The finite Minkowski space and the existence of ovoids in the orthogonal polar space

Let \F_q be a finite field with q elements, where q is odd, and let A=A^{\top}\in GL_n(\F_q) be an invertible symmetric matrix of size n with coefficients in \F_q. The talk will be about the maps \Phi :\F_q^n\to \F_q^n that satisfy the implication
(x-y)^{\top}A(x-y)=0,  x\neq y
\Longrightarrow
\big(\Phi(x)-\Phi(y)\big)^{\top}A\big(\Phi(x)-\Phi(y)\big)=0, \Phi(x)\neq \Phi(y)
for all column vectors x,y\in \F_q^n. The classification problem of such maps is partially related to some physics. The results and the techniques applied in the proofs are related to finite geometry and graph theory. Primarily, this is a typical `preserver problem’ studied in matrix theory.