In Calculus I, we learn about families of functions. For instance, y=t*x is a family of lines parametrized by the real line and y=x^2+a*x+b is a family of parabolas parametrized by the plane. It would be natural to ask if we can parametrize "all" lines, parabolas, or whatever geometric object we are interested in. We will discuss what it means to parametrize all objects of a given type using objects called moduli spaces. We will then discuss some of the interesting questions posed by moduli spaces and how a moduli space can be used to elegantly prove the fundamental theorem of algebra.
The Geometry of Families