Konrad Swanepoel

Title: Ordinary circles and extraordinary curves

Abstract: An ordinary circle of a set of n points in the plane is defined as a
circle that contains exactly three points of the set. We show that if the 
set is not contained in a line or a circle, then it spans at least 
n^2 / 4 - O(n) ordinary circles. For sufficiently large n, we determine 
the exact minimum number of ordinary circles and describe all point sets 
that come close to this minimum.

This description is based on a structure theorem that states that for large n,
if a set of n points spans at most Kn^2 ordinary circles, then all but 
O(K) points of the set lie on a special type of algebraic curve of degree 
at most four.

Joint work with Aaron Lin, Mehdi Makhul, Hossein Nassajian Mojarrad, Josef
Schicho, and Frank de Zeeuw.