**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