Semin Yoo


University of Rochester


Thursday, January 14, 2021 - 3:00pm to 4:00pm


Zoom: https://uci.zoom.us/j/94525934294
In this talk, we introduce a new isometric invariant of combinatorial type on the quadratic space $(\mathbb{F}_{q}^{n},x_{1}^{2}+\cdots+x_{n}^{2})$ over $\mathbb{F}_{q}$. First, we recall some basic facts about quadratic forms. In particular, we will restrict ourselves to the case, where the base field is finite. In order to define this new invariant, we introduce special types of lines, named after line types in Minkowski's geometry. We prove that counting lines of each type is an isometric invariant on the quadratic space $(\mathbb{F}_{q}^{n},x_{1}^{2}+\cdots+x_{n}^{2})$ over $\mathbb{F}_{q}$. In order to prove this theorem, we redrive Minkowski's formula for the size of spheres on finite fields in an elementary way. Only some elementary facts of number theory are required for this talk.