Diana Castaneda Santos, Department of Pure Mathematics, University of Waterloo
"Affine and Projective Planes"
As the basic ingredients of our discussion, we shall only require the notion of a set $\mathcal{P}$ of points, together with a collection $\mathcal{L}$ of subsets of $\mathcal{P}$, called lines, satisfying certain conditions. We will define and relate affine and projective planes and provide some numerical results in the finite case. We will show how a projective plane is constructed from a division ring and conclude that there exist finite projective planes of order $p^n$ for any prime $p$. Thanks to a result of Bruck-Ryser it is known that there are no projective planes of order $n$ if $n\equiv 1$ or $2$ mod $4$ and $n$ is not the sum of two squares. If time permits we will see another approach with Latin squares to find projective planes of order $n$.
M3 3103