# Colloquium abstracts

Nikolai Vavilov
St. Petersburg State University, Russia
January 27, 2011

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 $\phi(E(\Delta,A))$ 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 Borewicz 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. Petrov, the author, You Hong, Luzgarev, Stepanov, Ananievsky, Sinchuk succeeded in completely settling th