The Department of Algebra and Functional Analysis was established on the basis of the Departments of Functional Analysis and Algebra and Number Theory. Currently, the department is headed by Professor Omirov Bakhrom Abdazovich, Ph.D. The founder of the department of functional analysis is the famous mathematician, academician, statesman Tashmuhammad Alievich Sarimsakov.
    Tashmuhammad Alievich Sarimsakov paid great attention to organizing the problems of probability theory with the methods of functional analysis, as well as applied problems of mathematics. In particular, T.A. Sarimsakov received the State Prize for his scientific work in meteorology. He founded the theory of half-fields, one of the new scientific directions of modern mathematics, and created the Tashkent school of functional analysis. In 1968 and 1994, the scientist was awarded the State Prize of Uzbekistan named after Abu Rayhan Beruni for his scientific research in this direction. In 1990, he was awarded the title of Hero of Labor. Topological Boolean algebra monograph by T.A. Sarimsakov was published in the USA.
    The idea of ​​an algebraic approach to the organization of probability theory found little development in the later works of T.A. Sarimsakov and his students. Algebraic description of non-commutative (quantum) probability theory was given. Currently academician Sh. A. Ayupov, professors V. I. Chilin, R. N. Ganikhojhaev and their students are carrying out fruitful scientific research on the theory of operator algebras and its applications, the theory of symmetric spaces, the theory of quadratic operators. Representatives of the school have received State awards several times for the scientific and practical results and achievements in these areas. Many international scientific meetings and conferences were held on their initiative.
    The Department of Algebra and Number Theory was founded in 1956 by the initiative of Professor N.P. Romanov, and in the early years, the department carried out research on multiplicative functions and trigonometric series, and from the 70s, the works of T.M. Zuparov and his students on additive problems and the metric theory of Diophantine approximations were added to it.
    Later, scientific research was carried out on the topic of the theory of invariants and its applications. Academician J. Hojiev and his students on this subject obtained a number of results devoted to the classification of non-closing orbits for images of some groups in vector spaces, problems related to the 14th problem of Gilbert about the finite generation of the algebra of invariants, ideals of algebras of invariants, and other problems.
    In the Department of Algebra and Functional Analysis, the structural theory of non-associative algebras, considered one of the important directions of modern algebra, was taught by Academician Sh.A. It is organized under the leadership of Ayupov.
    The scientific centers of Spain, Germany, Russia, Kazakhstan highly appreciate the results obtained in the scientific school of operator algebras and its applications, founded by Academician Sh.A. Ayupov, and conduct research in constant cooperation with them.
    In 2003, Sh.A. Ayupov was the first among the scientists of Uzbekistan to be elected a full member of the Developing World Academy of Sciences (TWAS).