Fuchs received in 1964 his Russian candidate degree (Ph.D.) under
Albert S. Schwarz at
Moscow State University, where he taught thereafter. Schwarz conducted a seminar on algebraic topology with
Mikhail Postnikov and
Vladimir Boltyanski. Fuchs participated in the seminar and, as a student, published papers with Schwarz, as did
Askold Ivanovich Vinogradov a few years earlier. Fuchs received his Russian doctorate (higher doctoral degree) in 1987 at
Tbilisi State University. Since 1991 he has been a professor at the
University of California, Davis. With
Israel Gelfand he introduced in 1970 the Gelfand-Fuchs cohomology of Lie algebras. Gelfand-Fuchs cohomology has applications in the proof of the
Macdonald identities in combinatorics and in the calculation of characteristic classes of
foliations. With
Boris Feigin he determined the structure of
Verma modules in the
Virasoro algebra representation theory, which has applications in
string theory and
conformal field theory. His students include Boris Feigin (with whom he has collaborated extensively), Fedor Malikov,
Sergei Tabachnikov, and
Vladimir Rokhlin, as well as
Edward Frenkel for whom Fuchs was a second advisor. Frenkel, among many others, was affected by the Soviet antisemitism which flourished from 1954 to 1970 (described by Fuchs) and on into the 1980s. In 1978 Fuchs was an Invited Speaker with talk
New results on the characteristic classes of foliations at the
International Congress of Mathematicians in
Helsinki. ==Personal life==