MexSIAM Takes Lead in Forging Links Between Mexican and U.S. Researchers

January 30, 2003

Organizers of the SIAM-sponsored session at the 2002 annual meeting of the Sociedad Matematica Mexicana, left to right: Pedro Gonzalez Casanova, Barbara Keyfitz, and Pablo Barrera Sanchez. Not shown is Humberto Madrid, who chairs SIAM's Mexico Section, which coordinated arrangements for the session.

Last year once again, during the week of October 7, SIAM sponsored a session at the annual meeting of the Mexican Mathematical Society (Sociedad Matematica Mexicana--SMM--whose members include both core and applied mathematicians, as well as teachers and students). The origins of SIAM's co-sponsorship seem to have been lost in some institutional archive, but this is the third time that I have been involved. For the second year in a row, the session was coordinated by the SIAM Mexico Section (MexSIAM). The organizers--along with MexSIAM chair Humberto Madrid and myself--were Pedro Gonzalez Casanova and Pablo Barrera Sanchez of UNAM (Universidad Nacional Autónoma de México).

Such special sessions give colleagues on both sides of the U.S.-Mexico border the opportunity to make contact with each other and to learn about research priorities and accomplishments in both countries. The American Mathematical Society has sponsored a session at the SMM meeting for several years and, in fact, has now expanded the interaction to a joint meeting held every three years. This year's SMM meeting also featured sessions sponsored by the mathematical societies of Spain, Portugal, and Bolivia.

The meeting makes visible a phenomenon familiar to the older generation of U.S. mathematicians: the expansion of research capability and infrastructure from a few prestigious institutions to universities all around the country. By design, SMM meets every year in a different one of Mexico's 32 states. Research accounts for only a part of the meeting program--with education at various levels and short courses constituting the other parts--but it is the part most accessible to U.S. attendees, since many of the research talks are given in English. (Sometimes this is "English-on-demand": With unflappable deference to American linguistic inadequacies, people switch to English whenever even one non-Spanish-speaking person is present.) And the establishment's interest in this contribution to capacity-building is evident: Grants from CONACyT (the Mexican NSF) cover registration and hotel rooms for all the U.S. invited speakers in the SIAM session. This year, SIAM generously covered a portion of the airfare for the four U.S. speakers.

The conference might remind Americans of earlier, more gracious days in other ways as well: Each invited speaker in the session had an hour to speak, and there were several breaks as well as a 2 1/2-hour stop for lunch. Despite this old-fashioned schedule, though, we had modern overhead and data projectors.

The 2002 meeting was held in Durango. Americans know Durango as a center of the movie industry--indeed, the environs retain the flavor that we associate with the Wild West: high mesas and the fresh scent of desert vegetation. The city itself, with a population of 500,000, has a small but lively downtown, a prosperous appearance, and a very grand hotel, the Gobernador, where the speakers were housed. (We were told that it was once a jail, but we were skeptical.)

This year also provided an unexpected adventure: a freak thunderstorm in Mexico City Wednesday evening ("Tiempo loco," in the words of our taxi driver). While no one's flight went according to plan, all five Americans arrived by Thursday night to a warm welcome from their hosts, including a concert with a five-piece band at an excellent local restaurant.

As a consequence of the rearranged schedule necessitated by the storm, two of the SMM-SIAM talks were given on Thursday. Pedro Miramontes opened the session with an unusual challenge from mathematics to biology: The most widely accepted mechanism of biological evolution is the Darwinian proposal of natural selection. There is growing concern among theoretical biologists because the Darwinian orthodoxy seems unable to explain many facts in evolutionary biology. Miramontes presented strong evidence that DNA evolution may be the outcome of a dynamical system and proposed a novel alternative explanation to natural selection. The proposal is a mathematical model in the framework of cellular automaton theory. Miramontes is a mathematician who shares his time between a position in Mexico, at UNAM, and a visiting position at the University of Montreal.

Following Miramontes on the program was Susana Gomez, an applied mathematician at UNAM whose team has been working on the ill-posed parameter estimation problem for oil reservoir modeling. Applying their global optimization tunneling method, they have found several local minima with good match to the real data. This set of optimal solutions produces varied scenarios of oil production; Gomez presented results for both synthetic and real reservoirs.

The first talk on Friday was on a topic that--with the possible exception of weather forecasting--could not have been more timely: prediction of lava flows. Bruce Pitman, a mathematician at the University at Buffalo, has been working with colleagues in the geosciences and geography to model the dynamics of free-surface elastic-plastic flows; the group compares the models with laboratory experiments on granular flows and applies them with real data to "postdict" such events as the recent Colima eruption in Mexico.

The modeling of granular and plastic flows is an evolving art; Pitman made clear some of the issues of stress-strain and yield conditions, as well as some features of the numerical simulation, including the use of real data (how do you handle NASA data sets when you are doing adaptive mesh refinement?) with the goal of providing real-time information to manage hazard risk (which towns do you evacuate?).

With a talk on the prediction of lava flows, session speaker Bruce Pitman found an especially receptive audience given the recent activity of the Colima volcano.

Next to speak was Miguel Angel Moreles, an analyst at CIMAT (Centro de Investigacio en Matematicas, a government-run research institute in Guanajuato) who has been working with an interdisciplinary group on the recovery of coefficients of hydraulic conductivity and effective porosity in phreatic (unconfined) aquifers. This is a typical ill-posed inverse problem.

For background, Moreles described the analogous problem in confined aquifers, where techniques for identifying transmissivity and storativity parameters are available. The differential system method Moreles has developed produces excellent results with the synthetic data; the next step is the use of real data.

In a talk on algorithms for efficient computation of PDEs, David Keyes, then of Old Dominion University, ICASE, and Lawrence Livermore National Laboratory and now at Columbia University, considered two themes: recent progress in increasing peak performance of scientific computing applications, and the widespread importance of these applications in such areas as climate prediction and reservoir simulation. Keyes made the case that further improvements depend on increasing the amount of parallelism or concurrency that can be achieved in simulations, and then explained in detail why it is so difficult in practice to achieve theoretical peak performance. Before explaining why domain decomposition and multilevel iteration have become the dominant paradigms, he made it clear that today's computational scientist has to grapple with issues ranging from computer architecture (which, he pointed out, has occasionally been described as "hostile") to Courant number limitations.

Yue Liu, from the University of Texas at Arlington, spoke on instability of standing waves in the nonlinear Schrödinger equation with a spatially dependent potential function. This talk, on work done jointly with Xiao-Ping Wang and Ke Wang of Hong Kong, explained how techniques of nonlinear analysis, such as virial identities and associated energy estimates, can be used to obtain an intriguing result: Whereas with a homogeneous potential certain standing waves are always unstable, they can be stabilized by long-wavelength perturbations of the potential. This answers a question that has been posed by physicists about the propagation of laser beams.

U.S. participants in the session, from left: David Keyes of Columbia University (who is responsible for the photos reproduced here), Bruce Pitman of the University of Buffalo, Eun Heui Kim of California State University, Long Beach, Barbara Keyfitz of the University of Houston, and Yue Liu of the University of Texas at Arlington.

The last speaker in the session, Eun Heui Kim of California State University, Long Beach, continued the theme of applied analysis. Kim spoke on ongoing work she is doing with Suncica Canic of the University of Houston, in which I am also involved. The goal of the project is to establish existence theorems for two-dimensional self-similar flows in gas dynamics. Kim explained how, at least in prototype problems for simpler equations that have been solved so far, the most difficult analysis centers on free boundary problems for quasilinear degenerate elliptic equations, with nonlinear boundary conditions arising from the Rankine-Hugoniot conditions for shocks. She outlined the cases that have been solved, along with strategies for further progress.

Each talk in the special session was followed by questions and lively discussion. As the U.S. participants prepared for the (uneventful) trip home, we all thanked our MexSIAM hosts for their hospitality and the opportunity the session provided to learn about new areas and to meet other applied mathematicians.

Barbara Lee Keyfitz is a professor of mathematics at the University of Houston.

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