Summer Math Scholarship

Independent study and Research in Dynamics

In essence dynamics is the study of the solutions of differential equations. Differential equations are often used as models in many areas of science and technology. An example is the Newton Equation for the motion of two planets around a sun. A very simple situation. Nevertheless, nobody is able to write down the solutions of this equation. A reason for this difficulty is that there are orbits, i.e. solutions, which have a rather complicated shape and depend very sensitively on the starting point. The orbits are "chaotic". An example can be seen on this youtube video.

Poincare suggested to not even try to find explicit formulas for the orbits. In stead, he suggested we can study the orbits from a qualitative point of view. For example, it is more important to know whether an orbit is cyclic than exactly its shape.

This qualitative study of dynamics has been successful in low dimensional dynamics, i.e. involving models with not to many variables, one or two. An attractive aspect of dynamics is that one needs many branches of math to understand chaotic orbits. Of course, analysis plays a role. But also topology, geometry, probability theory, functional analysis, and even algebra and number theory.

The independent study part of this project will familiarize the student with the basic, and less basic, notions of the qualitative theory of dynamical systems. One or two students will have the opportunity to continue their study in dynamics by actually starting a small research project during the summer.