Semin Yoo (유세민), Combinatorics of Euclidean spaces over finite fields

$q$-analogues of quantities in mathematics involve perturbations of classical quantities using the parameter $q$, and revert to the original quantities when $q$ goes $1$. An important example is the $q$-analogues of binomial coefficients, denoted by ${(\genfrac{}{}{0ex}{}{n}{k})}_q$, which give the number of $k$-dimensional subspaces in ${\mathbb{F}}_q^n$. When $q$ goes to $1$, this reverts to the binomial coefficients which measure the number of $k$-sets in $\left[n\right]$.

In this talk, we add one more structure in ${\mathbb{F}}_q^n$, which is the Euclidean quadratic form: ${\text{dot}}_n:=x_1^2+x_2^2+\cdots +x_n^2$. It turns out that the number of quadratic subspaces of Euclidean type in $({\mathbb{F}}_q^n,{\text{dot}}_n)$ can be described as the form of the analogue of binomial coefficients. The main goal of this talk is to define the dot-analogues of the binomial coefficients and to study related combinatorics. No prior knowledge about the theory of quadratic form is required.