The theory of continua

This survey is devoted to divisions of the theory of continua associated with snake- like, tree- like, and circlelike bicompacta, homogeneous spaces, hyperspaces of continua, and Whitney maps. Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 29, pp. 63–119, 1991.     

New representations of algebraic domains and algebraic L-domains via closure systems

Closure systems (spaces) play an important role in characterizing certain ordered structures. In this paper, FinSet-bounded algebraic closure spaces are introduced, and then used to provide a new approach to constructing algebraic domains. Then, a special family of algebraic closure spaces, algebraic L-closure spaces, are used to represent algebraic L-domains. Next, algebraic approximate mappings are defined and serve as the appropriate morphisms between algebraic closure spaces, respectively, algebraic L-closure spaces. On the categorical level, we show that algebraic closure spaces (respectively, algebraic L-closure spaces,) each equipped with algebraic approximate mappings as morphisms, are equivalent to algebraic domains (respectively, algebraic L-domains) with Scott continuous functions as morphisms.

     

Another Approach to Topological Descent Theory

In the category T op of topological spaces and continuous functions, we prove that surjective maps which are descent morphisms with respect to the class E of continuous bijections are exactly the descent morphisms, providing a new characterization of the latter in terms of subfibrations E(X) of the basic fibration given by T op/X which are, essentially, complete lattices. Also effective descent morphisms are characterized in terms of effective morphisms with respect to continuous bijections. For classes E satisfying suitable conditions, we show that the class of effective descent morphisms coincides with the one of effective E-descent morphisms.     

Distributions on homogeneous spaces

A survey of results and ideas in the general analytic treatment of the theory of distributions on homogeneous spaces is presented.Translated from Itogi Nauki i Tekhniki. Problemy Geometrii, Vol. 8, pp. 5–24, 1977.     

Computation of weight lattices of <Emphasis Type="Italic">G</Emphasis>-varieties

Let G be a connected reductive group. To any irreducible G-variety, one assigns the lattice consisting of all weights of B-semiinvariant rational functions on X, where B is a Borel subgroup of G. This lattice is called the weight lattice of X. We establish algorithms for computing weight lattices for homogeneous spaces and affine homogeneous vector bundles. For affine homogeneous spaces of rank rkG we present a more or less explicit computation. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra, 2008.     

Generalized symmetric spaces,Yu. P. Solovyov’s formula,and the generalized Hermitian geometry

We collect some basic results on canonical affinor structures of classical type on generalized symmetric spaces. Yu. P. Solovyov’s stimulating influence on this topic during its initial stages is illustrated. Using special canonical f-structures on homogeneous k-symmetric spaces, we also present a new collection of homogeneous Hermitian f-manifolds. Dedicated to the memory of Yu. P. Solovyov Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 43–60, 2007.     

Problems of the theory of bitopological spaces. 2

This paper is the second part of a survey of recent results and new problems in the theory of bitopological spaces. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 5–67. Translated by A. A. Ivanov.     

A refinement of Stone duality to skew Boolean algebras

We establish two theorems that refine the classical Stone duality between generalized Boolean algebras and locally compact Boolean spaces. In the first theorem, we prove that the category of left-handed skew Boolean algebras whose morphisms are proper skew Boolean algebra homomorphisms is equivalent to the category of étale spaces over locally compact Boolean spaces whose morphisms are étale space cohomomorphisms over continuous proper maps. In the second theorem, we prove that the category of left-handed skew Boolean -algebras whose morphisms are proper skew Boolean -algebra homomorphisms is equivalent to the category of étale spaces with compact clopen equalizers over locally compact Boolean spaces whose morphisms are injective étale space cohomomorphisms over continuous proper maps.     

A Class of Groupoids

A new class of left distributive groupoids and their relationship with homogeneous spaces are studied. Bibliography: 5 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 305, 2003, pp. 136–143.     

Siegel coordinates and moduli spaces for morphisms of Abelian varieties

We describe the moduli spaces of morphisms between polarized complex abelian varieties. The discrete invariants, derived from a Poincaré decomposition of morphisms, are the types of polarizations and of lattice homomorphisms occurring in the decomposition. For a given type of morphisms the moduli variety is irreducible, and is obtained from a product of Siegel spaces modulo the action of a discrete group. Mathematics Subject Classification (2000) 14K20     

Harmonic maps and harmonic morphisms

A harmonic morphism is a map between Riemannian manifolds which preserves Laplace's equation. We compare the properties of harmonic morphisms with those of the better known harmonic maps, seeing that they behave in some ways “dual” to the latter. In particular, we give representation theorems for harmonic morphisms in low dimensions which suggest that the equations might be soluble in some cases by integrable-system techniques in a similar way to that used in harmonic map theory. Bibliography: 38 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 234, 1996, pp. 190–200.     

A Nonexistence Theorem of Proper Harmonic Morphisms Between Hyperbolic Spaces

We provide a nonexistence theorem of harmonic morphisms between hyperbolic spaces which are proper and C ²-smooth to the boundary.     

Vanishing and nonvanishing theorems for numerically effective line bundles on complex spaces

Summary We prove some vanishing and nonvanishing theorems for numerically effective line bundles, generalizing to complex spaces and morphisms between complex spaces results of Kawamata, Viehweg and Shokurov.A Guido Zappa per il suo settantesimo compleannoSupported by SFB 170 «Geometrie und Analysis», Göttingen (FRG), and by Ministero Pubblica Istruzione.     

Dimension of maps,universal spaces,and homotopy

This paper is a survey of some recent results in dimension theory. The main topics under consideration are: dimension of maps in the classical and extension dimension theories, universal spaces (in particular, universal compacta) in extension dimension theory, and [L]-homotopy. A number of theorems included in the survey are accompanied by proofs. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 48, General Topology, 2007.     

Complete minimal submanifolds of compact Lie groups

We give a new method for manufacturing complete minimal submanifolds of compact Lie groups and their homogeneous quotient spaces. For this we make use of harmonic morphisms and basic representation theory of Lie groups. We then employ our method to construct many examples of compact minimal submanifolds of the special unitary groups.     

A 2-groupoid Characterisation of the Cubical Homotopy Pushout

We give a 2-track-theoretical characterisation of the homotopy pushout of a 3-corner by recognising the mapping 2-simplex as an initial object in a coherent homotopy category of Hausdorff spaces under a 3-corner with morphisms expressed in terms of the 1-morphisms and 2-morphisms of a homotopy 2-groupoid.     

λ-Topologies on Function Spaces

This paper is devoted to the spaces C _λ(X) of all continuous real-valued functions on X endowed with arbitrary λ-topologies. This is a fairly complete survey of the results obtained by the author in the following domains of the theory of λ-topologies: cardinal functions; locally convex properties; weak and strong topologies; dual spaces; lattices of λ-topologies; completeness. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 2, pp. 3–56, 2003.     

On categorical notions of compact objects

Due to the nature of compactness, there are several interesting ways of defining compact objects in a category. In this paper we introduce and study an internal notion of compact objects relative to a closure operator (following the Borel-Lebesgue definition of compact spaces) and a notion of compact objects with respect to a class of morphisms (following Áhn and Wiegandt [2]). Although these concepts seem very different in essence, we show that, in convenient settings, compactness with respect to a class of morphisms can be viewed as Borel-Lebesgue compactness for a suitable closure operator. Finally, we use the results obtained to study compact objects relative to a class of morphisms in some special settings.Partial financial assistance by Centro de Matemática da Universidade de Coimbra and by a NATO Collaborative Grant (CRG 940847) is gratefully acknowledged.     

Separating functions and their applications

In the present paper, a survey of applications of separating functions in the theory of representations of quivers and algebras in linear and metric spaces is given. An application of these functions to problems in spectral graph theory is also demonstrated. Bibliography: 27 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 202–212.