Robert Garbary, Department of Pure Mathematics, University of Waterloo
"Parametrizing Points on Algebraic Curves"
Algebraic Geometry is the study of subsets of kn (here, k is a field) defined by the vanishing of polynomial equations - these objects are called algebraic varieties. For example, the circle S1 R2 is an algebraic variety (over R) since it is precisely the zero locus of the polynomial x2+y2 1 2 R[x; y]. The circle is an example of a plane curve - the vanishing of a single polynomial inside the plane k2. These objects get complicated very quickly. To illustrate this, I'm going to show a nice way to find all the rational points on a quadratic curve by parametrizing it using rational functions. I'm then going to show that the points of a cubic curve cannot be parametrized by rational functions. This reflects a very strong rigidity property that algebraic varieties have. Time permitting, I will talk about why cubic curves are so great, and about points on higher degree plane curves.