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^nto F_q^n that satisfy the implication
(x-y)^{top}A(x-y)=0,  xneq y
Longrightarrow
big(Phi(x)-Phi(y)big)^{top}Abig(Phi(x)-Phi(y)big)=0, Phi(x)neq Phi(y)
for all column vectors x,yin 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.