.
- A workshop at the University of Bristol, UK, sponsored by the Heilbronn Institute.
Technical details and and additional resources
Visit the project online at www.LMFDB.org
LMFDB team members have been involved in resolving two more instances of the Langlands Program beyond the case proved by Wiles. Samir Siksek and his collaborators proved the modularity of all real quadratic elliptic curves. And in a feat that combined both theoretical work and intricate computer algorithms, a team of four people proved that the hyperelliptic curve of conductor 277 is modular. Experimental evidence for many more cases is provided by the links on the LMFDB website.
More than 100 research papers reference the data in the LMFDB.
About the American Institute of Mathematics
Since 2002 AIM has been part of the National Science Foundation (NSF) Mathematical Sciences Institutes program. AIM receives funding from NSF to hold weeklong focused workshops in all areas of the mathematical sciences. In 2007 a program called SQuaREs which brings small research groups to AIM was developed. Each year twenty workshops are hosted at the institute and over thirty small research groups. The mission of AIM is to advance mathematical knowledge through collaboration, to broaden participation in the mathematical endeavor, and to increase the awareness of the contributions of the mathematical sciences to society.
David Farmer, AIM Director or Programs, 570 238-3290, farmer [at] aimath.org
Brian Conrey, AIM Director, 408-350-2088, conrey [at] aimath.org
Holly Swisher, Associate Professor of Mathematics, Oregon State University, 541-231-3257, swisherh [at] math.oregonstate.edu
John Jones, Professor of Mathematics, Arizona State University, 480-643-9734, jj [at] asu.edu
Kiran Kedlaya, Stefan E. Warschawski Chair in Mathematics, University of California, San Diego, 858-534-2629, kedlaya [at] ucsd.edu
Jon Keating, Henry Overton Wills Professor of Mathematics, University of Bristol, Bristol, UK, J.P.Keating [at] bristol.ac.uk
John Cremona, Professor of Mathematics, University of Warwick, UK., j.e.cremona [at] warwick.ac.uk
John Voight, Associate Professor of Mathematics, Dartmouth College, 612-702-3448, jvoight [at] gmail.com
McGill University contact:
Dr. Stephan Ehlen
CRM-ISM Postdoctoral Fellow
stephan.ehlen [at] mail.mcgill.ca