Algebraic Graph Theory Seminar - Eric Moorhouse

Title: Projective Planes, Finite and Infinite

Abstract:

A projective plane is a point-line incidence structure in which every pair of distinct points has a unique joining line, and every pair of distinct lines meets in a unique point. Equivalently (as described by its incidence graph), it is a bipartite graph of diameter 3 and girth 6. We also impose a nondegeneracy requirement (e.g. the incidence graph contains an 8-cycle). Thus in the finite case, we have a plane of order $n\geq2$ with $n^2+n+1$ points and the same number of lines; $n+1$ points on each line, and $n+1$ lines through each point.

My survey of the finite case will be very brief.  Mostly I will list some of my favourite problems which are meaningful in both the finite and infinite case, and and discuss how the status of the problem may differ there.  I will focus primarily on problems about embeddability of substructures; automorphisms; and the number and size of orbits.