turkmath.org

In linear algebra, a typical preserver problem demands a characterization of all maps $\Phi:M \rightarrow M$ on some set $M$ of matrices that preserve some given function, relation or a subset. Often there are some additional assumptions on $\Phi$ and/or $M$. Historically, first types of such problems assumed that $M$ is a vector space and that $\Phi$ is linear. Classical assumptions on $\Phi$ are also bijectiveness, injectiveness and surjectiveness.

One of the most influential results in this area are the Hua’s fundamental theorems on various types of matrices. In the context of the set

$M_{m\times n}(\mathbb{F}) = \lbrace m\times n \,\,\,$ matrices with coefficients from the field $\,\,\, \mathbb{F}\rbrace$

such a result characterizes all bijective maps $\Phi:M_{m\times n}(\mathbb{F}) \rightarrow M_{m\times n}(\mathbb{F})$ that satisfy

$rank(A-B)= 1 \Longleftrightarrow rank(\Phi(A)-\Phi(B))= 1$.

It turns out that many other preserver problems can be solved by applying an appropriate ‘Hua’s theorem’. Consequently, mathematicians started to generalize such theorems by reducing their assumptions.

Whenever $\Phi :M \rightarrow M$ preserves some relation (like in Hua’s theorem), the corresponding preserver problem demands the characterization of all endomorphisms of the corresponding graph. In the context of Hua’s theorem above, the problem actually asks to characterize all automorphisms of the graph with the vertex set $V=M_{m\times n}(\mathbb{F})$ and the edge set $E=\lbrace \lbrace A, B \rbrace : rank(A-B)= 1\rbrace$. Consequently, the theory of graph homomorphisms and cores becomes useful to study preserver problems. This is especially true in the case $\mathbb{F}=\mathbb{F}_{q}$ is a finite field, since graph theory is much more developed for finite graphs (as it is for infinite graphs). In the talk I will present the interplay between preserver problems and graph cores. I will also provide some examples where the two areas merge with finite geometry.