Title:  Projective Geometry
<br>
Abstract:
I will explain why doing geometry in the plane (or really, in any 
Euclidean
Space) isn't quite good enough, and we will introduce projective space and
some main ideas of Projective Geometry.  I will discuss the duality 
between
points and lines in the projective plane, and how theorems translate into
their "dual" counterparts.  Then we can move up a degree and talk about
conics in projective space.  Starting with conics over the real numbers, 
we
can move to conics over the complex numbers, and perhaps even finite 
fields
where we can ask to count the number of rational points a given conic has.