April is Math Awareness Month

March 15, 2011

With Complex Systems, the Joint Policy Board for Mathematics found an exciting, timely theme for Mathematics Awareness Month 2011. As the poster states and abundantly illustrates, complex systems are all around us. Living organisms, along with cells and protein-interaction networks within organisms, are complex systems, as are some familiar elaborate engineered systems--financial markets, the Internet, and electric power grids. Complex systems function at widely varying scales, making them especially interesting to many mathematical and computational scientists.

A necessary place to begin an exploration of complex systems is with a definition. The experts, many of whom are affiliated with the numerous institutes all around the globe dedicated to the field, agree that complex systems share certain properties. The field builds on the traditional focus on the parts of systems, the New England Complex Systems Institute points out, and "integrates the networks of relationships within and between systems. These relationships are responsible for the collective ‘emergent' behaviors we see in all physical, biological, socio-economic and technological systems."

Using quantitative foundations from physics, mathematics, and computer science, according to the NECSI website, researchers in complex systems "apply computer simulations and high dimensional data analysis, to characterize the patterns of behaviors in the world around us. Further, we can describe how complex systems arise through evolutionary processes, and how systems become capable of achieving their goals."

As readers receive this issue of SIAM News, the 2011 Complexity Conference will be under way at Northwestern University---a joint project of NICO (Northwestern Institute on Complex Systems) and SONIC (Science of Networks in Communities Research Group at Northwestern). Several of the scheduled speakers will be well known to regular readers of SIAM News.

Math Awareness Month, by definition, is directed to people outside the math sciences community---university undergraduates, a scientifically inclined general audience. An effective focus for the activity, consequently, must be an exceptionally active and dynamic research area.

How do complex systems measure up? A web search on "complex systems institute" quickly reveals NECSI and NICO to have a multitude of counterparts; many have informative websites, sponsor conferences, and publish papers and journals. Worth a look is the online Complexity Digest, which offers articles, links to published articles, and a calendar of events.

Readers are especially encouraged to visit mathaware.org, the website for Math Awareness Month. As this issue of SIAM News goes to press, two stimulating essays had been posted; the site also provides links to resources for interested faculty and their students. Responsible for assembling these materials, and more to come, is a committee chaired by Joceline Lega of the University of Arizona.

As to the essayists who donated their time and talent to the cause: John Guckenheimer of Cornell discusses living organisms as complex systems. Given the variability of the environment in which an organism lives, he writes in the introduction to the essay, the organism

"needs to be adaptable and resilient. How do we connect the evolution of biological molecules to the fitness of the organism as an integrated whole? That is a question whose solution prompts the search for general principles about complex systems."

He follows with a clear presentation of approaches for understanding morphogenesis, for understanding organisms via bifurcation theory, and for building dynamic models of complex systems.

Paul Hines and Christopher Danforth, with undergraduate Benjamin O'Hara and graduate student Eduardo Cotilla-Sanchez, all of the University of Vermont, offer an insightful perspective on cascading failures, emphasizing power grids and power-law distributions of blackout size in computing the risk of failure. "Cascading failures can cause large blackouts," they point out, "but cascading failures do not necessarily produce power-law probability distributions in blackout sizes." They summarize the "leading hypothesis" as follows:

"Competing pressures to both use the electricity infrastructure efficiently to reduce costs, and upgrade the infrastructure when it fails, give rise to a self-organizing process that makes large cascading failures more frequent than they might be otherwise. The real complexity in the system comes from not merely the physics of electricity systems, but from self-organization that results from interactions between the physics of cascading failure and the decisions of humans that operate the grid. Even without human involvement, self-organization produces power-law failure distributions in other complex systems, like earthquakes, forest fires and landslides."

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