Work
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 lead 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 lead 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.
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