Corresponding-member of the Academy of Sciences of Azerbaijan Arif Aliheydar oglu Babayev has finished the secondary school in 1952 with gold medal and joined “Physics-mathematics” faculty’s “Mathematics” department of Azerbaijan (Baku) State University and graduated from it in 1957 with honorary diploma. At the same year he has been a post graduate student of the chair “Functions Theory and functional analysis”. While studying at the post graduate course, he has proved some theorems on existence and uniqueness of the solution of nonlinear singular integral equations. In 1960, under the guidance of academician A.I. Huseynov he has defended PhD dissertation “Investigation on nonlinear singular integral equations”.
A.A. Babayev has studied a singular integral on a smooth closed integration curve and has given application of these researches to linear and nonlinear singular integral equations. He has offered a new method for studying the Cauchy kernel singular integral operator on an open curve, and the local theory of nonlinear singular integral equations was based on this method. These investigations were on the basis of his doctor’s degree dissertation “Investigations on singular integral equations” defended in 1966 in Tbilisi city at A.M. Razmadze Mathematics Institute. The outstanding mathematicians as N.I. Muskheleshvili, I.N. Vekua, A.N. Tikhonov, Z.I. Khalilov have highly evaluated his results.
The last years, A.A. Babayev and his followers had solved the classic Plemel-Privalov theorem on a singular integral on a Cauchy type integral and non-smooth, closed rectifiable on curve.
Discrete analogs of a singular-integral and of spaces acted by this integral, and also application of these notion to approximate solution of nonlinear singular integral equations was given by A. Babayev.
Under his guidance, at the chair of “Mathematical Analysis” of BSU the fundamental results were obtained on local theory of nonlinear singular integral equations, approximate solution methods of these equations, local theory of equations and multi-dimensional singular integral operators on domain, its application in inclusion theorems of functional spaces, nonlinear singular integral equations in different fields of natural science.
A.A. Babayev has supervised over 40 PhD degree dissertations and his followers V.V. Salayev, R.S. Sadirkhanov, S.K. Abdullayev, V.I. Musayev, E.G. Huseynov, V.S. Guliyev, R.M. Rzayev have become doctors of sciences.
For a long years A.A. Babayev has been a member of the Scientific Council, the dean of Mechanics-mathematics faculty of Baku State University (1979-1994), chairman of the Expert Commission on mathematics and mechanics at the Higher Attestation Commission under the President of the Republic of Azerbaijan.
For his scientific and social activity, A.A. Babayev was awarded the “Badge of Honour”, and the name of honoured Science Worker was conferred upon him.