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Ewing Hall, University of Delaware, Newark, DE 19716, USA
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Title: PTR polynomials for the Hughes planes
Abstract: Coordinatization was introduced by Marshall Hall in the 1940s, introducing algebraic methods to the study of projective planes. The resulting object, known as a planar ternary ring (PTR), is a trivariate function with specific properties. When the plane has prime power order, the PTR can be described as a polynomial over a finite field. While the PTR produced by the plane is coordinatization dependent, each PTR uniquely describes a projective plane. Thus, studying the PTR is equivalent to studying the projective plane. In this talk, we give the first description of a class of PTR polynomials for the Hughes planes, a historically important infinite class of projective planes first described in the 1950s. Unexpectedly, the polynomial forms have generalized Catalan numbers among their coefficients.
