Department of Mathematics
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Is it possible to define the concepts of "circle", "interior" and "exterior" of a circle, a tangent to the curve etc. in a general projective plane where we lack the tools of analysis like continuity and derivatives? One of the simplest properties of a circle is that no three points of it are collinear and a tangent at P contains no other point of the circle. Also, every projective conic is isomorphic to the corresponding projective line under a suitable birational mapping and hence any potential "conic" in a projective plane of order q must contain q+1 points. A surprising theorem, due to B. Segre, says that in finite projective planes these point-sets called ovals satisfying such simple "cardinality conditions" are algebraically constrained. In fact, they are the zeros of an irreducible second degree polynomial i.e. they are algebraic conics!

This talk is a survey of known results in this direction.