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Volume 77, Nº 4 (2022)
- Ano: 2022
- Artigos: 8
- URL: https://journals.rcsi.science/0042-1316/issue/view/7529
Equivariant completions of affine spaces
Resumo
We survey recent results on open embeddings of the affine space $\mathbb{C}^n$ into a complete algebraic variety $X$ such that the action of the vector group $\mathbb{G}_a^n$ on $\mathbb{C}^n$ by translations extends to an action of $\mathbb{G}_a^n$ on $X$. We begin with the Hassett–Tschinkel correspondence describing equivariant embeddings of $\mathbb{C}^n$ into projective spaces and present its generalization for embeddings into projective hypersurfaces. Further sections deal with embeddings into flag varieties and their degenerations, complete toric varieties, and Fano varieties of certain types.Bibliography: 109 titles.
Uspekhi Matematicheskikh Nauk. 2022;77(4):3-90
3-90
Classification of involutive commutative two-valued groups
Resumo
A complete classification of finitely generated involutive commutative two-valued groups is obtained. Three series of such two-valued groups are constructed: a principal series, a unipotent series, and a special series; it is shown that any finitely generated involutive commutative two-valued group is isomorphic to a two-valued group in one of these series. A number of classification results are obtained for topological involutive commutative two-valued groups in the Hausdorff and locally compact cases. The classification of algebraic involutive two-valued groups in the one-dimensional case is also discussed.Bibliography: 45 titles.
Uspekhi Matematicheskikh Nauk. 2022;77(4):91-172
91-172
Schubert calculus and intersection theory of flag manifolds
Resumo
Hilbert's 15th problem called for a rigorous foundation of Schubert calculus, of which a long-standing and challenging part is the Schubert problem of characteristics. In the course of securing a foundation for algebraic geometry, Van der Waerden and Weil attributed this problem to the intersection theory of flag manifolds.This article surveys the background, content, and solution of the problem of characteristics. Our main results are a unified formula for the characteristics and a systematic description of the intersection rings of flag manifolds. We illustrate the effectiveness of the formula and the algorithm by explicit examples.Bibliography: 71 titles.
Uspekhi Matematicheskikh Nauk. 2022;77(4):173-196
173-196
Weak solvability of motion models for a viscoelastic fluid with a higher-order rheological relation
Uspekhi Matematicheskikh Nauk. 2022;77(4):197-198
197-198
Bi-Lipschitz isomorphisms of self-similar Jordan arcs
Uspekhi Matematicheskikh Nauk. 2022;77(4):199-200
199-200
Topological classification of flows without heteroclinic intersections on a connected sum of manifolds $\mathbb{S}^{n-1}\times\mathbb{S}^{1}$
Resumo
In this paper, we announce a result on the possibility of obtaining sufficient conditions for topological conjugacy of gradient-like flows without heteroclinic intersections, given on a connected sum of products $S^{n-1}\times S^1$ in combinatorial terms.
Uspekhi Matematicheskikh Nauk. 2022;77(4):201-202
201-202
Monomial non-Golod face rings and Massey products
Uspekhi Matematicheskikh Nauk. 2022;77(4):203-204
203-204
Logarithmic Sobolev inequality and Hypothesis of Quantum Gaussian Maximizers
Uspekhi Matematicheskikh Nauk. 2022;77(4):205-206
205-206