Shavkat Arifjanovich Alimov

Shavkat Arifjanovich Alimov is a well-known scientist, a world recognized specialist in the field of mathematical physics and functional analysis, who made a significant contribution to the spectral theory of differential operators, the theory of boundary value problems for equations of mathematical physics, and harmonic analysis.

Sh. A. Alimov was born on March 2, 1945 in the city of Nukus, Uzbekistan. In years from 1952 to 1962 he studied at a public school in Tashkent. After graduating from Tashkent Public School No. 88 in 1962 with a gold medal, he was enrolled to the Physics Department of Moscow State University named after M. V. Lomonosov. He graduated from the Department of Mathematics of this University in 1968, receiving a diploma with honors. From 1968 to 1970 he was a graduate student in the same department under the supervision of Professor V. A. Ilyin. He defended his candidate dissertation (equivalent of PhD) on theory of functions and functional analysis at the Scientific Board on doctoral dissertations, chaired by Academician M.V. Keldysh, at the Institute of Applied Mathematics of the Academy of Sciences of the USSR in June 1970.

In May 1970 he began his career at Moscow State University named under M. V. Lomonosov at the newly opened Faculty of Computational Mathematics and Cybernetics as an assistant professor, then from 1972 to 1974 he worked as an associate professor.

In May 1973, at the age of 28, he defended his doctoral dissertation (highest degree) on the equations of mathematical physics at the Scientific Board of the Faculty of Computational Mathematics and Cybernetics of Moscow State University, chaired by academician A. N. Tikhonov. In 1973, for research on the spectral theory of equations of mathematical physics, he was awarded the country's highest youth prize. In 1974, at the age of 29, he received a position at the rank of Professor at the Faculty of Computational Mathematics and Cybernetics of Moscow State University.

For ten years, from 1974 to 1984, he worked as a professor in the Department of General Mathematics at the Faculty of Computational Mathematics and Cybernetics of Moscow State University. In the same period, he was a member of two Specialized Scientific Boards for defense of doctoral dissertations, namely the Specialty 01.01.01 - "Functional Analysis and Theory of Functions chaired by A. N. Kolmogorov, and the Specialty 01.01.02 - "Differential Equations and Mathematical Physics chaired by A. N. Tikhonov. From 1970 to 1984 together with V. A. Ilyin he led a research seminar on Functional Methods of Mathematical Physics at Moscow State University.

In September 1984, Sh. A. Alimov joined the Tashkent State University as a professor, and in January 1985 he accepted the position of Deputy Director of the Institute of Mathematics of the Academy of Sciences of Uzbekistan.

From 1985 to 1987 he worked as the Rector of Samarkand State University, from 1987 to 1990 - the Rector of Tashkent State University, from January 1990 to February 1992 - the Minister of Higher and Secondary Special Education of the Republic of Uzbekistan. From 1992 to 1994, he headed the Department of Mathematical Physics of the Faculty of Applied Mathematics and Mechanics of Tashkent State University.

From 1994 to 1995 he worked as the Deputy Minister of Foreign Affiairs of the Republic of Uzbekistan. From November 1995 to August 1998 he was assigned as the Ambassador Extraordinary and Plenipotentiary of the Republic of Uzbekistan to the People's Republic of China. From August 1998 to January 2003 he worked as a Vice-Rector (provost) responsible for research at the University of World Economy and Diplomacy.

From September 2000 to June 2001, he worked as a Visiting Researcher at the California Institute of Technology (Caltech), USA. After returning to Tashkent next year, until 2012, he worked as a professor in the department of Mathematical Physics at the National University of Uzbekistan. At the same time, from the first days of the opening of the Tashkent branch of Moscow State University in 2006, he worked as a professor in the Department of Applied Mathematics of the mentioned branch.

From 2012 to 2017, he leaded the Laboratory of Mathematical Modeling of the Malaysian Institute of Microelectronic Systems (MIMOS) in Kuala Lumpur, being at the same time the chief scientist of this institute.

From 2017 to 2019, he worked as a professor at the Department of Differential Equation and Mathematical Physics of the National University of Uzbekistan. From 2019 to the present, Sh. A. Alimov is a Scientific Consultant at the Center for Intelligent Software Systems, and an adviser to the Rector of the National University of Uzbekistan.

The main scientific activity of Sh. A. Alimov is connected with the spectral theory of partial differential equations and theory of boundary value problems for equations of mathematical physics, largely initiated by the work of his scientific adviser academician Vladimir Aleksandrovich Il'in.

At the beginning of the seventies of the 20th century, Sh. A. Alimov investigated the convergence and summability of spectral decompositions related to elliptic operators of arbitrary order with smooth coefficients. For the most important classes of differentiable functions he established exact conditions under which the spectral decompositions uniformly approximate the decomposing functions. The conditions found were new even for such classical operators of mathematical physics as the Laplace operator. At the same time, Sh. Alimov studied the problems of summability of the spectral decompositions of functions in classes Lp, and first constructed an example of a function from this class whose spectral decomposition cannot be summed by Riesz means at any point. The results obtained by Sh. A. Alimov during his student years, were included in his first paper published in the journal Doklady Mathematics of the Academy of Sciences. The second paper, written jointly with V.A. Ilyin, was published in Comptes Rendus de l'Academie des Sciences, Paris. In the end of the seventies, by the motivation of A. V. Bitsadze, Sh. A. Alimov studied degenerate boundary value problems with an oblique derivative for second-order elliptic equations. In particular, he found the exact order of loss of smoothness of a solution depending on the degree of degeneracy of the vector field defining boundary conditions. A little later, a similar result was obtained by the Swedish mathematician Bent Wintzel. It should be noted that the method developed by Sh. A. Alimov for solving this problem also made it possible to cover those cases when the degeneracy is arbitrary (not necessarily power-law) and the description of the corresponding loss of smoothness requires the use of classes of functions with generalized smoothness.

In the early eighties, Sh. A. Alimov investigated elliptic equations with singular coefficients. This class of equations includes the Schrodinger  equation with a potential that has singularities not only at individual points, but also on manifolds that can spread up to infinity. The most important example is the Schrodinger operator, which describes the quantum-mechanical system of many particles with Coulomb interaction. For such equations, exact estimates of the spectral function were obtained and the conditions for representation of an arbitrary function using spectral decompositions were found.

At the same time, Sh. A. Alimov studied the spectral properties of nonlocal boundary value problems in which the boundary values of the desired function on a certain section of the boundary relate to its values at certain internal points of the region. The problem of this type was first formulated and investigated by A.V. Bitsadze and A.A. Samarskiy. The feature of such problems is that they are not self-adjoint, and therefore their spectrum is more complex than in the classical case. For a number of nonlocal boundary value problems, Sh. A. Alimov was able to prove the existence of a complete set of eigenfunctions and the basis property of the corresponding system.

In the early nineties, Sh. A. Alimov investigated the stability problem of quantum-mechanical systems describing the behavior of many-electron atoms and ions. For a wide class of long-range potentials estimates were obtained for the number of particles stably held by a positive nucleus depending on the nuclear charge. Sh. A. Alimov also obtained more accurate estimates for the number of bound states with negative energy, i.e. the number of negative eigenvalues of the Schrodinger operator, in comparison with the known ones.

Sh. A. Alimov's another area of interest was the solvability of boundary-value problems of Dirichlet type for general second-order partial differential equations. To investigate such problems he introduced a new definition of a trace of a function and, as a result, a new definition of a solution from Lp, which includes, as a special case, the classical definition of a solution as well as a generalized solution in the Sobolev classes.

Sh. A. Alimov also conducted a research on the theory of discrete Schrodinger operator defined on a uniform lattice with a sufficiently small step. An important feature of his results is that in all the estimates the dependence on the size of the lattice step is controlled. In particular, for the discrete Schrodinger  operator describing a quantum-mechanical system with a long-range potential, similar to a multiply charged ion, an estimate is obtained for the number of negative particles stably retained by the nucleus, which when the lattice size decreases, becomes a well-known estimate for the continuous Hamiltonian. The advantage of the proposed approach is that one can obtain estimates for the discrete Schrodinger  operator with a short-range potential unknown in the continuous case.

The quantum-mechanical interpretation of these studies is related to elucidating the ratio of the attractive forces of electrons to the nucleus and the forces of their mutual repulsion in the stability problem of many-electron atoms and multiply charged ions. Since 2000, Sh. A. Alimov studied the behavior of spectral decompositions depending on the geometric properties of lines and surfaces on which the decomposed function has discontinuities. The first results on this problem for spectral decompositions associated with the Laplace operator were published in the fifties of the twentieth century, but in recent years interest in this problem has increased again, especially in the USA, France, and Italy. Moreover, the spectral decompositions of piecewise-smooth functions related to second-order elliptic operators were mainly studied. As a result of research Sh. A. Alimov was able to give a complete description of the set of divergence depending on the geometry of discontinuities for elliptic operators of arbitrary order with constant coefficients. In those same years he studied the spectral properties of the so-called partial integral operators. Equations with integral operators of this type are found in contact problems of the theory of elasticity, as well as in the theory of elasticity. Sh. A. Alimov undertook a study of the solvability conditions for general Volterra-type integral equations with a spectral parameter in abstract Banach spaces.

Since 2005, Sh. A. Alimov obtained important results in the theory of boundary control of the heat transfer process. In particular, conditions were found to ensure that a given temperature was obtained in a limited volume for a certain time and an estimate was made of the minimum time necessary for this, depending on the power and location of heat or cold sources.

At present, Sh. A. Alimov is conducting research on the mathematical problems of peridynamics related to the theory of hypersingular integrals. The methods of spectral theory developed by him earlier turned out to be an effective tool for studying the properties of hypersingular integro-differential equations of peridynamics, which allows one to find solvability conditions for these equations.

The scientific merits of Sh. A. Alimov were widely recognized. In 1984, he was elected a corresponding member of the Academy of Sciences of Uzbekistan, in 1991 an academician of the International Academy of Higher Education Sciences, and since 2000 Sh. A. Alimov is an Academician of the Academy of Sciences of the Republic of Uzbekistan.

In 1985, Sh. A. Alimov was awarded the title of Laureate of the Biruni State Prize of Uzbekistan for research in the field of mathematical physics.

In 2019, Sh. A. Alimov was awarded the Order of "Mehnat Shukhrati” for achievements in science and education.

Sh. A. Alimov has over 150 published scientific and a large number of educational works. Among his students are 10 doctors of sciences and more than 20 candidates of sciences (PhD) working at universities in Uzbekistan, Russia, USA, Finland, Malaysia, and at universities in other countries.

For about thirty years, Sh. A. Alimov has been actively involved in the reform of mathematical school education. In 1978-1984, he was the scientific secretary of the Commission for School Books and Curricula of the Mathematics Department of the USSR Academy of Sciences, one of the authors of algebra textbooks and began analysis for secondary schools, recognized in Uzbekistan, the Russian Federation, Ukraine, Belarus and other countries. In addition, Sh. A. Alimov is also actively involved in the development and publication of textbooks and teaching aids for students of higher educational institutions. So in 2012, he in co-authorship with his student Professor R. R. Ashurov prepared the textbook "Mathematical Analysis"in two volumes, second edition of which was published as a textbook in three volumes in 2018.

With the aim of conducting research and lecturing Sh. A. Alimov visited universities in the USA, Japan, Germany, Hungary, Poland and other countries. As a leader or member of official delegations, Sh. A. Alimov visited many cities and countries, such as Washington DC (USA), London (Great Britain), Paris (France), Rome (Italy), Brussels (Belgium), Vienna (Austria), Prague (Czech Republic), Bratislava (Slovakia), Sofia (Bulgaria), Skopje (Macedonia), Tokyo (Japan), Cairo (Egypt), Delhi (India), Jakarta (Indonesia) and others.

Sh. A. Alimov meets his seventy-five year anniversary in the prime of his life, and we heartily congratulate him on his jubilee and wish him good health, new successes in scientific and pedagogical activity, family well-being and long years of fruitful life.