The proposal focuses on infinite dimensional algebras and infinite residually finite groups and their interactions. We plan to study the recently discovered families of Lie algebras of infinite differential operators over fields of positive characteristics which are both nil and of polynomial growth. We intend to continue our work on asymptotic properties of Golod-Shafarevich groups and the Lubotzky-Sarnak conjecture. These groups naturally arise in 3-dimensional topology and number theory. This work is also related to the Wigderson's problem of expanding subspaces in the theoretical computer science. We will further develop the connections between the Specht Problem and linear representations of Golod-Shafarevich groups.
We hope that the results of the work on the proposal will lead to solutions of several long standing problems and will have implications for Theoretical Computer Science (fast expanding systems).
