Research in Combinatorial Group Theory

这位研究人员将致力于他的强大的分级图几何机制的进一步发展和应用，该机制是为了解决著名的Burnside问题(提出于1902年)而创建的，该问题关于具有足够大的偶数指数的周期群。特别是，他将扩展这一机器的主要归纳构造的框架，以获得关于定义关系子为n次方的群的更多结果。此外，他还将研究关于自由群的子群的Hanna Neumann猜想的推广版本和关于平凡群表示的长期存在的Stallings问题。这项研究是组合群论，研究通过生成元和定义关系子定义的群，处于低维拓扑、几何和数理逻辑群论的交叉点。群论是一种数学对称性理论，它与许多其他学科相互作用，例如数学之外的物理和化学，数学内部的编码论、数论、拓扑学和几何。

The investigator will work on further development and applications of his powerful geometric machinery of graded diagrams which was created to solve the famous Burnside problem (posed in 1902) on periodic groups of sufficiently large even exponents. In particular, he will extend the framework of the main induction construction of this machinery to obtain more results on groups whose defining relators are nth powers. In addition, he will work on a generalized version of the Hanna Neumann conjecture on subgroups of free groups and long-standing Stallings' problem on presentations of the trivial group.This research is in combinatorial group theory which studies groups defined by means of generators and defining relators and is at the intersection of the theory of groups with low dimensional topology, geometry, and mathematical logic. The theory of groups is a mathematical theory of symmetry which interacts with many other disciplines, for example, physics and chemistry outside of mathematics, coding theory, number theory, topology and geometry inside mathematics.