### INSTITUTE OF MATHEMATICS named after V.I.ROMANOVSKIY

** Year of creation**: November 3, 1943. After several transformations, the Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan in its current status was established on February 17, 2017 on the basis of the Institute of Mathematics at the National University of Uzbekistan.

**Requisites**: address: Uzbekistan, 100170, Tashkent, Mirzo-Ulugbek 81.; tel.: 262-56-94,262-75-44; fax: 262-73-57; e-mail: uzbmath@umail.uz

Director: Academician Ayupov Shavkat Abdullaevich.

Employees: academicians - 5, doctors of sciences - 22, candidates of sciences - 21, doctoral students - 2, basic doctoral students - 20, researchers - 72, total employees - 108.

**The main areas of research of the Institute:**

- Algebra and functional analysis;

- Differential equations and mathematical physics;

- Theory of Probability and Mathematical Statistics;

- Theory of dynamical systems;

- Applied and computational mathematics.

**History of the Institute.** During the period of its activity, the Institute has made a significant contribution to the development of mathematical science, the training of highly qualified personnel for the republic and has become one of the recognized centers of mathematical research at the international level. The path taken by the Institute of Mathematics’ personnel is vividly expresses in the development of mathematical science in Uzbekistan.

Initially, the Institute was created as part of the Academy of Sciences of Uzbekistan as the Institute of Mathematics and Mechanics with a computer center. In addition to mathematics, scientific research was also conducted in mechanics, theoretical geophysics, seismology, and computer technology. Over time, the directions of mechanics and seismic resistance of structures, cybernetics, and seismology separated as independent institutes.

Academicians T.N. Kary-Niyazov and T.A. Sarymsakov, the first presidents of the Academy of Sciences of Uzbekistan

In various periods, the Institute of Mathematics was led by prof. N.N. Nazarov (1943-1947), prof. V.A.Dzhoggio (1947-1950), Acad. V.I. Romanovskiy (1950-1952), Acad. V.A. Bugaev (1952-1957), Acad. S.Kh.Sirazhdinov (1957-1960 and 1962-1967), corresponding member V.I. Gubin (1960-1962), Acad. M.S.Salakhitdinov (1967-1985), Acad. T.D.Jurayev (1985-1992 and 1997-2003), prof. B.B.Rikhsiev (2003-2004) and Acad. Sh.A. Ayupov (1992-1997 and from 2004 to the present).

In 1954, the institute was named after V.I. Romanovskiy, the founder of the mathematical scientific school of Tashkent. The most significant achievements of the Institute can be listed by creation of scientific schools recognized by the world mathematical community on functional analysis, differential equations, probability theory and algebra thanks to the efforts of V.I. Romanovskiy, T.A. Sarymsakov, S.Kh. Sirazhdinov, M.S. Salakhitdinov, T.D. Juraev. Currently, these schools are run by academicians Sh.A. Ayupov, A. Azamov, Sh.A. Alimov, Sh.K. Farmonov, and J. Hadzhiev.

The scientists of the Institute also carried out fruitful research and made an enormous contribution to the study of the history of the development of mathematics in the Middle Ages (academicians S.Kh.Sirazhdinov, G.P. Matvievskaya and others).

Scientific research of the Institute employees was awarded five State Prizes of Uzbekistan. In 2017, the State Prize of the first degree in the field of science and technology of the Republic of Uzbekistan was awarded to a team of mathematicians - academician Sh.A. Ayupov and his students, doctors of sciences K. Kudaiberganov (KKSU), B .Omirov (NUUz) and U. Rozikov (IM AS RUz) for the fundamental research “Development of the theory of non-associative algebras, derivations, and nonlinear dynamical systems”.

In 2019, PhD student of the Institute, J. A. Karimov, was awarded the State Scholarship of the President of the Republic, and PhD student S. Shopulatov became the laureate of the “Mard O’g’lon” State Prize.

Academician Sh.A. Ayupov and prof. U.A. Rozikov was elected as Fellows of the World Academy of Sciences (TWAS).

The Institute has a scientific council for awarding PhD and Doctor of Science degrees. Over the past five years, 12 doctors of philosophy (PhD) and 13 doctors of sciences have been trained.

** The most important scientific research results:**

* In the field of algebra and functional analysis.*

Based on the theory of operator algebras and non-commutative integration, a new algebraic approach to the description of the space of observable of physical systems in statistical mechanics and quantum field theory is proposed.

The structural theory of finite-dimensional Leibniz algebras and their complete classification for small dimensions are constructed.

The theory of p-adic systems in statistical mechanics and nonlinear dynamical systems in population biology is developed.

Topical issues of ordered and non-associative algebras have been developed as applied to problems of non commutative integration, quantum theory and others, which have received worldwide recognition.

Scientific discussions of Sh.A.Ayupov with Fields medalist Professor E. Zelmanov (USA) and Professor of Sichuan University (China) F. Eshmatov at the Institute of Mathematics.

*In the field of differential equations and dynamical systems.*

Methods have been developed for solving basic boundary value problems for a third-order equation in multidimensional infinite domains. Exact conditions for the summability of multiple series and Fourier integrals associated with arbitrary elliptic operators of piecewise smooth functions are found.

The boundary value problems for parabolic-hyperbolic equations with fractional differentiation operators are studied.

A new one-step method for the approximate solution of the Cauchy problem for quadratic dynamical systems is developed. An algorithm for the numerical integration of a high degree of accuracy based on the Taylor formula is proposed and an explicit estimate of the remainder term is found. The connection between the Taylor formula and the generator of N. Chomsky grammar is established. The fractal structure of the terms of the Taylor formula is revealed.

Computer programs have been developed with the aim of automating research work on dynamical systems, allowing to build phase portraits of two-dimensional and three-dimensional dynamical systems through interactive control of the image of a phase portrait.

*In**probability theory and mathematical statistics.*

New methods have been developed for stochastic and statistical analysis of random processes, including models describing real phenomena. Integral-local versions of the central limit theorem for sums of independent random variables in the series scheme are proved. The asymptotic properties of branching processes with immigration and the limiting distributions for the total number of particles participating in such processes are studied. A new method for obtaining bootstrap versions of U-statistics from dependent observations has been created.

*In the field of computational mathematics.*

A theory is developed for constructing optimal cubature and interpolation formulas in Hilbert spaces, and optimal cubature and quadrature formulas are obtained for the approximate integration of strongly oscillating and singular integrals. Invariant cubature formulas with a minimum number of nodes on multidimensional spheres are constructed.

**Practical developments:**

Academician A. Azamov and his students developed a new mathematical model of the heat transfer process in rotating regenerative air heaters (RVAH) of heat power stations. Based on the model, a computer program was created that controls the operation of the RVAH.

Professor F.T. Adylova, together with her students, conducted research on new methods for assessing the model's ability to sample when constructing a structure-activity model in drug development. They got new predictive activity models for a number of alkaloids.

A software tool for monitoring and forecasting the state of the atmosphere in industrial regions developed by scientists from the department of “Mathematical Modeling of Nonlinear Systems” was introduced in the activities of the Samarkand Regional Department of Agriculture and Water Management. As a result of implementation, the economic effect amounted to 72.3 million soums per year.

An experiment was conducted to study the distribution of dust and salt from the drained part of the southern Aral Sea region in the surface and boundary layers of the atmosphere.

Based on the results obtained over the past 5 years, scientists of the Institute have prepared and published 9 monographs on modern problems of the mathematical sciences, including 6 in foreign publishing houses and 11 proceedings of scientific conferences held by the Institute have been published. The following monographs have been published.

- B. Russo, A. Aksoy,R. Ashurov, Sh. Ayupov (editors) Topics in Functional Analysis and Algebra. Contemp. Math. 672. AMS. 2016.
- Azamov A., Bunimovich L., Dzhalilov A., Zhang H.K. Differential Equations and Dynamical Systems. USUZCAMP 2017. Springer Proceedings in Mathematics & Statistics, 2018, Vol. 268. Springer, Cham. 195 .
- Rozikov U.A. An introduction to mathematical billiards. World Sci. Publ. Singapore.2018, (220 pages).
- Ibragimov Z., Levenberg N., Rozikov U., Sadullaev A. (Eds). Algebra, Complex Analysis, and Pluripotential Theory. Springer Proceedings in Mathematics and Statistics. 2018, V. 264, Springer Nature Switzerland.
- Векслер А.С. Статистические эргодические теоремы в перестановочно-инвариантных пространствах измеримых функций. –Lambert Academic Publishing, ISBN 978-3-659-5047-0, 2018,156 c.
- Sh. Ayupov, B. Omirov, I. Rakhimov. Leibniz Algebras, Structure and Classification. CRC Press, Taylor & Francis Group, 2019.

Monographs published abroad in recent years by institute staff

**Integration of science and education **

In this direction, the Institute has always carried out active and fruitful work. Recently, deepening the integration of mathematical science in the education system has become one of the Institute's priority tasks, which are being successfully implemented.

In recent years, employees of the Institute have prepared and published a textbook and 10 teaching aids for students and pupils of schools and lyceums. 24 employees of the Institute give ordinary and special courses at universities (NUUz, a branch of Moscow State University in Tashkent, TFI, UMED, TIIZhT, KKSU, FerSU, NamSU, BukhSU, KarSU, etc.), lyceums and colleges, conduct scientific seminars, supervise research of the undergraduates and master’s students.

Academician A. Azamov with students after an open lesson

The Institute has concluded and is implementing multi-year cooperation agreements with the National University of Uzbekistan, Karakalpak, Samarkand, Andijan, Bukhara, Gulistan, Namangan, Ferghana, Karshi, Urgench and Termez universities. These agreements provide for the participation of the faculty of universities at scientific seminars led by leading scientists of the Institute, conducting joint research, passing undergraduate studies at the Institute for Qualifying Practices, teaching gifted youth in the doctoral studies of the Institute, and conducting joint scientific conferences. The scientists of the Institute regularly travel to lecture and conduct scientific consultations and transfer research experience to universities in the regions of the republic.

Academician Sh.A. Ayupov is the chief editor of the journal "Physics, Mathematics, Computer Science", intended for secondary schools and academic lyceums.

There is a school of academician Sh.A. Ayupov at the Academic Lyceum "Computer Technologies" at Tashkent State Technical University.

Academician A. Azamov maintains the rubric "Matematika Jozibasi" in the scientific and methodological journal "Physics, Mathematics, Computer Science", (intended for students of secondary schools and academic lyceums). Together with his student A.Tilavov, he prepared and published a three-volume textbook “Truly Entertaining Mathematics” for schoolchildren.

Jury members and winners of the 1st International Mathematical Olympiad among students named after Al-Khwarizmi

**International scientific cooperation**

Institute staff established fruitful scientific relations with specialists from the USA (University of San Diego, NewHaven), Germany (Bonn, Bielefild, Jena), Italy (Trieste, Rome), Great Britain (Cambridge), Spain (Seville, Santiago de Compostela), China ( Sichuan University) , Malaysia, Australia (Adelaide, Sydney), France (Strasbourg, Mulhouse, Bures sur Yvette, Marseille, Nantes, Paris), Serbia, as well as Moscow, Kiev, Kharkov, St. Petersburg, Ufa, Kazan, Novosibirsk and with many other scientists.

About 70 scientific trips of scientists of the Institute have been invited to foreign centers for joint research and lecturing over the past 5 years. As a result, about 100 scientific articles jointly with foreign scientists were published in prestigious international journals.

Scientific communication is effectively used in the preparation of highly qualified personnel. In particular, over the past 3 years, young scientists of the Institute K. Masutov, H. Karimjanov, R. Turdibaev and S. Murodov successfully defended their dissertations and received PhD degrees at the University of Santiago de Compostela (Spain).

The Institute has organized many international conferences in the field of mathematics, such as the First World Congress of the Bernoulli Society (Tashkent, 1986), the Third Soviet-Japanese Symposium (Tashkent, 1975), the Uzbek-French Meeting (Tashkent, 1997), the French-Uzbek Meeting (Mulhouse , 1995), Uzbek-American Mathematical Conferences (Fullerton, 2014, Urgench, 2017), the CIMPA international scientific seminar for young scientists (Tashkent 2018), the first Shina-Central Asian Meeting of Mathematicians at Sichuan University (China, 2019) and others.

A number of well-known foreign mathematicians are members of the editorial board of the Uzbek Mathematical Journal, published by the Institute in English, and actively help in reviewing articles, which is one of the measures taken to enhance the prestige of the journal:

**Aernoutvan Enter **– University of Groningen (Netherlands); **Fumio Hiroshima **– Kyushu University (Japan); **M.Ladra **– University of Santiago de Compostella (Spain); **Lingmin Liao** – University Paris-Est (France); **Arnaud Le Ny** – University Paris-Est (France); **Lorenzo Ramero **–University of Lille (France); **Sukochev** –University of South Wales (Australia); **J.P.Tian **– New Mexico State University (USA); **Efim Zelmanov ** – University of California, San Diego (USA).

Under a grant of USA-Uzbekistan collaboration for International Research Experience of Students (IRES) in June-August 2017, a group of 9 students from USA spent summer at the Institute and gave talks at the second Uzbek-American conference of the obtained results.

In 2018-2019 (July-August) also at the Institute, according to the same program, groups of 10 American students were engaged in scientific research.

**Research Prospects.**

The strategy for the further development of the Institute of Mathematics covers research in the following areas:

The structural theory of nonassociative algebras, the study of derivations of operator algebras and their application in studies of dynamics in biological systems;

The study of boundary and inverse problems for partial differential equations of integer and fractional orders and their applications;

Development of new methods for the study of mathematical models of dynamical processes and their applications;

Stochastic analysis of random processes and their applications;

Optimization of computational methods in functional spaces;

Construction and research of mathematical models of physical, chemical and biological processes occurring in complex structures;

Development of a computational strategy for creating new drugs using methods of artificial intelligence.

From January 1, 2020, the Institute of Interns-Researchers in the Field of Mathematical Sciences is being created.

The prospects for the development of research at the Institute are determined by a decree of the President of the Republic of Uzbekistan “On measures of state support for the further development of mathematical education and science, as well as radical improvement of the work of the Institute of Mathematics named after V.I. Romanovskiy of the Academy of Sciences of the Republic of Uzbekistan” dated July 9, 2019, No. PP-4387 and are directly related to the deepening of relevant fundamental and applied scientific research, the solution of a number of practically important tasks for economic and social sectors , improving the system of teaching mathematics in universities, intensifying the training of highly qualified scientific personnel - mathematicians and developing international scientific and technical cooperation.

**Thesis defense**

The following fellows of the Institute of Mathematics successfully defended their dissertations of the doctor of philosophy (PhD) on Physical and Mathematical sciences at the Scientific Council DSc.27.06.2017.FM.01.01 National university of Uzbekistan, Institute of Mathematics:

**Khakimov Otabek Norbuta ugli****.**

**Theme of dissertation****:** «*p*-Adic Gibbs measures on a Cayley tree».

**Speciality:** 01.01.01. – Mathematical analysis.

**Scientific** **supervisor:** prof. Rozikov Utkir Abdulloevich.

**Nurillayev Muzaffar Eshnazarovich**.

**Theme of dissertation****:** «Classification of nuclear real *W**– factors».

**Speciality:** 01.01.01. – Mathematical analysis.

**Scientific supervisor:** prof. Rakhimov Abdugafur Abdumadjidovich.

**Mamadaliyev Nodir Kamoldinovich**.

**Theme of dissertation****:** «Cardinal and topological properties of the space of semi-additiv functionals».

**Speciality:** 01.01.04. – Geometry and topology.

**Scientific supervisor:** dotsent Beshimov Ruzinazar Bebutovich.