Asyptoptic and Combinatorial Methods in Infinite Groups and Algebras

本文主要研究无限维代数和无限剩余有限群及其相互作用。我们计划研究最近发现的具有无穷微分算子的李代数族，这些李代数族在既有零又有多项式增长的正特征域上。我们打算继续研究gold - shafarevich群的渐近性质和Lubotzky-Sarnak猜想。这些群自然出现在三维拓扑和数论中。这一工作也与理论计算机科学中扩展子空间的Wigderson问题有关。我们将进一步发展光谱问题与gold - shafarevich群的线性表示之间的联系。我们希望该提案的工作结果将导致解决几个长期存在的问题，并将对理论计算机科学（快速扩展系统）产生影响。

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 haveimplications for Theoretical Computer Science (fast expanding systems).