Word maps on simple algebraic groups and related topics

Let $G$ be a group and let $F_n$ be the free group of rank $n$. For any word $w\in F_n$ we can define a word map $\widetilde{w}:G^n\to G$ by formula $\widetilde{w}((g_1,\dots,g_n))=w(g_1,\dots,g_n)$. The investigation of word maps is a popular topic during last 10-15 years, especially, in the case when $G=\mathcal{G}$ or $G=\mathcal{G}(K)$ for a simple algebraic group $\mathcal{G}$ which is defined over a field $K$. In this talk we suppose to give a brief survey of the results and problems of the theory of word maps on simple algebraic groups and the relations of this theory to the Representation Theory.