His interests are
algebra,
algebraic geometry and
number theory. He solved the Strong approximation problem, developed the reduced
K-theory and solved the
Tannaka–Artin problem. He solved the
Kneser-Tits and Grothendieck problems. Together with F. Grunewald he solved the arithmeticity problem for finite extensions of arithmetic groups and the rigidity problem for arithmetic subgroups of algebraic groups with radical. Platonov solved the rationality problem for
spinor varieties and the Dieudonne problem on
spinor norms. Platonov was an invited speaker of the International Congresses of Mathematicians in Vancouver (1974), Helsinki (1978) and the European Congress of Mathematicians in Budapest (1996). He is a member of the
Canadian Mathematical Society and from 1993 to 2001 was a Professor of the
Faculty of Mathematics of the
University of Waterloo in
Waterloo, Ontario,
Canada. He is the author, with Andrei Rapinchuk, of
Algebraic Groups and Number Theory. He currently works as a Chief Science Officer of
Scientific Research Institute of System Development (NIISI RAN). == Assault conviction ==