He has received his PhD at the Landau Institute for Theoretical Physics.
His supervisor was Prof. A. Polyakov, however, at the time of his PhD courses ("aspirantship") Vadim was already a first-class physicist, and supervising him was a pure formality.
At the secondary school he won twice the USSR national physics olympiad.
He wrote his first scientific article (in collaboration with Prof. L. Andreev) as a student in 1982.
[1] Knizhnik died of a heart attack in Moscow at the age of 25.