New results on Subgroups of Classical Groups

We give an account of some recent results on description of subgroups of a [classical] Chevalley group $G{\Phi, R)$ of type $\Phi$
over a commutative ring $R$, containing an elementary subgroup of $\Phi(E(\Delta, A))$ in a rational representation $\Phi$. A natural context to specify broad classes of large semi-simple subgroups in classical groups is provided by Aschbacher's  subgroup structure theorem and its generalisation to exceptional groups by Liebeck and Seitz. Until recently, little was known on description of subgroups from Aschbacher classes. The only case which was completely settled in the 1980-ies, originally by Borewics and the author, were overgroups of subsystem subgroups, class C_1+C_2.  Generalisations of these results to other classes were widely discussed, but no definitive results were in sight until 2000. Over the last decade the situation changed dramatically.   ...........contd.....
..................Here, one should from the very start take account of the effects from the theory of algebraic groups, whereas the proofs heavily rely on the power of localisation methods, such as Quillen - Suslin - Vaserstein localisation and patching, or Bak's localisation-completion.