Spectral Sequences for Cyclic Homology
We prove the non-commutative Hodge-to-de Rham Degeneration Conjecture of Kontsevich and Soibelman.
Let Fλ be a generalized flag variety of a simple Lie group G embedded into the projectivization of an irreducible G-module Vλ. We define a flat degeneration Fλa, which is a GaM variety. Moreover, there exists a larger group Ga acting on Fλa, which is a degeneration of the group G. The group Ga contains GaM as a normal subgroup. If G is of type A, then the degenerate flag varieties can be embedded into the product of Grassmannians and thus to the product of projective spaces. The defining ideal of Fλa is generated by the set of degenerate Plüker relations. We prove that the coordinate ring of Fλa is isomorphic to a direct sum of dual PBW-graded g-modules. We also prove that there exist bases in multi-homogeneous components of the coordinate rings, parametrized by the semistandard PBW-tableux, which are analogues of semistandard tableaux.
We present a combinatorial monomial basis (or, more precisely, a family of monomial bases) in every finite-dimensional irreducible so2n+1-module. These bases are in many ways similar to the FFLV bases for types A and C. They are also defined combinatorially via sums over Dyck paths in certain triangular grids. Our sums, however, involve weights depending on the length of the corresponding root. Accordingly, our bases also induce bases in certain degenerations of the modules but these degenerations are obtained not from the filtration by PBW degree but by a weighted version thereof.
After the breakup of the Soviet Union, the behaviour of Russians in all spheres of life began to change; the demographic dimension is not an exception. Nowadays youth born and raised in modern Russia have another vision of the sequence of demographic events than that held by Soviet generations. According to examples of other countries and the theory of the Second Demographic Transition, it is possible to assume that differences in norms, values, and views could lead to a shift in calendars and intensity of births and matrimonial events. Have these value transformations resulted in transformations of actual reproductive and matrimonial behaviour of Russians? If so, what is the direction of changes? We tried to answer these and accompanying questions in the paper. Using Event History Analysis (EHA) we revealed a relationship between matrimonial and reproductive behaviour across generations, defined zones of their basic distinctions, and made assumptions concerning the reasons and consequences of these transformations.