On the 75th birth anniversary of Marat Mirzaevich Arslanov

I. Sh. Kalimullin ; V. L. Selivanov

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 161 (2019) no. 1, pp. 152-160

Résumé

The paper describes the biography and the main scientific achievements of Marat Mirzaevich Arslanov, the Head of the Department of Algebra and Mathematical Logic of Kazan Federal University, Professor, who is the founder of the Kazan School of Mathematical Logic and Computability Theory. The main scientific finding of M.M. Arslanov that gave him international fame, is Arslanov’s fixed point theorem (also known as Arslanov’s completeness criterion), which was first formulated and proved by M.M. Arslanov in 1977. Numerous generalizations and unexpected applications of this theorem have been found by many authors in various areas of mathematics and computer science. M.M. Arslanov was one of the first of Russian mathematics who actively participated in the investigations of the local theory of (Turing) unsolvability degrees, i.e., the degrees reducible to the degree of creative sets. This research area is closely related to the investigation of Turing degrees in Ershov’s hierarchy. He has solved a whole series of problems in this area. In particular, he found a structural difference between the semilattice of computably enumerable degrees and the semilattice of $n$-c.e. degrees, $n > 1$. One of the most important results of M.M. Arslanov's research activity is the solution of Downey's problem on the elementary non-equivalence between the degrees of finite Boolean combinations of computably enumerable sets, which was obtained by him jointly with I.Sh. Kalimullin, his former postgraduate student, and S. Lempp, Professor of Wisconsin University, United States.