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Over the last few years, a large group of mathematicians have been developing an online database to systematically collect the known facts, numerical data, and algorithms concerning some of the most central types of objects in modern number theory, namely the L-functions associated to various number fields, curves, and modular forms, as well as further data about these modular forms.  This of course includes the most famous examples of L-functions and modular forms respectively, namely the Riemann zeta function $\zeta(s)$ and the discriminant modular form $\Delta(q)$, but there are countless other examples of both. The connections between these classes of objects lie at the heart of the Langlands programme.

As of today, the “L-functions and modular forms database” is now out of beta, and open to the public; at present the database is mostly geared towards specialists in computational number theory, but will hopefully develop into a more broadly useful resource as time develops.  An article by John Cremona summarising the purpose of the database can be found here.

(Thanks to Andrew Sutherland and Kiran Kedlaya for the information.)

Almost a year ago today, I was in Madrid attending the 2006 International Congress of Mathematicians (ICM). One of the many highlights of an ICM meeting are the plenary talks, which offer an excellent opportunity to hear about current developments in mathematics from leaders in various fields, aimed at a general mathematical audience. All the speakers sweat quite a lot over preparing these high-profile talks; for instance, I rewrote the slides for my own talk from scratch after the first version produced bemused reactions from those friends I had shown them to.

I didn’t write about these talks at the time, since my blog had not started then (and also, things were rather hectic for me in Madrid). During the congress, these talks were webcast live, but the video for these talks no longer seems to be available on-line.

A couple weeks ago, though, I received the first volume of the ICM proceedings, which is the one which among other things contains the articles contributed by the plenary speakers (the other two volumes were available at the congress itself). On reading through this volume, I discovered a pleasant surprise – the publishers had included a CD-ROM on the back page which had all the video and slides of the plenary talks, as well as the opening and closing ceremonies! This was a very nice bonus and I hope that the proceedings of future congresses also include something like this.

Of course, I won’t be able to put the data on that CD-ROM on-line, for both technical and legal reasons; but I thought I would discuss a particularly beautiful plenary lecture given by Étienne Ghys on “Knots and dynamics“. His talk was not only very clear and fascinating, but he also made a superb use of the computer, in particular using well-timed videos and images (developed in collaboration with Jos Leys) to illustrate key ideas and concepts very effectively. (The video on the CD-ROM unfortunately does not fully capture this, as it only has stills from his computer presentation rather than animations.) To give you some idea of how good the talk was, Étienne ended up running over time by about fifteen minutes or so; and yet, in an audience of over a thousand, only a handful of people actually left before the end.

The slides for Étienne’s talk can be found here, although, being in PDF format, they only have stills rather than full animations. Some of the animations though can be found on this page. (Étienne’s article for the proceedings can be found here, though like the contributions of most other plenary speakers, the print article is more detailed and technical than the talk.) I of course cannot replicate Étienne’s remarkable lecture style, but I can at least present the beautiful mathematics he discussed.
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