Buchstaber's first research work was in
cobordism theory. He calculated the differential in the
Atiyah-Hirzebruch spectral sequence in
K-theory and
complex cobordism theory, constructed
Chern-Dold characters and the
universal Todd genus in cobordism, and gave an alternative effective solution of the
Milnor-Hirzebruch problem. He went on to develop a theory of double-valued formal groups that led to the calculation of cobordism rings of
complex manifolds having symplectic coverings and to the explicit construction of what are now known as
Buchstaber manifolds. He devised filtrations in
Hopf algebras and the
Buchstaber spectral sequence, which were successfully applied to the calculation of stable
homotopy groups of spheres. He worked on the
deformation theory for mappings to groups, which led to the solution of the
Novikov problem on multiplicative subgroups in operator doubles, and to construction of the
quantum group of complex
cobordisms. He went on to treat problems related both with
algebraic geometry and
integrable systems. He is also well known for his work on sigma-functions on universal spaces of
Jacobian varieties of
algebraic curves that give effective solutions of important integrable systems. Buchstaber created an algebro-functional theory of
symmetric products of spaces and described algebraic varieties of
polysymmetric polynomials. ==Academic career==