Algorithm

By Richard Taylor 

In modular arithmetic, one thinks of the whole numbers arranged around a circle, like the hours on a clock, instead of along an infinite straight line. Here we have seven “hours” on our clock—arithmetic modulo 7. To add 3 and 5 modulo 7, you start at 0, count 3 clockwise, and then a further 5 clockwise, this time ending on 1. To multiply 3 by 5 modulo 7, you start at 0 and count 3 clockwise 5 times, again ending up at 1.

Modular arithmetic has been a major concern of mathematicians for at least 250 years, and is still a very active topic of current research. In this article, I will explain what modular arithmetic is, illustrate why it is of importance for mathematicians, and discuss some recent breakthroughs.

For almost all its history, the study of modular arithmetic has been driven purely by its inherent beauty and by human curiosity. But in one of those strange pieces of serendipity which often characterize the advance of human knowledge, in the last half century modular arithmetic has found important applications in the “real world.” Today, the theory of modular arithmetic (e.g., Reed-Solomon error correcting codes) is the basis for the way DVDs store or satellites transmit large amounts of data without corrupting it. Moreover, the cryptographic codes which keep, for example, our banking transactions secure are also closely connected with the theory of modular arithmetic. You can visualize the usual arithmetic as operating on points strung out along the “number line.”

By Helmut Hofer and Derek Bermel 

This image (produced with a Java applet by Alec Jacobson at http://alecjacobåson.com/programs/three-body-chaos) shows color­ful trackings of the paths of satellites as they evolve from a simple single orbit to a complex multicolored tangle of orbits.

I can’t understand why people are frightened of new ideas. I’m frightened of the old ones.—John Cage

Helmut Hofer, Professor in the School of Mathematics, writes:

Last September, the School of Mathematics launched its yearlong program with my Member seminar talk “First Steps in Symplectic Dynamics.” About two years earlier, it had become clear that certain important problems in dynamical systems could be solved with ideas coming from a different field, the field of symplectic geometry. The goal was then to bring researchers from the fields of dynamical systems and symplectic geometry together in a program aimed at the development of a common core and ideally leading to a new field—symplectic dynamics.

Not long before, in my 2010 inaugural public lecture at IAS, “From Celestial Mechanics to a Geometry Based on the Concept of Area,” I had described the historical background and some of the interesting mathematical problems belonging to this anticipated field of symplectic dynamics. The lecture began with a computer program showing chaos in the restricted three-body problem. This problem describes the movement of a satellite under the gravity of two big bodies, say the earth and the moon, in a rotating coordinates system in which the earth and the moon stay at fixed positions. The chaos in the system is illustrated by putting about ten satellites initially at almost the same position with almost the same velocity.

When the system starts evolving, the program shows colorful trackings of the paths of the satellites as they evolve from a simple single orbit to a complex multicolored tangle of orbits, once the orbits of the different satellites start separating.