Lennaert van Veen recently sat down with Crawford Prize winner Arnd Scheel to discuss his work on pattern formation, future directions for dynamical systems, and carrying on the traditions associated with this award.
LvV: I distinctly remember you giving a plenary lecture at Snowbird in which you showed many natural examples of pattern formation. Was that the year you received the prize?
AS: No, that was two years earlier. The work I presented then was not exactly what was mentioned in the citation. In the talk, I focused on the relation between the physical observations and more technical aspects, such as manifolds and connecting orbits. Some of that work was done in collaboration with Bjorn Sandstede. After that, I worked with Mariana Haragus on defects in two-dimensional media, grain boundaries, corners in interface propagation. One goal was to avoid using comparison principles, but rather use such techniques as singular perturbation and bifurcation theory. There are many challenges, like how to discuss defects or coherent structures in problems with overdamped elasticity, for instance modeled by nonlinear heat equations. You have to carefully define what defects are. One would ideally phrase the existence problem through an implicit function theorem, like in bifurcation theory. But setting the problem on an infinite domain, continuous spectra make the analysis difficult.<br>
LvV: Any recent work in this direction?
AS: In recent work, we study Turing patterns, looking for instance at the relation between strain and displacement. We considered reaction-diffusion, Cahn-Hilliard, Ginzburg-Landau equations and such. We are trying to use dynamical systems theory to compress information. Just like in the classification of bifurcations, we try to classify defects, here depending on boundary conditions. Moving domain boundaries are interesting, too, in particular where they are selecting patterns. More generally, one goal is to find out what universal effects one can expect from defects? When studying defects in 2D systems, it is not so clear what a dislocation is or how the dynamical systems approach helps. We are still working out the links, like what are multiplicities, what is the proper implicit function theorem to work from?
LvV: Always more questions...
AS: Yes, the fun part is that you have to add to what you thought was the whole story when going out to deal with problems people actually face.
LvV: What other open challenges do you see?
AS: Many, for instance, space-time propagation of instabilities, wave-number selection, the role of initial conditions in simulations and experiments in the generation of, for instance, invasion fronts. One only has to read over recent reviews by Wim van Saarloos. Another question of interest is the analysis of nonlocal coupling. These are of interest, for instance, in neuroscience. I have done some work in that direction recently with Gregory Faye.
LvV: We are meeting at a workshop on rigorous computing, are you getting into that field?
AS: Several people in the field of rigorous computing are interested in nonlinear wave problems, and I was invited to present some open problems that elements of rigorous computing could be applied to. Rigorous computing has achieved a lot, but so far mostly as a proof-of-concept. There are plenty of open problems that go back to the 1800s, one could really have a lasting impact if one manages to solve some of these problems using these new techniques.
LvV: It is tradition that Crawford Prize winners are on the committee for their successor. How did you find being on the committee?
AS: It was hard, comparing apples to oranges. I believe the new rules put in place have helped. There is certainly nothing to complain about if you look at the prize winners. I suppose there's many different views on what the prize should accomplish, and those may determine how satisfied you are with the outcome.
LvV: What would you think of a Crawford lecture instead of a Moser lecture?
AS: Well, this has been discussed. Sometimes you hardly see the Crawford winners, but I've always appreciated the Moser lectures as a part of the program, they often give a different perspective on the field. It'd be worthwhile to find a slot for a Crawford lecture as well, but it might be hard to squeeze it into the busy Snowbird program.