Spring 2007

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Math Matters: IMA Public Lectures

IMA Public Lecture Series" Math Matters" features distinguished mathematicians and scientists who are also superb expositors and are able to illuminate the role mathematics is playing in understanding our world and shaping our lives.

Public Lectures make an impact on high school students

We had four superb, educational and entertaining public lectures delivered during the Algebraic Geometry year. For all four lectures we had over 300 people in the audience of which more than 100 were high school students. After each lecture several of the students who were inspired by these lectures initiated talks with the speakers, asked many questions and had their pictures taken. The math teachers from Mounds view high school, where most of these students are from, have indicated how useful these public lectures are to their students and have already built it in their curriculum to attend next year's public lectures.

Summary of the talks

Margaret H. Wright
The first public lecture of the year 2006-2007 How Hard Can It Be?, November 2, 2006 was delivered by Margaret H. Wright, Silver Professor of Computer Science, Courant Institute of Mathematical Sciences, New York University. Margaret was the first woman to speak at these public lectures. Wright opened her talk by sharing with her audience some amazing examples of "hard" mathematical problems that were turned into easy problems to solve. She emphasized how mathematicians play a role in deciding which problems are hard and are considered to be NP-problems (problems that are easy to check, but hard to solve with our current tools). She showed for example that completing a jigsaw puzzle with n pieces, when n is large is NP. She then turned to a discussion of Cryptography and showed how Diffie and Hellman revolutionized this field by using modular arithmetic and factorization of large primes in number theory. Finally she talked about linear programming and gave a practical example on how eating a good breakfast relates to linear programming and showed how algorithmic improvements in both simplex and interior-point methods, plus faster computers, have enabled us to solve huge linear programs very quickly.

Christopher J. Budd
The second public lecture of the year was given by Christopher J. Budd,Professor in Applied Mathematics and Director of the Centre for Nonlinear Mechanics, University of Bath, England, on Making Sense of a Complex World , on January 18, 2007. Budd started his lecture by explaining what we actually mean by a complex system and how much of the natural and human behavior is complex. He gave several examples of complex systems, such as the stock market, the weather and turbulence and also some examples of complicated behaviors that are not complex such as tides. He then astonished the audience with his "Double Pendulum" demonstration which is an example of complex behavior in a simple coupled system. Although each part of the system is relatively simple, it is the coupling which leads to new complex emergent behavior and forms a chaotic motion. Other examples of chaotic behavior are aircraft undercarriage and motion of the asteroids. Budd also reviewed how emergence arises from the way the components interact with each other in a system and not just from their individual properties by giving further examples of complexity. In conclusion, he mentioned that complexity may apply to many other situations including spread of disease, customer behavior, transport networks and chemical reactions and pointed out that much still remains to be discovered.

Martin Golubitsky
The third public lecture was delivered by Martin Golubitsky, the Cullen Distinguished Professor of Mathematics at University of Houston on Patterns Patterns Everywhere, March 7, 2007. Golubitsky's lecture started with amazing patterns of mud plains, leopard spots, convention cells, sand dunes,and zebra stripes. He justified why we should study patterns and how they are useful in our everyday life. For example, it is important to understand patterns that emerge in fingerprints, atmospheric jet streams, and geological formations. He gave a brief history of Navier-Stokes equations and showed how mathematics provides a menu of patterns by giving two examples on stripes and spots in planar systems and oscillations with circle symmetry. Golubitsky then described symmetry, and showed how planar dynamics creates symmetric chaos. He further shared with the audience how the Logo of the IMA was created using such techniques. He continued with showing several fascinating images of symmetric chaos and discussed their sensitive dependence and continued with some examples of the Faraday experiments, patterns in neuroscience, and geometric visual hallucinations. He concluded his presentation by stating that the patterns created are not only pretty but also provide us with models that will allow us to understand the patterns that appear in physical and biological systems and often have their genesis in symmetry.

Jennifer Tour Chayes
The final public lecture of the year was presented by Jennifer Tour Chayes, Manager of the Theory Group and Research Area Manager for Mathematics and Theoretical Computer Science, Microsoft Research,on Epidemics in Technological and Social Networks: The Downside of Six Degrees of Separation, April 18, 2007. Chayes started with a discussion of the Internet, WWW and spread of viruses and worms and described how we get epidemics on networks. She then explained how we can view the Internet and WWW as graphs where the nodes are autonomous systems and the edges are the connections and how for the Internet the connections are undirected whereas for the WWW the connections are directed. She further talked about clustering of networks, and explained that the sights that have been around longer and have better quality have more connections, such as "my space". Chayes compared these networks with social networks. She discussed the history of power-law degrees and how the link farms were formed to distort the web and how mathematicians are trying to resolve the distortions caused by these link farms using mathematical models. She also referred to other social examples of networks including the famous acquaintance networks: six degrees of separation and the experiment done by Stanley Milgram to prove it empirically. Chayes turned to the models of the Internet and WWW and other power-law models and discussed some of the rigorous work done in this area. She talked about the viruses and worms on computer networks, the difference between viruses and worms, discussed the code-red worm saga, and finally talked about how to control epidemics on the networks and the recent progress made in finding and implementing the best models to fight these epidemics.