Our researchers study algebraic geometry, both classical and derived noncommutative, arithmetic geometry, and algebra. The subjects of their interest include Mirror Symmetry, the theory of Landau-Ginzburg models, birational geometry and the Minimal Model Program, automorphisms of algebraic varieties and algebraic groups, the theory of arithmetic lattices, the study of zeta-functions of algebraic varieties, the theory of higher-dimensional adeles, algebraic K-theory, and the theory of algebraic cycles.
Research areas: algebraic commutative and noncommutative geometry, derived and triangulated categories, (quasi-)coherent sheaves, motives, K-theory, Mirror Symmetry, birational geometry, Minimal Model Program, Fano varieties, algebraic groups, arithmetic varieties, higher-dimensional adeles.
Alexander Efimov solved several fundamental hard problems in non-commutative geometry: he proved a conjecture of M. Kontsevich on homotopy finiteness of derived categories of coherent sheaves on separated schemes of finite type over a field of characteristic zero [1], he found counterexamples to conjectures of Kontsevich about generalized Hodge-to-de Rham degeneration for dg-categories [2], and, in particular, he obtained a negative answer to a question of Bertrand Toen, a world known top expert in this field.
Sergey Gorchinskiy and Denis Osipov completed a local theory of the higher-dimensional Contou-Carrere symbol, in particular, characterizing the symbol by a universal property. One-dimensional theory has been developed by a number of authors, including P. Deligne, A. Beilinson, S. Bloch, H. Esnault. Two-dimensional theory was partially treated by Osipov and Xinwen Zhu, but already in the two-dimensional case, one needs completely different methods, which were finally developed by Gorchinskiy and Osipov with the help of a novel geometric approach to iterated Laurent series.