In 1972 he received his Candidate of Sciences degree (PhD) with thesis Стабильность действия алгебраических групп и арифметика квазиоднородных многообразий (Stability of the action of algebraic groups and the arithmetic of quasi-homogeneous varieties).

In 1984 he received his Russian Doctor of Sciences degree (habilitation) with thesis Группы, образующие, сизигии и орбиты в теории инвариантов (Groups, generators, syzygies and orbits in the theory of invariants).

[2][3] He is a member of the Steklov Institute of Mathematics and a professor of the National Research University – Higher School of Economics.

[6] In 2006, with Nicole Lemire and Zinovy Reichstein, Popov published a solution to a problem posed by Domingo Luna in 1973.

[7] In 2012, he was elected a member of the inaugural class of Fellows of the American Mathematical Society[8] which recognizes mathematicians who have made significant contributions to the field.