Grant: Discovery Grants Program - Individual

Physical systems take on many various configurations. A simple example is given by the tip of a pendulum as it moves along the circle given by its radius. As another example, the space of all possible positions of a double pendulum is described by an inner tube (mathematicians call this a torus). An important goal is to understand the motions of physical systems under forces such as gravity. Mathematicians describe these configurations as the zeros of a collection of polynomials, In the example of the circle, the famous equation is x^2 + y^2 -1 = 0. Mathematicians and physicists have developed sophisticated techniques for understanding physical systems through an analysis of the algebraic and geometric properties of the associated equations. These fascinating and sophisticated techniques have developed over centuries of attention. In many instances the configurations exhibit symmetries, and these are best detected by symmetries of the associated polynomials. This is the subject of invariant theory which has its roots in the time of Newton and Vandermonde. These symmetries reflect periodic phenomena which can therefore be understood and described in a wide variety of settings. Modern day applications have been made to such examples as the positioning of satellites through to the detection of appropriate use of stamps by the post-office. In each instance the problem involves the comparison of an image which is rotated or translated, even upside-down, to an existing template picture using the techniques of invariant theory.