Calculations in Exceptional Groups over Rings

One of the major discoveries of the late XIX and XX century was  construction, alongside with the familiar classical linear, symplectic, and orthogonal groups, of the 5 exceptional groups of types $E_6, E_7, E_8, F_4$ and $G_2$. These groups form an inalienable part of the classification of simple Lie groups, simple algebraic groups, finite simple groups, etc.  It is easy to calculate in classical groups using their small degree matrix representations. Calculations in exceptional groups over fields are based on short canonical forms, such as Bruhat or Gauss decomposition.  However, over rings no such canonical forms can possibly exist, in general. Thus, one has to use representation theory.
In the present series of seminars we describe a major project carried forward by the author and his students, which allows direct matarix calculations in exceptional Chevalley groups as matarices of degrees 27x27, 56x56,248x248, 27x27 and 8x8 respectively. We describe several key reductions and simplifications proposed by the author, jointly with Stepanov and Plotkin, and then jointly with Gavrilovich, Nikolenko and Luzgarev, which make such computations feasible.  I will discuss also some typical applications,such as structure theorems, description of [various classes of] subgroups, and lower K-theory. ............contd........ To make these seminars accessible to students, I plan to start with briefly recalling some basic definitions and methods used for groups over fields and for classical groups.