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 plays in understanding our world and shaping our lives.

Summary of the talks

He showed by using several examples how simple mathematics can explain complex natural phenomena such as the growth of sunflower seeds in spiral patterns descrived by Fibonacci numbers and the golden section. He then discussed how recent multiscale geometrical techniques are used to set up mathematical models and how we go about solving them using variety of geometrical and numerical methods. After finding the solution, We must check and see if the solution to our mathematical model is close enough to reality. He gave several examples of mathematical modeling, including modeling aircrafts using geometrical methods, modeling the environment - specifically one on theory on tides, and pollutants in Venice Lagoons. He talked about mathematical models for conservation of the architectural heritage, discovering the geometry of artery of patients using segmentation and simulating the blood flew in the vessels, using numerical methods. He showed the mathematical equations behind each example which lead to the simulations in each case. He continued with a few examples from mathematics of vorticity and turbulence which were used to design cars for racing, and swimsuits to speed up the swimmer's speed.

Finally he turned to the Olympic rowing and concluded the talk with a brief review of mathemtics used to build and race the recent yachts in America's cup which started in 1851 and was won by Americans for 132 year. In 1983, Australians used mathematics to design their yacht and won the cup and since then most teams have been using modeling in their race. He then explained the details of the mathematical models with 162 million unknowns and 10^6 operations which were used to design Alinghi which has lead to the swiss team winning the last two cups. They used partial differential equations to model the flow of the sea around the hull, the dynamics of the air and the sails, and the turbulent interaction of the ocean, wind, and boat and applied advanced numerical algorithms to solve these equations on high performance computers. This allowed them to optimize such things as hull and keel design, sail geometry and placement.

Ivar Ekeland

Galileo was the first scientist to say that the universe is a book written in mathematical language. Many scientists tried to reconcile the language of bible and the language of creation by trying to show that science could help us discover blueprint of creation. The idea of optimization is to find the best possible solution. Monreau believed that God optimizes and optimization applies to all of physics which he introduced through "the principle of least action". Euler, who was a devout Calvinist, agreed more or less with the theology of Maupertuis whose basic approach, turned out to be essentially correct and has become one of the pillars of modern science.

Nature does not optimize but do human beings maximize? Ekeland discussed the rational behavior, ability to make coherent choices, collective decision making of human beings and talked about Arrow's theorem: any collective decision depends as much on the procedure as on the individual preference. He used an example to show that groups have circularity problem and do not optimize and for groups the process is important and small groups acting for special interests are much more likely to influence public. He concluded by talking about the Stern review on the economics of climate change and the need for the use of feedback in the process.