Modern Algebra and Its Application

• We have solved the problem of dimensional subgroups formulated in the 1950s and the 1960s.
• We have produced functorial description of the Hopf algebra H_*(A,Z/2).
• Our researchers have provided a complete solution of the Bousfield problem.
• We have achieved new results in the theory of limits in categories of free corepesentations, in particular new descriptions of fr-proposals encoding functors in categories of groups.
• Our researchers have found a series of three-dimensional manifolds that are pairwise not homologically equivalent, but their groups have isomorphic additions.
• We have proven that for any oriented theory of cohommologies motivic decomposition is the same as the decomposition of some (infinite dimensional) Hecke algebra that is built using formal group law into the direct sum of its ideals.
• Our researchers have proved the theorem concerning increase of the decomposition of the tensor co-algebra functor to the level of unstable Adams spectral sequences for spaces that are suspensions.
• We have been continuing research of connection between the spectral graph theory of group actions and the quasi-crystal theory. This research was commenced by R.I. Grigorchuk, D. Lenz and T.V. Nagniveda in 2013. Using new combinatorial methods we have localized the theorem concerning the Cantor spectrum of Lebesgue null-measure for Schrödinger and Jacobi operators on a wide class of symbolic dynamical systems including Sturm shift and simple Teplitz shifts. By expansion of the class of dynamic systems for which a theorem concerning Cantor spectrum of Lebesgue null-measure is known we solved the problem of non isotropic Markov operators over Schreier graphs of all the groups of uncountable family of groups of intermediate growth group introduced by R. I. Grigorchuk in 1984. We have obtained results indicating spectral type non-rigidity for Laplace operator over Grigorchuk groups depending on choice of generator systems.
• For some quasi-splittable reductive G groups over the common F field we have built automorphism i of the G group over F that is correctly defined as an element of the factor on inner automorphisms. Automorphism i generalizes (only for quasi-splittable groups) involution built by Moeglinniy, Vignéras and Waldspurger for classical groups that sends irreducible allowable representation G(F) for the classical G group and the local field F into dual (contragredient) representation.
• We have formulated a hypothesis concerning the contragradient representation of the irreducible allowable representation G(F) for the reductive algebraic group G over the local field F in terms of (increased) Langlands parameter of the representation.
• We have shown that the Schellach and Barner theorem is not true if n=2.

Education and career development: 2 candidate dissertations have been defended