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Vol 77, No 4 (2022)

Year: 2022
Articles: 8
URL: https://journals.rcsi.science/0042-1316/issue/view/7529

Equivariant completions of affine spaces

Arzhantsev I.V., Zaitseva Y.I.

Abstract

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.

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Uspekhi Matematicheskikh Nauk. 2022;77(4):3-90

3-90

(Rus)

(JATS XML)

Classification of involutive commutative two-valued groups

Buchstaber V.M., Veselov A.P., Gaifullin A.A.

Abstract

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.

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Uspekhi Matematicheskikh Nauk. 2022;77(4):91-172

91-172

(Rus)

(JATS XML)

Schubert calculus and intersection theory of flag manifolds

Duan H., Zhao X.

Abstract

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.

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Uspekhi Matematicheskikh Nauk. 2022;77(4):173-196

173-196

(Rus)

(JATS XML)

Weak solvability of motion models for a viscoelastic fluid with a higher-order rheological relation

Zvyagin V.G., Orlov V.P.

Uspekhi Matematicheskikh Nauk. 2022;77(4):197-198

197-198

(Rus)

(JATS XML)

Bi-Lipschitz isomorphisms of self-similar Jordan arcs

Galay I.N., Tetenov A.V.

Uspekhi Matematicheskikh Nauk. 2022;77(4):199-200

199-200

(Rus)

(JATS XML)

Topological classification of flows without heteroclinic intersections on a connected sum of manifolds $\mathbb{S}^{n-1}\times\mathbb{S}^{1}$

Grines V.Z., Gurevich E.Y.

Abstract

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.

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Uspekhi Matematicheskikh Nauk. 2022;77(4):201-202

201-202