Address
100174, Tashkent city,
Olmazor district, University
street, house 9
Tel.: +998712079140
Fax: +998712625236
Web site: www.mathinst.uz
Email: uzbmath@umail.uz
Exat: math@exat.uz

Staff
Omirov Bakhrom Abdazovich
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academic degree: DSc,
academic title: Professor,
position: Chief Researcher,
speciality: 01.01.06Algebra
room number: 218,
phone: +998908054288,
email: omirovb@mail.ru
The main scientific direction: Algebra (theory of invariants and its application in geometry and physics, theory of Igroups, theory of aggregation functions) Differential geometry (differential and integral invariants of curves and surfaces) Functional analysis (topological halffield theory, Fourier series theory in Banach spaces, integral theory)


Rakhimov Isamiddin Sattarovich
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academic degree: DSc,
academic title: Professor,
position: leading researcher,
speciality: 01.01.06Algebra
room number: ,
phone: +60173192396,
email: risamiddin@gmail.com
The main scientific direction: Non associative algebras, Lie (super)algebras, Leibniz (super)algebras, structure theory of algebras, evolution algebras and their applications, algebraic geometry


Mukhamedov Farrukh Maqsudovich
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academic degree: DSc,
academic title: Professor,
position: Leading researcher,
speciality: 01.01.01Mathematical analysis
room number: ,
phone: ,
email: far75m@gmail.com
The main scientific direction: Quadratic operators, operator theory, genetic algebras, noncommutative ergodic theory, nonlinear functional analysis


Eshmatov Farkhod Khasanovich
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academic degree: DSc,
academic title: Senior Research Fellow,
position: Leading researcher,
speciality: 01.01.06Algebra
room number: 218,
phone: +998974211510,
email: olimjon55@hotmail.com
The main scientific direction: Ktheory, Cohomology, Homology, Algebraic geometry


Jamilov Uygun Umurovich
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academic degree: DSc,
academic title: Senior Research Fellow,
position: Leading researcher,
speciality: 01.01.01Mathematical analysis
room number: 315,
phone: +998901277237,
email: jamilovu@yandex.ru
The main scientific direction: Genetic and population dynamics, nonlinear dynamic systems, evolutionary algebras


Dadakhodjayev Rashidkhon Asadullayevich
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academic degree: PhD,
academic title: Senior Research Fellow,
position: Senior researcher,
speciality: 01.01.01Mathematical analysis
room number: 217,
phone: +998935817298,
email: rashidkhon@mail.ru
The main scientific direction: Functional analysis, algebra and topology, operator algebras, structure of Jordan algebras


Adashev Jobir Qodirovich
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academic degree: DSc,
academic title: Senior Research Fellow,
position: Senior researcher,
speciality: 01.01.06Algebra
room number: 103,
phone: +998909279921,
email: adashevjq@mail.ru
The main scientific direction: Nonassociative algebras, Lie algebras, Liebniz algebras, Zinbiel algebras, structural theory of algebras


Khakimov Otabek Norbo‘ta o‘g‘li
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academic degree: PhD,
academic title: Senior Research Fellow,
position: Senior researcher,
speciality: 01.01.01Mathematical analysis
room number: 103,
phone: +998977841600,
email: hakimovo@mail.ru
The main scientific direction: Gibbs measure, dynamic systems, padic number theory.


Turdibaev Rustam Mirzalievich
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academic degree: PhD,
academic title: ,
position: Senior researcher,
speciality: 01.01.06Algebra
room number: 218,
phone: +998998631833,
email: r.turdibaev@yahoo.com
The main scientific direction: Leibniz algebras, nonassociative algebras. CalogeroMoser space.


Abdurasulov Qobiljon Komiljon o‘g‘li
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academic degree: PhD,
academic title: ,
position: Senior researcher,
speciality: 01.01.06Algebra
room number: 312,
phone: +998911655953,
email: abdurasulov0505@mail.ru
The main scientific direction: Leibniz and Lie algebras, nonassociative algebra


Boxonov Zafar Saydimahmudovich
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academic degree: ,
academic title: ,
position: Junior researcher,
speciality: 01.01.01Mathematical analysis
room number: 301,
phone: +998946443238,
email: z.b.x.k@mail.ru
The main scientific direction: Dynamic systems.


Djavvad Khadjiev
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academic degree: DSc,
academic title: Academician,
position: chief researcher,
speciality: 01.01.06Algebra
room number: 218,
phone: +998972669747,
email: khdjavvat@gmail.com
The main scientific direction: Algebra (theory of invariants and its application in geometry and physics, theory of Igroups, theory of aggregation functions) Differential geometry (differential and integral invariants of curves and surfaces) Functional analysis (topological halffield theory, Fourier series theory in Banach spaces, integral theory)


History of the laboratory
The history of the laboratory of “algebra and its applications” goes back to the department
of “mathematical analysis and mechanics”, which began its work when the Institute of Mathematics was
founded. The department of “mathematical
analysis and mechanics” was
founded on January 16, 1944. From 1944 to 1960 the department was staffed by
such scientists as N.N. Nazarov, M.T.Urazbaev, D.K. Karimov, I.S. Arjanikh, M.S. Areshev, M.F. Shulgin.
In the early years, the
department of “mathematical analysis and
mechanics” was headed by M.S.Areshev, but in 1956 the department was divided into “mathematical analysis” and “theoretical mechanics”. The department of “mathematical
analysis” was headed by S.Kh.Sirojiddinov, and the department
of “theoretical mechanics” was headed
by I.S. Arjanikh. In 19561958 S.Kh.Sirojiddinov, in 19581960 N.P. Romanov, in 19601980 I.S.
Arjanikh, and in
19801985 G.P. Matviyevskaya worked in the department of "Mathematical
analysis" as its
director.
In 1979, the department of "functional
analysis" was established at the Institute
of Mathematics, headed by T.A. Sarimsakov from 1979 to 1986. In 1986, “mathematical
analysis” and “functional analysis” were
merged to form “algebra and analysis”. In
19861988 T.A. Sarimsakov
was the head of the department "algebra
and analysis", in 19882018 Sh.A. Ayupov was
the head of the department. From 2018 to 2019, U.A. Rozikov headed the
department.
On July 9, 2019, the
President of the Republic of Uzbekistan issued a decree “On state support for the further development of mathematics
education and science, as well as measures to radically improve the activities
of V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences.” According to the PP4387, the department
“algebra and analysis” was
renamed to the laboratory “algebra and
its applications”.
At present, in the
laboratory “algebra and its applications” acad. Sh.A.Ayupov, acad. Dj.Khadjiev, prof. U.A.Rozikov,
prof. B.A.Omirov, DSc. I.S.Rakhimov, DSc. U.U.Jamilov, DSc. F.Kh.Eshmatov, DSc.
A.Kh.Khudoyberdiyev, PhD. R.A.Dadaxodjaev, PhD. J.Q.Adashev, PhD. O.N.Khakimov,
PhD. R.M.Turdibaev, PhD. K.K. Abdurasulov carry out scientific activities.
Research directions
Researchers in the Laboratory of “Algebra and its applications” are currently working in the following areas:
operator algebras, Jordan algebras of
joint operators and Jordan Banach algebras, derivations in operator algebras,
derivations and automorphisms in unlimited operator algebras in Gilbert spaces
and their applications in quantum dynamics, nonassociative algebras, Lie and
Jordan algebra algebras and superalgebras, Zinbiel algebras, deformations and
cohomologies of algebras, padic analysis, evolution algebras and their
applications, algebraic geometry, group and graph theory, Gibbs measures,
random variables in disordered fields, padic pop static physics, padic
statistical physics.
The main results
Operator
algebras:
·
The derivations of the algebra of all
local dimensional operators, the algebra of all dimensional operators, Arens
algebras associated with von Neuemann algebras and the exact normal trace are
classified.
·
In the algebra of Tcompact operators
connected to a halffinite infinite specific background Neumann algebra, the
operation of derivation is proved to be automatically continuous.
·
It is shown that derivation is formed by
the elements of the algebra of all Tdimensional operators connected to
vonNeumann algebras.
·
Description of local derivations of
regular commutative algebras and algebras of dimensional operators connected to
type I vonNeumann algebras are given.
·
Conditions for the existence of
nonderivation local derivation in commutative unit regular algebras are found.
·
Local derivations of algebras of
dimensional operators with respect to type I vonNeumann algebras without Abel
component is described.
Dynamic
systems:
·
All extreme Volterra operators identified
in three and fourdimensional simplexes are classified as ergodic or
nonergodic. It is shown that quadratic stochastic operators corresponding to
graphs have a single fixed point, and that the arbitrary trajectory of such
operators tends to a fixed point faster than any geometric progression.
·
The arbitrary trajectory of quadratic
stochastic operators corresponding to any finite Abel group or the
approximation of a periodic or fixed point is proved. It is shown that an
arbitrary fixed no Volterra operator in a twodimensional simplex has a single
fixed point, and that such operators have two and threeperiod trajectories.
·
The quadratic stochastic operator Volterra
with a twoperiod trajectory of a bisexual population is constructed. A
sufficient condition for the inertia of quadratic stochastic operators has been
found for such a population, and this condition, found in a smallsized
simplex, is also shown to be a necessary condition for inertia.
·
The infinitedimensional simplex includes
the quadratic stochastic operators Volterra and the block Volterra, for which a
set of limit points on the strong and weak approximations of the trajectory is
defined. The existence of nonergodic Volterra operators has been proven using
the classification of these two types of limit points.
·
Criteria have been found for the
relationship between the susceptibility of Markov operators and the
preservation of orthogonality in infinitedimensional simplex. It has been
proven that such properties of Markov's operators do not overlap with those of
finitedimensional simplex. It is also shown that the subjectivity of Markov
operators has applications such as finding positive solutions of some integral
equations.
·
Quadratic stochastic operators in space have
been identified and their relationship to discrete quadratic stochastic
operators in infinitedimensional simplex has been studied. It is shown that
the projective surrogate of quadratic stochastic operators is derived from the
surrealism of discrete operators.
Leibniz
algebras:
·
Consequences such as Engel's theorem,
Lie's theorem, Cartan's solvability criteria, and the addition of Cartan
subalgebras, which are classical results in the theory of finitedimensional
nilpotent Lie algebras, are also valid for Leibniz algebras.
·
Classification of zerofiliform, naturally
graded filiform, quasifiliform and pfiliform Leibniz algebras are obtained.
·
All finitedimensional nilpotent Leibniz
superalgebras with a maximum nilindex and nilindex equal to the dimension of
the superalgebra are classified.
·
Several classes of solvable Leibniz
algebras with nilradicals have been classified, and it is proved that solvable
Leibniz algebras with codimension is equal to the number of generators are
cohomologically rigid.
·
Lowerlevel algebras are studied and
classification of algebras of level one and two are obtained.
·
Algebraic and geometric classifications of
4dimensional nilpotent Novikov, right commutative and left symmetric algebras
are obtained.
·
It is proved that all local and 2local
derivations of simple and semisimple Lie algebras are derivations in the
simplest sense.
·
It is shown that an arbitrary nilpotent
Lie algebra has 2local derivation, which is not a derivation.
Gibbs
measures:
·
The Gibbs measures theory was developed
for the models of statistical physics given in the Cayley tree, such as Potts,
SOS, HC, XY, IsingVannimenus. In particular, all translationalinvariant Gibbs
measures were classified for the Potts model.
·
Periodic, weak periodic Gibbs measures
were built for HC, SOS, and Potts models. Critical temperature points are
defined so that the constructed measurements (for a given model) are the
endpoints of all Gibbs sets of measures.
·
Gibbs measures were constructed for models
with infinite spin values and phase shifts were determined
depending on the temperature. The Gibbs gradient measures were constructed for
an SOS model with a finite spin value. For the SOS model given in the 2nd order
Kelly tree, all translationinvariant and 4 periodic gradient Gibbs set of
measures were described in terms of temperature variation.
·
Introduced the concept of quantum Markov
states in a tree, which is a generalization of the Cayley
tree.
·
In the theory of noarhimed Gibbs measures,
the concept of a generalized padic Gibbs measure was introduced. Generalized
padic Gibbs measures for IsingVannimenus, Potts models.
·
In contrast to the actual Gibbs measures
for the Potts model given in the Cayley
tree, it has been shown that a set of Gibbs measures with a padic value can
include any periodic measure. The translationalinvariant Gibbs measure
classification for this model also proved to differ sharply depending on the
tree order in the field of real and padic numbers.
Algebraic
geometry:
•
For
Koszul CalabiYau
algebras, the existence of a shifted bisimple structure meaning
CravleyBoeveyEtingofGinzburg is shown.
•
Negative
cyclic homology of several classes of algebra or gravity of algebraic structure
by cyclic cohomology.
•
Relationships
between CalabiYau
algebras, symmetric Frobenius algebras, Poisson algebras of the same modulus,
and Frobenius Poisson algebras of the same modulus have been identified.
Awards
•
In 2017, Sh.A.Ayupov, K.K.Kudaybyergenov,
B.A.Omirov U.A.Rozikov were awarded the State Prize of the Republic of Uzbekistan
of the 1st degree.
•
In 2021, Sh.A.Ayupov was awarded the title
of "Hero of Uzbekistan" and the highest award  "Gold Star"
medal.
•
U.A. Rozikov 2017 Springer Nature Top
Author International Award; In 2018 he was elected an academician of the World
Academy of Sciences (TWAS) and in 2020 was awarded the COMSTECH International
Award "Best Scientific Article" (Rozikov UA Jour. Math. Biology,
2017. V.75, No. 67, p.1715—1733). ) was awarded.
International relations
The department actively cooperates with many
institutes and universities, including:
•
University of San Diego (USA)
•
University of São Paulo (Brazil)
•
University of Bonn (Germany)
•
Bielefeld
University (Germany)
•
Rhein
University (Germany)
•
University of
Trieste
(Italy)
•
University of Cambridge
(Great Britain)
•
University of
Leeds (Great Britain)
•
University of Seville
(Spain)
•
University of Santiago
de Compostela (Spain)
•
University of Granada
(Spain)
•
University of Adelaide
(Australia)
•
University of Sydney
(Australia)
•
University of
Strasbourg (France)
•
AixMarseille University (France)
•
University of Nantes
(France)
•
University of Paris (France)
•
Kazan Federal University
(Russia)


