The Department of Algebra and Functional Analysis was established on the basis of the former Departments of Functional Analysis and Algebra and Number Theory.
Currently, the Department is headed by PhD, Associate Professor Rustamjon Qahramonovich Gaybullayev.
The founder of the Department of Functional Analysis was well-known mathematician, academician, and statesman Tashmukhamed Aliyevich Sarymsakov. T.A. Sarymsakov devoted significant attention to the study of problems in probability theory using methods of functional analysis, as well as to applied problems in mathematics. In particular, for his scientific contributions in meteorology, he was awarded the State Prize. He laid the foundations of the theory of semi-fields – one of the emerging areas of modern mathematics – and established Tashkent school of functional analysis. For his achievements in this field, he received the State prize of Uzbekistan named after Abu Rayhon Beruniy in 1968 and 1994. In 1990, Sarymsakov was awarded the title Hero of Labor. His monograph titled “Topological Boolean Algebras” was published in the USA.
The idea of an algebraic approach to probability theory was further developed in Sarymsakov’s later works and in the research of his students. An algebraic formulation of noncommutative (quantum) probability theory was established. At present, Academician Sh.A.Ayupov, Professors V.I.Chilin and R.N.Ganikhodzhaev, together with their students, are conducting productive research in operator algebras and their applications, the theory of symmetric spaces, and quadratic operators. For achievements in these areas, representatives of the school have repeatedly been awarded State Prizes. At the initiative of T.A. Sarymsakov, numerous international scientific conferences and symposia were organized.
The Department of Algebra and Number Theory was founded in 1956 on the initiative of Professor N.P.Romanov. In its early years, research focused on multiplicative functions and trigonometric series. Beginning in the 1970s, this work was complemented by research conducted by T.M.Zuparov and his students on additive problems and the metric theory of Diophantine approximation. Subsequently, research was carried out in invariant theory and its applications. In this area, Academician Dj.Khadjiev and his students obtained significant results concerning the classification of non-closed orbits in representations of certain groups on vector spaces, problems related to Hilbert’s 14th problem on the finite generation of invariant algebras, ideals of invariant algebras, and related topics.
    Nowadays, under the leadership of Academician Sh.A.Ayupov, the Department of Algebra and Functional Analysis conducts research in the structural theory of non-associative algebras, one of the important directions of modern algebra.
Research centers in Spain, China, Turkey, Malaysia, Germany, Russia, and Kazakhstan highly appreciate the results obtained within the scientific school of operator algebras and its applications founded by Academician Sh.A.Ayupov, and maintain ongoing collaborative research.
In 2003, Academician Sh.A.Ayupov became the first Uzbek scientist elected as a Full Member of The World Academy of Sciences (TWAS). Later, other faculty members were elected Full Members of this prestigious academy: Academician Utkir Abdullayevich Rozikov in 2018 and Professor Bakhrom Abdazovich Omirov in 2022.
Among the graduates of the Department of Algebra and Functional Analysis, there are numerous recipients of the Presidential Scholarship, the “Shuhrat” Medal, the named scholarships “Ulugbek” and “Beruniy,” as well as winners of national and international academic Olympiads.
The Department regularly hosts the research seminar “Modern Algebra and Its Applications” led by Academician Sh.A.Ayupov, and the seminar “Functional Analysis and Its Applications,” named after Academician T.A.Sarymsakov and led by Professor A.A.Rakhimov.
In recent years, faculty members of the Department have published textbooks entitled “Linear Algebra” and “Functional Analysis and Integral Equations”, as well as study guides including “Algebra and Number Theory”, “Abstract Algebra”, “Independent Studies in Functional Analysis”, “Independent Studies in Algebra”, “Linear Algebra”, and “Functional Analysis”.