On July 20, 2021, Semin Yoo (유세민) from the University of Rochester gave a talk on an analogue of q-binomial coefficients, counting subspaces having the Euclidean quadratic form, and its applications at the Discrete Math Seminar. She will move to KIAS next month as a postdoc. The title of her talk was “Combinatorics of Euclidean spaces over finite fields“.

## 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 $\binom{n}{k}_{q}$, 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.