Russian Mathematical Surveys, 2024, Volume 79, Issue 2, Pages 369–373 DOI: https://doi.org/10.4213/rm10168e (Mi rm10168)

DOI: https://doi.org/10.4213/rm10168e

Bibliographic databases:

Zoom inZoom out

On 7 Fedruary 2024 the well-known Russian Mathematician, academician of the Academy of Sciences of the Republic of Tatarstan, doctor of physical and mathematical sciences, professor, head of the Department of Algebra and Mathematical Logic of Kazan Federal University, organizer and scientific advisor of the Scientific and Educational Mathematical Center of the Volga Federal District Marat Mirzaevich Arslanov observed his 80th birthday.

He was born in the village of Imen’kovo in the Laishevo district of the Republic of Tatarstan. After graduating from the local seven-year school he continued his education in the Tatar school no. 35 in Kazan. On passing excellently the entrance examinations, in September 1961 he became a student of the Faculty of Mechanics and Mathematics of Kazan University. Since his first year at the university he majored at the Department of Algebra, with the head of the department V. V. Morozov as his scientific advisor. He had an individualized learning plan prepared for him by the department. In accordance with this plan, he spent the last two years of his graduate studies in Novosibirsk University. In 1970 he defended his Ph.D. thesis “On the structure of recursively enumerable sets” and, in 1988, the D.Sc. thesis “Completeness in the arithmetic hierarchy and $\Delta_2^0$-sets”. Since 1969 his career has been connected with Kazan University, where he has gone from an assistant professor to a professor emeritus, and the last 35 years (since 1989) he has been the head of the Department of Algebra and Mathematical Logic of Kazan Federal University.

Arslanov founded and is now the head of the internationally renowned Kazan scientific school in mathematical logic. His research activities are related to the field of recursion theory. He obtained major results on the algebraic structure of the ordering of the degrees of unsolvability and on the development of the hierarchical structure of the computable functions. He developed general methods enabling one to describe classes of arithmetic sets which are complete (with respect to all main reductions) on the corresponding level of the arithmetic hierarchy. These methods are now widely used and are known as Arslanov’s completeness criteria in the literature [1]–[3], [5], [7]. These results summarized years of research by many authors on the effectivization of classes of simple and hypersimple sets.^1[x]1For slightly more details, see the note by I. Sh. Kalimullin “Professor M. M. Arslaanov and the fixed point theorem” (Uchen. Zapiski Kazan. Univ. Ser. Fiz.-Mat. Nauk, 156, no. 1, 2014, Pubishing House of Kazan University, Kazan, pp. 154–156; in Russian). In recent years some Russian and foreign authors found a number of generalizations of Arslanov’s completeness criteria and applied them to various areas of mathematics and theoretical cybernetics [14]–[16], [22]–[24].

Arslanov was the first Russian mathematician who was actively involved in the development of the local theory of (Turing) degrees of unsolvability, that is, of degrees reducible (in the sense of Turing) to the stopping problem [4], [6], [8], [11], [13]. A close line of research deals with Turing degrees in the Ershov hierarchy [9], [10], [12], [18]. Arslanov also solved a number of open problems in recursion theory, in particular, he found an $\exists\forall\exists$-formula [4], [6] the distinguishing of the semilattice of recursively enumerable degrees from the semilattices of $n$-recursively enumerable degrees for $n>1$. For many experts in the field this was an unexpected result, which prompted many new publications on this subject. Arslanov’s most striking result in this direction was his solution, in conjunction with his student I. Sh. Kalimullin and prof. S. Lempp from the University of Wisconsin in USA, of a problem well known in the literature and concerning pairwise distinguishing elementary theories of $n$-recursively enumerable degrees [17]. These results are widely known and highly regarded worldwide. Subsequently, they were developed by Arslanov, his students (see [19]–[21]), and many Russian and foreign authors and presented in monographs and textbooks.

Arslanov gave lectures at universities in the USA, Great Britain, Germany, France, Italy, China, Singapore, Iran, Greece and spoke at All-Union, All-Russian, and international conferences, symposia, seminars, and schools, giving plenary talks in algebra, computability theory, and mathematical logic. Very important for the development of computability theory in Russia were a number of major Russian and international conferences and symposia that he organized, where leading world experts were invited. These evens, as well as several periods of Arslanov’s extended stay in the USA, allowed him to establish good connections and cooperation with leading authors in mathematical logic and the theory of algorithms, such as G. Sacks, C. Jockusch, R. Soare, R. Shore, T. Slaman, and their students. We can also mention in this connection the Russian translation of the well-known monograph Recursively enumerable sets and degrees by R. Soare, which was performed at Arslanov’s suggestion and under his editorship and became a desk book for experts in the area.

Arslanov is a brilliant teacher and organizer of science, who pays particular attention to the work with young researchers. He and his students were scientific advisors of 15 Ph.D. theses, six of whose authors, V. L. Selivanov, V. D. Solov’ev, Sh. T. Ishmukhmetov, I. Sh. Kalimullin, M. Kh. Faizrakhmanov, and A. N. Frolov, defended subsequently also their D.Sc. theses. As a result, the Kazan school of computability theory emerged and then developed into one of the world centres of research in this field. Furthermore, Arslanov’s scientific and organizational activities are invaluable for the development and preservation of the rich traditions of Kazan University in algebra and mathematical logic. Since 1989, when he became the head of the Department of Algebra, later transformed into the Department of Algebra and Mathematical Logic, he has given a strong impetus to the development of this department. Presently, the Department of Algebra and Mathematical Logic of Kazan University is one of the best Russian university departments in this field, where apart form mathematical logic and computability theory, also more traditional lines of research in algebra, going back to N. G. Chebotarev, are successfully developed.

Also today Marat Arslanov is one of the key figures in the Institute of Mathematics and Mechanics of Kazan University, who participates strongly in many major projects of the university. His enthusiastic engagement, enormous energy, and skills in uniting people for the solution of large-scale problems are invaluable for the life of the institute and the whole university. For the past seven years he has presided over the jury of the International Competition of the N. I. Lobachevskii Medal and Prize. This prestigious award with more than 100 years of history is inextricably connected with Kazan University on the one hand, and the progress of mathematical sciences in this country and worldwide on the other. Arslanov was among the initiators of the resumption of awarding the Lobachvskii Medal and Prize for results on fundamental and applied mathematics and one of the organizers of the re-established competition in 2017–2023. The central role of Arslanov in the establishment and development of the Scientific and Educational Center of the Volga Federal District, of which he is the head since its foundation, should also be mentioned.

Arslanov is the member of the editorial boards of the international journals Asian-European Journal of Mathematics and Journal of Universal Computer Science, and of the Russian journals Lobachevskii Journal of Mathematics, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika,^2[x]2Translated in Journal of Mathematical Sciences. and Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki. He is the editor-in-chief of the new journal Matematika i Teoreticheskie Komp’uternye Nauki, which he founded at Kazan Federal University. He is the chairmen of the dissertation council for D.Sc. and Ph.D. degrees in two fields of fundamental and applied mathematics. He is a member of the American Mathematical Society and Association for Symbolic Logic, and for many years was a member of the international commission for translations into Russian and other East-European languages of the American Mathematical Society and Association for Symbolic Logic.

Marat Arslanov is a professor emeritus of Kazan Federal University, a member of the praesidium of the Russian Professors’ Association, a member of the Academy of Science of the Republic of Tatarstan (since 2016: a corresponding member since 1995), a Honoured Scientist of the Republic of Tatarstan (1998), and a Honoured University Worker of the Russian Federation (2007).

We wish Marat Mirzaevich Arslanov good health, long life, and further creative achievements for the benefit of science and education.

Bibliography
1.	M. M. Arslanov, “On effectively hypersimple sets”, Algebra and Logic, 8 (1969), 79–85
2.	M. M. Arslanov, R. F. Nadyrov, and V. D. Solov'ev, “Criterion for the completeness of recursively enumerable sets and several generalizations of the fixed-point theorem”, Izv. Vyssh. Uchebn. Zaved. Mat., 1977, no. 4, 3–7 (Russian)
3.	M. M. Arslanov, “Some generalizations of a fixed-point theorem”, Soviet Math. (Iz. VUZ), 25:5 (1981), 1–10
4.	M. M. Arslanov, “Lattice properties of degrees below $O'$”, Soviet Math. Dokl., 32 (1985), 58–62
5.	M. M. Arslanov, “The recursion theorem, approximations, and classifying index sets of recursively enumerable sets”, Fundamentals of computation theory (Kazan' 1987), Lecture Notes in Comput. Sci., 278, Springer-Verlag, Berlin, 1987, 34–37
6.	M. M. Arslanov, “The lattice of the degrees below $0'$”, Soviet Math. (Iz. VUZ), 32:7 (1988), 43–53
7.	M. M. Arslanov, “Completeness in the arithmetical hierarchy and fixed points”, Algebra and Logic, 28:1 (1989), 1–9
8.	M. M. Arslanov, “On the structure of degrees below $0'$”, Recursion theory week (Oberwolfach 1989), Lecture Notes in Math., 1432, Springer-Verlag, Berlin, 1990, 23–32
9.	M. M. Arslanov, S. Lempp, and R. A. Shore, “Interpolating d-r.e. and REA degrees between r.e. degrees”, Ann. Pure Appl. Logic, 78:1-3 (1996), 29–56
10.	M. M. Arslanov, S. Lempp, and R. A. Shore, “On isolating r.e. and isolated d-r.e. degrees”, Computability, enumerability, unsolvability, London Math. Soc. Lecture Note Ser., 224, Cambridge Univ. Press, Cambridge, 1996, 61–80
11.	M. Arslanov, “Degree structures in the local degree theory”, Complexity, logic, and recursion theory, Lecture Notes in Pure and Appl. Math., 187, Marcel Dekker, Inc., New York, 1997, 49–74
12.	M. M. Arslanov, G. L. LaForte, and T. A. Slaman, “Relative enumerability in the difference hierarchy”, J. Symb. Log., 63:2 (1998), 411–420
13.	M. Arslanov, “Open questions about the $n$-c.e. degrees”, Computability theory and its applications (Boulder, CO 1999), Contemp. Math., 257, Amer. Math. Soc., Providence, RI, 2000, 15–22
14.	M. M. Arslanov, “Table complete sets of Kolmogorov complexity of conputations”, Collection of selected papers by members of the Academy of Science of the Republic of Tatarstan, Foliant, Kazan, 2002, 199–209 (Russian)
15.	M. M. Arslanov, “Truth-table complete computably enumerable sets”, Computability and models, Univ. Ser. Math., Kluwer Acad./Plenum Publ., New York, 2003, 1–10
16.	M. M. Arslanov, “Generalized tabular reducibilities in infinite levels of Ershov difference hierarchy”, Logical approaches to computational barriers (CiE' 2006), Report Series, Swansea, 2006, 15–23
17.	M. M. Arslanov, I. Sh. Kalimullin, and S. Lempp, “On Downey's conjecture”, J. Symb. Log., 75:2 (2010), 401–441
18.	M. M. Arslanov, “The Ershov hierarchy”, Computability in context. Computation and logic in the real world, Imperial College Press, London, 2011, 49–100
19.	M. M. Arslanov, “Structural theory of degrees of unsolvability: advances and open problems”, Algebra and Logic, 54:4 (2015), 342–346
20.	M. Arslanov, “Splitting and non-splitting in the difference hierarchy”, Math. Structures Comput. Sci., 28:3 (2018), 384–391
21.	M. M. Arslanov and M. M. Yamaleev, “On the problem of definability of the computably enumerable degrees in the difference hierarchy”, Lobachevskii J. Math., 39:5 (2018), 634–638
22.	M. M. Arslanov, “Fixed-point selection functions”, Lobachevskii J. Math., 42:4 (2021), 685–692
23.	M. M. Arslanov, “On a general method of constructing post reducibilities and the corresponding completeness criteria”, Lobachevskii J. Math., 43:12 (2022), 3430–3434
24.	M. M. Arslanov, “Completeness criterions for a class of reducibilities”, Russian Math. (Iz. VUZ), 66:10 (2022), 62–66

Citation in format AMSBIB

\Bibitem{AbyBekGon24}
\by A.~N.~Abyzov, L.~D.~Beklemishev, S.~S.~Goncharov, Yu.~L.~Ershov, I.~Sh.~Kalimullin, V.~L.~Selivanov, A.~L.~Semenov, A.~Tuganbaev, M.~Kh.~Faizrahmanov
\paper Marat Mirzaevich Arslanov (on his eightieth birthday)
\jour Russian Math. Surveys
\yr 2024
\vol 79
\issue 2
\pages 369--373
\mathnet{http://mi.mathnet.ru//eng/rm10168}
\crossref{https://doi.org/10.4213/rm10168e}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4782817}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2024RuMaS..79..369A}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=001306112700007}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85201000823}