Gutman Alexander Efimovich

January 7, 1966, Novokuznetsk (Kemerovo region)

1983	Graduated with gold medal honors from Secondary School N 11 in Novokuznetsk.
1988	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.)
1991	Completed a postgraduate course at the Institute of Mathematics SB AS USSR.
1991	Defended a Ph.D. thesis in Mathematical Analysis at the Institute of Mathematics SB AS USSR.
1995	Defended a D.Sc. thesis in Mathematical Analysis at the Sobolev Institute of Mathematics SB RAS.
2001	Awarded the title of Associate Professor for Department of Mathematical Analysis.
2005	Awarded the title of Professor in Mathematical Analysis.

1987–1992	NSU, Sunday and Summer Mathematical School, Teacher of Mathematics
1991–1992	IM SB AS USSR, Laboratory of Functional Analysis, Junior Scientific Officer
1991–1994	NSU, Department of Mathematical Analysis, Assistant Professor
1992–1993	IM SB AS USSR, Laboratory of Functional Analysis, Scientific Officer
1993–1998	IM SB RAS, Laboratory of Functional Analysis, Senior Scientific Officer
1994–2000	NSU, Department of Mathematical Analysis, Associate Professor
1998–2009	IM SB RAS, Laboratory of Functional Analysis, Leading Scientific Officer
2000–…	NSU, Department of Mathematical Analysis, Full Professor
2006–2009	IM SB RAS, Laboratory of Functional Analysis, Acting Head of the Laboratory
2009–…	IM SB RAS, Laboratory of Functional Analysis, Head of the Laboratory

functional analysis
Boolean-valued analysis
infinitesimal analysis

Complete list of publications (108 publications, 1988–2024)
Order analysis (20 publications, 1988–2004)
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 publications, 1995–2008)
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.
Banach bundles (8 publications, 1999–2014)
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 publications, 2000–2013)
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 (16 publications, 1999–2023)
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.
- The technique is developed of partial elements of a Boolean-valued system and the corresponding ascents.
- Predicative Boolean-valued classes are studied which admit quanification.
- Logial interrelations are described between the basic principles of Boolean-valued systems: transfer, mixing, and maximum.
- The notion of transitive Boolean-valued system is introduced and a version of Lévy's Lemma on absolute formulas is proven for such systems.
- The Boolean algebras are characterized for which the classes of regular and σ-regular Boolean-valued systems coincide.
- The universe over an arbitrary extensional Boolean-valued algebraic system is introduced.
- The structure of the Boolean-valued universe is described by means of various cumulative hierarchies.
- It is proven that the Boolean-valued universe is unique up to a unique isomorphism.
- It is shown that Venn diagrams and truth tables serve as proof tools for Horn sentences in the theory of arbitrary (not necessarily complete) Boolean algebras.
Polyverse (8 publications, 1998–2005)
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.
Transition functions (2 publications, 2003–2004)
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 publications, 2007–2013)
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 publications, 2008–2017)
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 publications, 2011–2015)
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 publications, 2013–2014)
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.
Ordered locally convex spaces (11 publications, 2015–2024)
Description of locally convex spaces which include nonclosed Archimedean cones; study of maximal cones in vector spaces
- 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.
- The notion of (pre)lexicographic structure on a vector space is introduced and studied.
- A new proof is presented for the theorem on isomorphic embedding of a totally ordered vector space into a lexicographically ordered space of real-valued functions with well-ordered supports.
- A criterion is obtained for denseness of a maximal cone with respect to the strongest locally convex topology.
- The class is characterized of vector spaces in which there exist nonbasic maximal cones.
- The notion is introduced and studied of quasidense subset in a locally convex space.
- An exhaustive description is provided for the class of locally convex spaces in which all Archimedean cones are closed.
- Criteria are established for the closedness of Archimedean cones in countable-dimensional locally convex spaces in terms of projective parallelotopes and projective automorphisms.
Problem formalism (10 publications, 2016–2021)
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.
Metascience (7 publications, 2003–2017)
Surveys, bibliographies, and other metascientific publications