Projective Planes
Speaker
Navin Singhi
Tata Institute of Fundamental Research India
http://www.math.tifr.res.in/
Description
A finite projective plane is finite geometry in which any two points determine a line and any two lines determine a point, and (to exclude degenerate examples) there are four points such that no line is incident with more than two of them. These can exist only for point sets of order n2+n+1 with the additional properties that every point has n+1 lines through it, and very line contains n+1 points. For example, projective plane of order two is the well known Fano plane. These geometries have been widely used to give (optimal) constructions in combinatorics, among others.
Two well known conjectures on projective planes state that the order of every finite projective plane is a power of a prime number and that a finite projective plane with no proper subplanes is isomorphic to a plane coordinatized by a prime field. In this talk we will present a structure theory for free semiadditive rings, which provide an analogue of the ring of integers and polynomials for the ternary rings. Application of these methods to the above mentioned conjectures will be discussed.
Two well known conjectures on projective planes state that the order of every finite projective plane is a power of a prime number and that a finite projective plane with no proper subplanes is isomorphic to a plane coordinatized by a prime field. In this talk we will present a structure theory for free semiadditive rings, which provide an analogue of the ring of integers and polynomials for the ternary rings. Application of these methods to the above mentioned conjectures will be discussed.
Event Topic
Discrete Applied Math Seminar