Manin's early work included papers on the arithmetic and
formal groups of
abelian varieties, the
Mordell conjecture in the
function field case, and
algebraic differential equations. The
Gauss–Manin connection is a basic ingredient of the study of
cohomology in families of
algebraic varieties. He developed the
Manin obstruction, indicating the role of the
Brauer group in accounting for obstructions to the
Hasse principle via
Grothendieck's theory of global
Azumaya algebras, setting off a generation of further work. Manin pioneered the field of
arithmetic topology (along with
John Tate,
David Mumford,
Michael Artin, and
Barry Mazur). He also formulated the
Manin conjecture, which predicts the asymptotic behaviour of the number of rational points of bounded height on algebraic varieties. In mathematical physics, Manin wrote on
Yang–Mills theory,
quantum information, and
mirror symmetry. He was one of the first to propose the idea of a
quantum computer in
1980 with his book
Computable and Uncomputable. He wrote a book on
cubic surfaces and
cubic forms, showing how to apply both classical and contemporary methods of algebraic geometry, as well as
nonassociative algebra. == Awards ==