The Graph Isomorphism (GI) problem is celebrated and a central open problem in theoretical computer science and mathematics. Given two graphs as input, GI asks whether they are isomorphic. The best-known algorithm for GI runs in quasi-polynomial time is due to Babai. Further improvement is still an open question. Thus, it is interesting to study some special cases of GI, e.g., the Group Isomorphism (GrI) problem. Given two groups as input by their multiplication table, GrI asks whether they are isomorphic. The problem admits a trivial quasi-polynomial time algorithm, which is due to Miller attributed to Tarjan. Over the past few years, GpI has barely seen any asymptotic improvement. 

In computational complexity theory, roughly speaking, reduction is an algorithm for transforming one computational problem into another problem. There are certain problems in computational group theory which admit trivial quasi-polynomial time algorithms when the input group is given as a multiplication table. Such problems might help us to study the complexity of GrI via reduction. Currently, I am studying problems from computational group theory, which admits easy quasi-polynomial time algorithms, but it is unknown if they admit better algorithms.

I am also studying algorithmic problems on groups where the input group is given by a generating set. In this setting, it is more challenging to design an efficient algorithm as the size of the group could be much larger than the input size. Problems defined in this model have many application in solving GI.