Researchers announce new way to explore mathematical universe

Online resource reveals deep relationships and provides a guide to previously uncharted territory

May 10, 2016

An international group of mathematicians at Arizona State University and other institutions have released a new kind of online resource to help discover uncharted mathematical worlds.

The “L-functions and Modular Forms Database,” or LMFDB, is an intricate catalog of mathematical objects and the connections between them. Both beautiful and functional like an atlas, the LMFDB reveals deep relationships 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.  A Maass form, one of the 20 different types of objects in the LMFDB A Maass form, one of the 20 different types of objects in the LMFDB. Image by Fredrik Streoemberg Download Full Image

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

Massive computations

The scale of the computational effort that went into creating the LMFDB is staggering: a total of nearly a thousand years of computer time spent on calculations by multiple teams of researchers. One recent computation used more than 72,000 cores of Google's Compute Engine to complete a tabulation in one weekend that would have taken more than a century on a single computer. The objects in the LMFDB are all connected to the area of mathematics known as number theory. Related computations are often amenable to massive parallelization, and this allows for scaling to the cloud. Some of the calculations are so intricate that only a handful of experts know how to do them, and some are so big that it makes sense to run them only once, and then share the (verified) results.

"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," said LMFDB project member John Jones, a professor of mathematics in the School of Mathematical and Statistical Sciences at Arizona State University.

Professor John Jones
ASU professor John Jones is a pioneer in providing mathematical data via the web. Starting in the 1990s, he wrote a series of papers with David Roberts of the University of Minnesota at Morris on techniques for novel number theoretic computations, and would provide results on his website. Since 2011, he has been a regular contributor to the LMFDB. Photo by Rhonda Olson/ASU

Mathematical interconnections

Large­-scale cloud computing is just one of the ways in which the project is changing the way mathematics research is done. 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.”

Just like explorers, mathematicians become excited when they discover a connection between two areas that at first appear to be unrelated. The discovery leads to further breakthroughs in research when those connections are made explicit. Researchers following different paths may be heading 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 includes researchers from more than a dozen areas, all of whom are building connections between their seemingly separate specialties.

“Each mathematician is an expert in their own area and knows one path really well, but the LMFDB helps to demystify the other paths by providing the possible interconnections in clear and navigable terms. It opens up a whole new world of possibilities for mathematical research,” said Jones.

The LMFDB also includes an integrated knowledge database that further clarifies relationships to mathematicians by explaining the contents of the site and the mathematics behind it.

Uses for today’s high-tech world

All mathematics is eventually useful. One of the great triumphs in mathematics of the late 20th century was achieved by British number theorist Andrew Wiles with his proof of Fermat’s Last Theorem, a famous proposition by Pierre de Fermat that went unproven for more than 350 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 simple cubic equations; 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.

A team effort

There will be several simultaneous events happening May 10 in North America and Europe to celebrate the launch of the LMFDB, including public presentations and lectures at the American Institute for Mathematics in San Jose, California; Dartmouth College in Hanover, New Hampshire; and the University of Bristol in the United Kingdom.

The LMFDB project has been funded by the U.S. National Science Foundation, the UK Engineering and Physical Sciences Research Council, the American Institute of Mathematics, the EU 2020 Horizon OpenDreamKit Project, and the Institute for Computational and Experimental Research in Mathematics, and it involves researchers from Arizona State University, Dartmouth College, MIT, Oregon State University, the University of Bristol, the University of California at San Diego, the University of Paris, the University of Sydney, the University of Warwick, the University of Waterloo and other institutions. 


Rhonda Olson

Manager of Marketing and Communication, School of Mathematical and Statistical Sciences


Thunderbird team wins international business ethics case competition

May 10, 2016

While cynics may say that business ethics is an oxymoron, it’s alive and well at ASU's Thunderbird School. Case in point: A team of five T-birds bested such b-school heavyweights as the University of St. Thomas, the University of Oxford-Saïd School of Business and the University of Miami to win top spot in two out of three categories in the 20th International Business Ethics Case Competition (IBECC) held April 20-22, in Cambridge, Massachusetts.

In true Thunderbird fashion, the winning quintet included two Indian students (Vyoma Mehra and Gaurav Shetty), one Hungarian (Zoltan Nagy), one Australian (Christopher White) and one American (Kimberly Hergenrader). All five are Master of Global Management candidates who began their programs in spring of this year. This was their first case competition — and the first time Thunderbird took part in the IBECC. The Lincoln Center for Applied Ethics at Arizona State University, which is headed by T-bird David C. Lincoln and family, funded the team’s participation. The winning Thunderbird team. Download Full Image

The IBECC, which is jointly sponsored by the Center for Ethics and Business at Loyola Marymount University, the Opus College of Business of the University of St. Thomas, the Center for Business Ethics at Bentley University and the Ethics & Compliance Initiative, asked teams of graduate and undergraduate business students to prepare three presentations on a business-ethics case of their choosing: a 25-minute presentation, a 10-minute presentation and a 90-second presentation.

Thunderbird’s team won both the 10-minute and 90-second rounds for their presentations on the 2015 privacy vs. protection dilemma Apple faced when asked by the U.S. Federal Bureau of Investigation to provide back-door access to the iPhones used by San Bernardino shooting assailants Tashfeen Malik and Syed Rizwan Farook.

“The ethical issue Apple faced was that to open access like this to the U.S. government would set a legal precedent that would have to be followed when any country’s government made a similar demand,” explained Mehra, the team’s lead. “Our stance was that Apple had a responsibility to both itself and the worldwide internet community to refuse this request because it could not risk being forced to comply with future governmental requests that might not be ethically motivated.”

The team had three months to prepare its position and presentations. Melissa Samuelson, clinical professor at Arizona State University’s W. P. Carey School of Business and former clinical professor of ethics at Thunderbird, acted as faculty coach and mentor.

“The Thunderbird team picked a case that was not only relevant for a major multinational company based in the US, but they were able to show and draw connections to global privacy and security issues in the technology industry,” said Samuelson. “This kind of analysis and understanding of global issues set this team above their peers. They approached the dilemma as if they were truly consulting with Apple on the most ethical response — not just preparing for a competition.”

That consultant mentality, however, got the team into a bit of a quandary when they chose to approach their 25-minute presentation (which was delivered first) as if the judges knew the ins and outs of the Apple situation and needed little background to follow the team’s position. They learned afterward that wasn’t the case.

“After the 25-minute round, we basically changed our presentation approach overnight and delivered the 10-minute and 90-second speeches from a more layman perspective,” Mehra said. The team also adjusted their speaker roles for the following two presentations after learning that the judges were challenged by all of their varying accents.

“At Thunderbird we hear different accents all the time so it isn’t unusual for us. We adjusted our approach so we would not have that problem again,” Mehra said.

Accent issues aside, the global nature of the Thunderbird team had extraordinary benefits.

“Our diversity allowed us to bring a global perspective that other teams could not,” said Mehra. “The cultures represented in our group — and the international dimension that added — brought so much to our discussions and our solution.”

How did the team propose Apple handle issues of this nature moving forward? Create a detailed, hacker-proof process by which the Apple organization can access iPhone data stored on iCloud (which, for legal purposes, belongs to Apple) — rather than a process for accessing data stored on the phone itself (which belongs to the individual). By doing so, Apple can share its own data without overstepping its ethical bounds.

As the first business school to adopt a professional Oath of Honor pledging commitment to ethical business practices, ethics is a fundamental pillar of the Thunderbird experience. These students are a perfect example of that.

“The IBECC was an amazing experience,” said Mehra. “We look forward to participating again next year — and bringing an undergraduate Thunderbird team with us.”