University of Nottingham researchers are part of an international team mapping the mathematical objects behind prime numbers; used to encrypt online bank accounts and protect web security.
The online database of more than a billion mathematical items provides a one-stop shop for big data - an invaluable research tool to solve mathematical problems in physics and computer science.
The database, supported by a grant of £2,246,114 from the Engineering and Physical Sciences Research Council (EPSRC), and additional support from the US National Science Foundation, was officially launched on May 10.
Similar to elements in the periodic table, the fundamental objects in mathematics fall into categories, including L-function, elliptic curve, and modular form.
Elliptic curves arise naturally in many parts of mathematics, and can be described by a simple cubic equation. They 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.
Elliptical curves and modular forms are connected via their L-functions. The L-functions play a special role, acting like ‘DNA’ which characterises the other objects.
During the past 50 years, modular forms and their L-functions have been at the forefront of number theory. This project pulls together all of the computations that have been done with these objects.
Academics from 80 countries, in more than a dozen research areas, including mathematicians at Nottingham have worked on the research initiative, known as the L-Functions and Modular Forms Database Project.
The group has also used an extensive network of computers to develop algorithms and perform calculations to map the underlying relationships between the objects. These details are now available on the LMFDB website, for everyone to explore and perhaps discover something new.
The full research team on the UK grant-funded part of LMFDB is led by Principal Investigator Professor John Cremona, University of Warwick, together with support from colleagues at the University of Washington, Seattle, the Abdus Salam ICTP, the American Institute of Mathematics and the University of Waterloo.
One of the leading UK contributors to the project, Dr Fredrik Strömberg, from the School of Mathematical Science at University of Nottingham, said: “One thing that amazes me with the project is that in the beginning, back in 2010, we mainly aimed to create an online mathematical table — like the periodic table used by chemists - to help mathematicians catalogue and find research data.
“However, as the work continued we realised that we were actually creating a whole encyclopedia which charts a large part of mathematics and makes it accessible for anyone with an A-level understanding of maths to learn about and explore millions of objects which are both beautiful in their own ways and plays a hidden role in most of our everyday lives.
“I hope that the project will contribute to increasing the public’s awareness of the richness and beauty of pure mathematics in general and number theory in particular.
“LMFDB is still growing, with the addition of both new objects to study as well as new areas of mathematics steadily being added through the joint efforts of a large community of researchers and enthusiasts from around the world,” adds Dr Strömberg, an expert in number theory.”
The LMFDB database was built to provide tools that could help tackle the “Riemann hypothesis” - a conjecture that the Riemann zeta function the Riemann zeta function has its zeros only at the negative even integers and the complex numbers with real part 1/2.
The “Riemann hypothesis” is a 157-year-old problem which many consider to be the most important outstanding problem in mathematics and one of the $1m Millennium Prize Problems nominated by the Clay Mathematics Institute.
Commenting on the project, Dr Kristin Lauter, head of the Cryptography Group at Microsoft Research, said: “LMFDB provides a valuable resource for both pure and applied research mathematicians. For example, minimal polynomials of algebraic special values of a variety of modular forms are useful in generating curves for use in cryptography. In addition to cataloguing data which can be useful in applications, the database will also be a rich source of new research problems and directions.”
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