He was a professor at the Moscow State University and was known for his work in group theory, especially on the Burnside problem.
This is one of the most remarkable, beautiful, and general results in algorithmic group theory and is now known as the Adian–Rabin theorem.
In spite of numerous attempts, nobody has added anything fundamentally new to the results during the past 50 years.
Adian's result was immediately used by Andrey Markov Jr. in his proof of the algorithmic unsolvability of the classical problem of deciding when topological manifolds are homeomorphic.
This just demonstrates that before their work nobody even came close to imagining the nature of the free Burnside group, or the extent to which subtle structures inevitably arose in any serious attempt to investigate it.
An approach to solving the problem in the negative was first outlined by P. S. Novikov in his note, which appeared in 1959.
However, the concrete realization of his ideas encountered serious difficulties, and in 1960, at the insistence of Novikov and his wife Lyudmila Keldysh, Adian settled down to work on the Burnside problem.
Completing the project took intensive efforts from both collaborators in the course of eight years, and in 1968 their famous paper appeared, containing a negative solution of the problem for all odd periods
The solution of the Burnside problem was certainly one of the most outstanding and deep mathematical results of the past century.
In many respects the work was literally carried to its conclusion by the exceptional persistence of Adian.
In that regard it is worth recalling the words of Novikov, who said that he had never met a mathematician more ‘penetrating’ than Adian.