Title: Nondesarguesian planes from bijections of the desarguesian plane

Abstract: I shall consider bijections of the projective plane $\mathbb{P}^2(\mathbb{F}_q)$. I will show that certain bijections (called \emph{semiquadratic}) of $\mathbb{P}^2(\mathbb{F}_q)$ can be used to define non-desarguesian planes, i.e., projective planes of the same order as $\mathbb{P}^2(\mathbb{F}_q)$ that are non-isomorphic to $\mathbb{P}^2(\mathbb{F}_q)$. It turns out that these planes are coordinatized by commutative semifields of order $q^3$. A vast family of bijections of  $\mathbb{P}^2(\mathbb{F}_q)$ we construct leads under this connection to a large family of commutative semifields of order $q^3$ that contains and extends known families of semifields found by Zha, Kyureghyan, Wang; and Bierbrauer. Our result also shows that the classification of bijective semiquadratic transformations of $\mathbb{P}^2(\mathbb{F}_q)$ is much more complicated than the corresponding classification of bijections of the projective line $\mathbb{P}^1(\mathbb{F}_q)$ which was recently independently achieved by G¨olo˘glu and Ding/Zieve.

This is based on joint work with Faruk G¨olo˘glu