SIAM conference on Applications of Dynamical Systems 2011: Impressions from Snowbird

By Steve Schecter
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The biannual SIAM `Snowbird' Conference on Applications of Dynamical Systems was held May 22 - 26, 2011 in Snowbird, Utah, outside of Salt Lake City.

Steve Schecter gives his impressions.


Snowy Snowbird.

Arrival

I arrived at Snowbird on Saturday afternoon, May 21, to find far more snow on the ground than one expects in late May. Snowbird is among the snowiest places in the U.S., with 500 inches annually, but this year over 730 inches had fallen. Sunday morning through the big Cliff Lodge windows we could watch crowds skiing down the mountain, as if it were midwinter. Snow fell again during the conference, covering the trees and softening the slopes.

A reception Saturday evening featured a free drink for each attendee. I had to show ID to prove age, which hasn't happened in a while. (I'm 63.) In honor of arrival in Utah, from the available options I chose a Polygamy Porter. Its slogan: "Why have just one?"

  
Impressions from the reception.

I ran into Barry Cipra, the well-known mathematical journalist who often writes for SIAM News, and told him I was playing journalist for this meeting. He said, "A journalist is someone who tries to explain to other people something that he doesn't understand himself."

I'll first describe a few talks, then turn to discussions about the future of the meeting and of applied dynamical systems.

Jones on Climate Change

Sunday morning Chris Jones of the University of North Carolina (full disclosure: he's a friend) gave a remarkable overview of the place for mathematics in the study of climate change. SIAG-DS chair Alan Champneys introduced Jones, a former chair, as the rare individual whose work "straddles rigorous nonlinear analysis and genuine grand-challenge problems." Jones has spent the past decade getting more and more deeply involved in the science of climate change, and trying to jump-start new kinds of mathematical work in the area. He directs the Mathematics and Climate Change Network of faculty at 13 universities, which was funded by NSF in 2010.

The three legs of most sciences, Jones reminded us, are observation, theory, and experiment. In the study of climate, experiment is impossible. In its place are the big climate models, the latest generation of which incorporate so many variables that they are called "earth system models." In climate science, the predictions output by these models are treated much as experimental results are in other sciences: they must be explained by theory. A remarkable fact is that early calculations of the effect of increased carbon dioxide on global temperature have survived all subsequent improvement in the models. It is generally believed that the most important variable not yet included is clouds, whose effect is not yet understood.

The big climate models are the Hollywood blockbusters of climate science: they are created by major centers, and everyone knows about them. By contrast, small models, which have just a few variables, are the indy films of climate science, created by a few collaborators. However, they provide the theory to explain the output of the big models. Raymond Pierrehumbert, a geophysicist at the University of Chicago, said in another plenary talk that all the important ideas in climate science have come from small models. Jones added that climate tipping points can be due to effects that are more easily incorporated in small models than big ones. He cited as an example recent work of Peter Ashwin and collaborators on the compost-bomb instability, in which a warming climate leads to faster decomposition of peatlands by soil microbes; heat is created more rapidly than it can escape into the atmosphere, leading to a catastrophic release of carbon.

Jones sees a two-fold role for our community in the study of climate change: first, aiding in the development and analysis of small models; and second, using dynamical systems perspectives to improve data assimilation, the process by which computer simulation and data from observations are combined to make predictions. Reciprocally, he predicts that the needs of climate science will lead to a change in the practice of applied mathematics, whereby models and data are treated in a more coordinated way.


During a coffee break.

Yorke receives Jürgen Moser Prize

Sunday evening James Yorke of the University of Maryland was awarded the Jürgen Moser Prize, SIAG-DS's career award for contributions to nonlinear science.

In his talk, Yorke said that chaos is a real phenomenon that can't be completely captured by a mathematical definition. He pointed out that his 1975 paper with T.-Y. Li, "Period Three Implies Chaos," which introduced the term "chaos" into mathematics, does not define it. He and Li just prove that certain hypotheses imply certain specific complicated behavior.

Yorke showed a picture of the convoluted basin of attraction of a fixed point in a dynamical system. If one strays too near its boundary, one risks landing outside the basin (due to a random perturbation or numerical error), in which case one flies off to infinity. So one should stay well away from the boundary, which still allows big excursions within the basin. From this Yorke derived a principle of administration. When he became chair of the Maryland Mathematics Department, he did not try to plan the department's future. Instead he asked, what is the worst that can happen?, and tried to make sure that it didn't.


Marty Golubitsky, Jim Yorke and Alan Champneys

Pujals on robustness

Enrique Pujals of IMPA, Rio de Janeiro, gave an overview Tuesday morning on progress in understanding what dynamic behavior can be robust to perturbation. In low dimensions, robust behavior is hyperbolic or partially hyperbolic (such as the Lorenz attractor, which cannot be hyperbolic because of an equilibrium within the attractor). "Wild" dynamics is caused by homoclinic tangencies. In higher dimensions results are not yet so satisfactory. Pujals added that trying to understand the variety of dynamical behavior he had encountered at Snowbird was a humbling experience. He suggested that perhaps theorists such as himself should concentrate more on the types of behavior seen in applications.

Hobbies

Several talks I heard grew out of interests that the speakers described as hobbies.

Tuesday afternoon Jeff Moehlis of the University of California at Santa Barbara described his work with a then-undergraduate, Ronald Votel, on animal territoriality. The work was suggested by observations of Tilapia mossambica in a rectangular pool with a sandy floor. The males dig pits by spitting sand away from the pit centers. The resulting ridges produce a pattern of pentagonal and hexagonal regions.

In Moehlis's model, each animal's territory is the set of points in a planar region that are closer to it than to any other animal. The animal then moves to the centroid of its territory, and territories are recalculated. Equilibrium configurations are centroidal Voronoi tesselations. Moehlis and Votel calculated these patterns for various rectangles and numbers of animals. At this point in their work, Moehlis happened to hear a talk by a robotics researcher at the Tokyo Institute of Technology, whose group had built robots that could be used to mimic the animals in the model. Moehlis treated the audience to movies of the robots moving around as the aspect ratio of the rectangle changed, occasionally jumping to new positions when the possible centroidal tesselations changed abruptly.

Wednesday afternoon Carmen Chicone of the University of Missouri (full disclosure: also a friend) discussed his fascination with a standing wave in a decorative fountain on his campus. Water flows along a high flat surface, then over the edge into a pool, from where it is pumped back. There is a standing wave clearly visible near the front edge of the surface. Chicone showed a movie in which he tried to disrupt the wave with his hand, but it instantly reformed. Using the Boussinesq approximation of shallow water theory with potential flow, Chicone showed that the standing wave corresponds to a closed orbit of a Duffing-like equation.

Stephen Morris's experiments

At noon Thursday Stephen Morris of the University of Toronto concluded the conference by describing three experimental efforts to reproduce pattern formation seen in nature. The experiments involved icicle formation, the development of "washboarding" in sandy roads due to traffic, and formation in lava flows of patterns of columns with hexagonal cross-section. In the last case, an experimental analogue is drying corn starch. Washboarding turns out to be caused not by wheels falling into low spots and packing them down, as one might expect, but by wheels flying over ridges and kicking up sand as they hit the next ridge.

Should Snowbird change?

Monday evening at the SIAG-DS business meeting, Jonathan Dawes of the University of Bath, a conference co-chair, explained changes in this year's conference. Invited presentations were reduced from 60 minutes to 45, and minisymposium presentations from 25 minutes to 20. These changes were needed because of the growth of the conference, which had over 800 participants this year, and because the conference was half a day shorter. It ended with a bang with Stephen Morris's talk at noon Thursday, rather than petering out with an afternoon of poorly-attended contributed talks, which was the case in 2009.

The ensuing discussion featured complaints about minisymposia. There are minisymposia on similar topics featuring different groups of friends who could benefit by being in the same room; people aren't invited just because no one thought of them; contributed talks have low attendance. One alternate suggestion was a small number of invited minisymposia, and contributed talks or posters for everyone else, grouped into coherent sessions by conference organizers. Another suggestion was to combine proposed minisymposia on related topics.

  
Impressions from the poster session.

The Big Picture

Wednesday evening SIAG-DS chair Alan Champneys assembled a panel---Carson Chow, Yannis Kevrekidis, Mason Porter, Mary Silber, and James Yorke---and peppered them and the audience with provocative questions.

Is it a problem that 40% of the meeting is biology?

Yorke: No, that's what we don't understand, we're just at the beginning of biology. And it's what people want to do. Porter: We shouldn't expect every talk to make a contribution to dynamical systems. We also want to hear about problems to which we can make a contribution. From the audience: This is a bottom-up meeting, the talks are about what our community wants to talk about, so the question whether a subject is over-represented is nonsensical. But it's good to use invited talks to extend our range.

Should we have more talks about climate change and sustainability?

Kevrekides: Sure, this meeting is an opportunity to get educated about areas we might be able to contribute to. Porter: Inviting people in new areas is a strength of the meeting. John Guckenheimer: The UN's sustainability issues are WEHAB---water, energy, health, agriculture, biodiversity. These areas don't have global models as climate does. These are new questions for us. If we're to help we need to reach out to the communities studying these issues. This meeting is a good place to do that. From the audience: There is already a lot we're doing that is relevant. Talks on Lagrangian coherent structures relate to the study of oceans, talks on inertial particles relate to clouds, talks on plankton relate to removing carbon from the atmosphere.

What are the right tools for understanding the dynamics of complex networks?

Porter: The subject is here to stay, although a large percentage of the work is BS. Kevrekides: It's pattern formation in a different topology. Chow: Each problem is treated individually, but general approaches will develop. Marty Golubitsky: Networks are a third paradigm between ODEs and PDEs. Network models will be produced whether we like it or not. There are too many ideas about networks floating around for the subject to go away.

Are we inbred? Does anyone outside our community care what we do? Should we be more evangelical?

Kevrekides: Many of us go to other meetings, but this is where we feel at home. Yorke: When you're writing a paper, write for one community only. If you want another community to know about your work, write in its journal and write for it. Chow: I worry that people in parallel minisymposia are actually working on similar problems but don't know of each other's work. Kevrekides: We try to expose students to everything they might need, but that makes it hard for them to get the bedrock knowledge they need to do anything.

What will you be working on in ten years?

Porter: Networks. It's still the Wild West out there. Kevrekides: Integrating data mining with models. Silber: Climate. Chow: How do you go from genotype to phenotype? Yorke: I don't think that far ahead.

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