Word maps of simple algebraic groups

Let $F_m$ be the free group of rank $m$. Then for any word ${w=w(x_1,\dots ,x_m)\in F_m}$ and for any group $G$ one can define the word map $\tilde{w}\colon G^m\rightarrow G$ by the formula: $\tilde{w}((g_1, \ldots, g_m)) := w(g_1, \ldots, g_m)$. Word maps have been intensely studied over at least two past decades in various contexts. In this talk we deal with the case where $G=\mathcal G(K)$ is the group of $K$-points of a simple linear algebraic group $\mathcal G$ defined over a field $K$. Here we consider the problem of surjectivity of word maps and also some related questions.