Date and place of birth
January 7, 1966, Novokuznetsk (Kemerovo region)
||Graduated with gold medal honors from Secondary School N 11 in Novokuznetsk.
||Graduated with honors from Novosibirsk State University. Defended a thesis at the Department of Mathematical Analysis. (The winner of the All-USSR contest of student theses, Diploma N 1.)
||Completed a postgraduate course at the Institute of Mathematics SB AS USSR.
||Defended a Ph.D. thesis in Mathematical Analysis at the Institute of Mathematics SB AS USSR.
||Defended a D.Sc. thesis in Mathematical Analysis at the Sobolev Institute of Mathematics SB RAS.
||Awarded the title of Associate Professor for Department of Mathematical Analysis.
||Awarded the title of Professor in Mathematical Analysis.
||NSU, Sunday and Summer Mathematical School, Teacher of Mathematics
||IM SB AS USSR, Laboratory of Functional Analysis, Junior Scientific Officer
||NSU, Department of Mathematical Analysis, Assistant Professor
||IM SB AS USSR, Laboratory of Functional Analysis, Scientific Officer
||IM SB RAS, Laboratory of Functional Analysis, Senior Scientific Officer
||NSU, Department of Mathematical Analysis, Associate Professor
||IM SB RAS, Laboratory of Functional Analysis, Leading Scientific Officer
||NSU, Department of Mathematical Analysis, Full Professor
||IM SB RAS, Laboratory of Functional Analysis, Acting Head of the Laboratory
||IM SB RAS, Laboratory of Functional Analysis, Head of the Laboratory
Main areas of research
- functional analysis
- Boolean-valued analysis
- infinitesimal analysis
- Complete list of publications(96)
- Order analysis(20)
Development of the theory of lattice-normed spaces and dominated operators
- The concept is developed of order approximation in lattice-normed spaces.
- Four types of order boundedness of an operator are introduced and studied, examples are presented which demonstrate that the types are different.
- Lattice-normed spaces are represented as spaces of extended continuous sections of ample Banach bundles.
- The decomposition is obtained of an arbitrary order-bounded disjointness preserving operator into a strongly disjoint sum of weighted shift operators.
- Function representations are obtained for a wide class of disjointness preserving operators, their properties are interpreted in terms of the representation.
- The bands are studied which are generated by disjointness preserving operators in lattice-normed spaces.
- The Wickstead problem(6)
Description of the vector lattices E for which all band-preserving operators T: E → E are regular
- The conjecture is disproved on coincidence of locally one-dimensional and discrete K-spaces.
- It is shown that a K-space is locally one-dimensional if and only if its base is σ-distributive.
Function representation of a Boolean-valued universe
- The notion of polyverse is introduced and studied, which is a continuous bundle of models of set theory.
- The function representation is obtained of a Boolean-valued universe as the class of continuous sections of a polyverse.
- It is shown that the notion of standard number can be introduced in the stalks of a polyverse so that the main facts of infinitesimal real analysis are valid.
- A criterion is obtained for completeness of the nonstandard hull of a normed space in a stalk of a polyverse.
- Banach bundles(8)
Development of the theory of continuous and measurable Banach bundles
- The theory of ample continuous Banach bundles is created.
- The theory of measurable Banach bundles is created.
- The notion of lifting in the space of measurable sections is introduced, and the corresponding theory is created.
- Banach bundles of operator spaces are defined and studied.
- The notion of dual Banach bundle is introduced, and the corresponding duality is studied.
- It is shown that the stalks of ample Banach bundles inherit finite representability of a normed space in “adjacent” stalks.
- Topological properties are studied of the sets of points at which the stalks of an ample Banach bundle are finite-dimensional or separable.
- Bundles of Banach lattices(4)
Development of the theory of continuous and measurable bundles of Banach lattices and the theory of Banach–Kantorovich lattices
- The notions of continuous and measurable bundle of Banach lattices are introduced and studied.
- Banach–Kantorovich lattices are represented as spaces of continuous sections of bundles of Banach lattices.
- An analytic representation is obtained of the conditional expectation by means of stalkwise integration in a bundle of measure spaces.
- It is shown that every positive lifting in a measurable bundle of Banach lattices is a lattice homomorphism.
- Every monotone operator acting from a vector lattice into a normed space is represented as the composition of a lattice homomorphism and a linear isometry.
- Boolean-valued analysis(9)
Development of the theory of Boolean-valued models and their applications in functional analysis
- By means of Boolean-valued methods, exact analogs are obtained of the uniform boundedness principle or Banach–Steinhaus theorem in lattice-normed spaces.
- As applied to Boolean-valued analysis, the syntactic technique is developed related to the notion of Δ₁ term.
- Every Boolean-valued model of set theory which satisfy the ascent principle is shown to have multilevel structure analogous to the von Neumann cumulative hierarchy.
- Transition functions(2)
Study of the spaces of transition functions and their relations to other objects of functional analysis
- Order, metric, and algebraic properties are studied of the set of finitely additive transition functions endowed with the structure of an ordered normed algebra.
- Interconnections are revealed between finitely additive transition functions, linear operators, vector measures, and measurable vector-valued functions.
- The question is examined of splitting the space of transition functions into the sum of the subspaces of countably additive and purely finitely additive transition functions.
- Continous-discrete functions(5)
Study of the spaces constituted by the sums of continuous and discrete functions
- The space of CD₀-sections of a continuous Banach bundle is studied.
- The representation is clarified of the space of CD₀-sections as the space of continuous sections.
- The space of CD₀-homomorphisms of Banach bundles is introduced and studied.
- Infinitesimal analysis(5)
Critical analysis of some papers related to formalization of infinitely small and infinitely large numbers
- Within the contemporary infinitesimal analysis, a trivial formalization is given for the basics of a series of papers on a positional numeral system with an infinitely large base.
- Algorithmic obstacles are specified to computer implementation of a positional numeral system with an infinitely large base.
- A trivial solution is proposed to the problem of reducing the lexicographic order to a numerical order, which does not require any extension of the classical set theory.
- The comparative logical and algorithmic status is described of the informal postulates underlying the positional numeral system with an infinitely large base.
- The mathematical insignificance is demonstrated of the version of nonstandard analysis based on the quotient of the algebra of sequences by the Fréchet filter.
- Rewriting systems(4)
Representation of object-oriented data by means of prefix rewriting systems, and development of the theory
- In the framework of prefix rewriting systems, the notions are introduced typical for the object-oriented approach to data organization. Namely, the following are introduced and studied:
- the notion of attribute (an analog of property and method of a class or object);
- the notion of inheritance (an analog of inheritance of classes and object instantiation);
- conceptual dependence and consistence, conceptual scheme;
- the notions of type and subtype.
- The questions are studied in detail of algorithmic verification of key “object-oriented” properties of rewriting systems.
- Fixed-point theorems(2)
Critical analysis of some papers devoted to generalization of fixed-point theorems
- A general notion of convergence space is considered.
- A description is obtained for convergence preserving mappings acting in convergence spaces.
- Sequentially and subsequentially convergent functions are described.
- It is shown that the main results of 28 articles devoted to generalizations of fixed-point theorems for mappings in special spaces are direct consequences of a single general theorem which translates previously known facts to the spaces under consideration.
- Nonclosed Archimedean cones(8)
Description of locally convex spaces which include nonclosed Archimedean cones
- The notion of Archimedean convex set is introduced and studied.
- It is shown that every locally convex space of uncountable dimension includes a nonclosed Archimedean cone.
- It is proven that existence of a nonclosed linearly independent set implies existence of a nonclosed Archimedean cone.
- An example is constructed of a countably-dimensioned locally convex space which admits discontinuous linear functionals but has no nonclosed linearly independent subsets.
- A criterion is provided for a wedge to be Archimedean which is based on the notion of incoming direction.
- The question is answered of when a wedge is included in a half-space.
- Problem formalism(8)
Use of binary correspondences for formalization of the notion of problem and related notions
- It is shown how binary correspondences can be used for formalization of the notion of problem, definition of the basic components of problems, their properties, and constructions.
- The following notions are formalized: the condition of a problem, its data and unknowns, solvability and unique solvability of a problem, inverse problem, composition and restriction of problems.
- In terms of correspondences, topological problems are described, as well as problems with parameters and the related notions of stability and correctness.
- Binary correspondences are applied to the study of systems of differential equations which describe processes in chemical kinetics, as well as the inverse problems.
- Within the clarification of the notion of inverse problem, a criterion is established for linear independence of functions in terms of finite sets of their values.
Surveys, bibliographies, and other metascientific publications
|The papers are presented here for academic purposes and are not intended for mass dissemination or copying.
December 9, 2018