The proposal deals primarily with the construction and classification of finite semifields, and the construction and enumeration of unitals embedded in semifield planes and regular nearfield planes. A semifield is an algebraic structure satisfying all the axioms of a field, except possibly commutativity and associativity under multiplication. In recent years there has been renewed interest in semifields, which have fascinating connections to many areas of finite geometry and combinatorics, such as translation planes, flocks of a quadratic cone, q-clans, translation generalized quadrangles, skew Hadamard difference sets, and pseudo-Paley graphs. A major part of the proposal is to use certain subspaces of linearized polynomials to produce new infinite families of semifields, and then use the associated "linear sets" to classify all finite semifields that are two-dimensional over their left nucleus and six-dimensional over their center.

A unital is the natural design theoretic generalization of a Hermitian curve in a Desarguesian square order projective plane. Many infinite families of non- Desarguesian square order planes also are known to contain embedded unitals, and it seems that unitals play a significant role in the basic understanding of all square order projective planes. F. Buekenhout has developed two general techniques for constructing unitals in certain translation planes, but the only enumeration of embedded Buekenhout unitals occurs in the Desarguesian plane. A second major part of the proposal involves the investigation and enumeration of Buekenhout unitals embedded in the regular nearfield planes. Hopefully, any techniques developed will naturally generalize to other translation planes, such as the Hall planes and certain families of semifield planes.

Article created: July 20, 2008