Functions Too Cool For Facebook. But Don’t Worry, We’ve Got You Covered

This map captures the web of contributions from over 80 participants in the LMFDB project over several years.

This map captures the web of contributions from over 80 participants in the LMFDB project over several years.

Today is the official launch of the L-functions and modular forms database. The LMFDB is a database containing all the relevant information about millions of mathematical objects. Set up like a Facebook for mathematical objects — by objects I mean curves, functions, special equations and structures — the LMFDB lets us see which objects are related to each other, which ones share a common ancestor, and which ones can at least play nice.

But maybe you, like nearly all people who aren’t seeped in a daily brew of number theory, wouldn’t recognize an L-function if it walked into the room right now. Even so, I promise this database has some exciting implications for you. Yeah, you. Understanding how the social network of all these millions of objects looks can give a huge kick in the pants to the famous Riemann Hypothesis. But even for those of us who don’t run around muttering about zeroes on the critical strip, we still profit, perhaps unwittingly, from this and other really hard number theory problems every day when we use the internet. Knowing more about he universe of the LMFDB can help find vulnerabilities in encryption, keeping our private data and transactions safe.

The connection between elliptic curves and modular forms is just a small part of the L anglands Program, a vast web of conjectures proposed by Robert Langlands, at the Institute for Advanced Study, in the late 1960s. Image courtesy of David Dumas, Timothy Boothby, and Andrew Sutherland.

The connection between elliptic curves and modular forms is just a small part of the L anglands Program, a vast web of conjectures proposed by Robert Langlands, at the Institute for Advanced Study, in the late 1960s. Image courtesy of David Dumas, Timothy Boothby, and Andrew Sutherland.

But much more broadly — and perhaps more importantly — one of the motivating goals of so much mathematics of the last century has been to find a so-called grand unifying theory of mathematics which we call the Langlands Program. In the mathematical universe we deal with all kinds of seemingly unrelated objects, like those curves and functions and other things I mentioned earlier. “The connections between these classes of objects lie at the heart of the Langlands program,” explained the Fields Medalist Terry Tao in a blog post about the LMFDB today. The LMFDB teases out a lot of surprising relationships between theoretical objects, ones that aren’t so easy to see when you look at these things one at a time.

And even if you aren’t chasing the grand unified theory, if you work in certain areas of math, these objects come up all the time, and having an atlas to this mathematical universe can be incredibly helpful. As Emmanuel Kowalski wrote on his blog today, the LMFDB can help us understand their “random and possibly spooky” behavior.

Another huge boon of the LMFDB is that it stores billions of time intensive calculations for immediate retrieval — literally thousands of years worth of computations — saving our future selves huge time and effort. Tim Gowers, Fields Medalist and proponent of effort-saving tools, wrote about the LMFDB on his blog today, saying “I rejoice that a major new database was launched today.” This frees us up to do other things, like prove deep results.

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