## The L-functions and modular forms database

With each passing decade, mathematics grows substantially. As it grows, mathematicians are forced to become more specialized — in the sense of knowing a smaller fraction of the whole — and the time and effort needed to get to the frontier of what is known, and perhaps to contribute to it, increases. One might think that this process will eventually mean that nobody is prepared to make the effort any more, but fortunately there are forces that work in the opposite direction. With the help of the internet, it is now far easier to find things out, and this makes research a whole lot easier in important ways.

It has long been a conviction of mine that the effort-reducing forces we have seen so far are just the beginning. One way in which the internet might be harnessed more fully is in the creation of amazing new databases, something I once asked a Mathoverflow question about. I recently had cause (while working on a research project with a student of mine, Jason Long) to use Sloane’s database in a serious way. That is, a sequence of numbers came out of some calculations we did, we found it in the OEIS, that gave us a formula, and we could prove that the formula was right. The great thing about the OEIS was that it solved an NP-ish problem for us: once the formula was given to us, it wasn’t that hard to prove that it was correct for our sequence, but finding it in the first place would have been extremely hard without the OEIS.

I’m saying all this just to explain why I rejoice that a major new database was launched today. It’s not in my area, so I won’t be using it, but I am nevertheless very excited that it exists. It is called the L-functions and modular forms database. The thinking behind the site is that lots of number theorists have privately done lots of difficult calculations concerning L-functions, modular forms, and related objects. Presumably up to now there has been a great deal of duplication, because by no means all these calculations make it into papers, and even if they do it may be hard to find the right paper. But now there is a big database of these objects, with a large amount of information about each one, as well as a great big graph of connections between them. I will be very curious to know whether it speeds up research in number theory: I hope it will become a completely standard tool in the area and inspire people in other areas to create databases of their own.

### 7 Responses to “The L-functions and modular forms database”

1. Emmanuel Kowalski Says:

There’s actually at least one paper (by Fouvry, Michel and myself) where Gowers norms meet (implicitly) L-functions over function fields, through trace functions of ell-adic sheaves…

http://www.math.ethz.ch/~kowalski/gowers-prime-fields.pdf

2. Anonymous Says:

Something of the same flavour, for classes of graphs : http://www.graphclasses.org/

3. Markus Schepke Says:

Reblogged this on Understanding the Riemann Hypothesis.

4. James Smith Says:

I’ve been getting very interested in the dissemination of mathematics this last six months or so. To me the arxiv started the process off, with open access journals then following either as arxiv overlays or otherwise.

To me though this seems like just the start. I typed ‘dissemination of mathematics’ into Google about six months ago and I’ve just typed it in again. Much the same has happened this time, although there are several new sites of interest. Basically, you get resources and initiatives concerned with mathematics education, some of which like http://iccams-maths.org/ seems like great stuff at a glance. Another site I found the other day in a similar vein was https://undergroundmathematics.org/, although I came to this a different way.

This kind of thing isn’t about the dissemination of mathematics amongst professional mathematicians themselves, however. Dissemination sideways if you like. Rather it’s about dissemination from the top down, if that’s the right way of putting it. It’s worth mentioning that this is a very laudable goal and anyone who’s ever taught mathematics at any level should appreciate these initiatives.

But what about dissemination sideways? To me to be able to fully disseminate mathematics sideways means that it must be formalised, however unfashionable amongst the majority of mathematicians this continues to be. The reason is that with formalised mathematics you have a corpus that is discoverable and usable, although I’m searching for the words here. I don’t think that formalisation is an end in itself and nor do I think that a formalised proof is necessarily any more worthy just because it has been verified by a computer. It just seems that it’s the only true place to start.

I think that mathematics that has been formalised has a structure and nature that somehow makes it much more amenable to being discoverable, reusable, searchable, etc. I got very excited when I watched a talk https://www.youtube.com/watch?v=Is_lycvOkTA by Thomas Hales again recently, since he seemed to be striking similar chords in places.

The arxiv, open-source web-based journals and an increasing number of on-line mathematics databases are fantastic but to me they still seem, because of their interfaces, to some degree like mathematics at a distance, again I’m searching for the words. They’re like Wikipedia on steroids. The content is more refined and detailed, so much so that it’s hardly a comparison, but the means for interactivity and collaboration are still limited. These sites might serve as resources for a mathematician taking part in a Polymath project, for example, but they do not serve as its context or arena. They afford the learning of mathematics amongst mathematicians, which is great obviously, but I think they are limited in affording the *doing* of mathematics. Both of these are needed for discovery of new mathematics or improvement of existing mathematics, I think.

5. Discoverable Math Requires Structure | Delightful & Distinctive COLRS Says:

[…] Tim Gowers blog, May […]

6. Tom Copeland Says:

Which OEIS entry?

7. Tom Copeland Says:

Which OEIS entry?