Joint Maths Meetings 2010

by Willie Wong

Last week I was in San Francisco, California, at the AMS/MAA/SIAM Joint Maths Meetings 2010. This year it was held at the Moscone Center. I was there in a position of quasi-responsibility: with Paul Allen, Michael Eichmair, Gustav Holzegel, and Jared Speck, I helped organize the AMS Special Session on Mathematical Challenges of Relativity, which is something that grew out of the Mathematical Research Communities program from last summer.

I saw many familiar faces there, though it was a shame that Michael could not make it in person. Unlike the MRC program from last year, where it was mostly focused on learning the tricks of the trade, the special session is more about presentation of new research work by young mathematicians. And I picked up quite a few things for ruminating later. Paul moderated the morning session and I did the afternoon (I hope I didn’t butcher the names of too many people!). Rather unfortunately there was still the usual hyperbolic versus elliptic split in the community.

The study of mathematical general relativity branched into several rather specialised disciplines in the past century. On the physics side there are of course studies of matter models and cosmology and whatnot, but even on the mathematical side there are deep divides. For starters there are numerical simulations: for a long while the field is largely ridiculed by more theory minded mathematicians, since a stable and robust method of simulation was lacking–when given the initial data for a static black hole, some of the numerical recipes will evolve it into all sorts of craziness. But recently (in the past 5 or 6 years) a lot of progress has been made, and we can speak of bona fide predictions based on numerical data now. But putting the theory versus numerics split aside, there is still the big divide between the hyperbolic/Lorentzian camp and the elliptic/Riemannian camp. One of the interesting features of general relativity is that, due to the strong coupling between gravitational effects and matter models, not all configurations of matter distribution and space-time gravitational field are admissible. There are some constraints to what an instantaneous snapshot of the space-time may look like. So unlike the study of many other equations from physics, where “now” can be freely prescribed and the only interest is in finding out about the “future”, general relativity has two aspects: the evolutionary problem of predicting what the “future” may be given information about “now”, and the constraint problem of what “now” can possibly be. The partial differential equations of the former are of hyperbolic type, while for the latter they are elliptic; the geometry for the former is Lorentzian, while for the latter is Riemannian.

As it so happens, in many ways the development of elliptic partial differential equations (and by extension Riemannian geometry) is eons ahead of that of hyperbolic equations. There exist more robust methods and more diverse tools. This unfortunately lead to the two branches becoming disjoint; and some of the current open questions asked by the Riemannian geometers, while many generations ago may have been inspired by a question in general relativity, have now only tenuous relations to the physics. (This is not to say that those problems are not interesting; just that they are not really motivated by physics.) This fact was somewhat reflected in the program for our special session. (As a side note, this was one of the major failures for the MRC program. An unstated goal I think was for the young mathematicians educated in the two different disciplines to intermingle and learn about each others’ fields. In actual implementation, the two groups more-or-less self-segregated.)

Now a word about the organization of the conference: our approach was to focus on making a more compelling narrative, and hence we grouped talks on similar topics together. We started the morning with two talks on decay estimates for wave equations on Schwarzschild background, one talk about wave equation on spaces with cosmological constant, and then two talks about the breakdown criterion in general relativity … you get the idea. The main advantage I found was that this keeps the focus of the audience. Of course, a downside is that shifts in the numbers of audience members is more noticeable: some people just stayed for the “block” that interested them. Perhaps we could’ve also done the grouping while mixing the talks so that there were some elliptic talks and some hyperbolic talks in each session, rather than having what ended up more as a hyperbolic morning plus an elliptic afternoon. This is something I’ll have to think about if I organize a conference again in the future.

An unfortunate thing about the JMM (and many AMS conferences actually) is that the non-plenary talks are restricted to something like 20 – 30 minutes. While it is not a problem in certain fields, for disciplines in which analysis plays a heavy role (I’m sure it is also the case for other fields, it is just that I am more familiar with analysis), such as differential geometry, partial differential equations, etc. the time limit really makes the presentation a short announcement of result plus an advertisement of why we should care: something like the 5 minute talks in other sciences. (Incidentally, the general contributed papers sessions limit their talks to ten minutes. I learned at JMM that it is not possible to be fashionably late to one of those talks. A previous talk ended a couple minutes late, I had a bit of difficulty finding where the next talk I wanted to hear was happening, plus a lack of completely synchronized watches [nothing to do with relativity, mind you] resulted in my walking into the talk when the audience started applauding.)

In a separate post I will write about the interesting things I learned at the JMM while listening to talks not in my field. (In particular, the Friday morning session on mathematical origami was spectacular.)