# Department of Algebra and Mathematical Logic

**Department of Algebra of Kazan University** was founded in 1934 by Nikolai Grigoryevich Chebotarev, prominent algebraist and corresponding member of the USSR Academy of Sciences. It happened when Department of Mathematics of Faculty of Physics and Mathematics had been devided into the three departments: Mathematical Analysis, Geometry and Algebra. Heyday of Department of Algebra fell on the 30th and 40th years of the last century. It was the time of the N.G.Chebotarev's researches when he headed the Department of Algebra up to 1947. The Kazan school conducted fundamental researches in algebra during these years and received worldwide recognition. It had developed, along with Fields Theory and Galois Theory, the theory of Lie Groups and Lie Algebras (by N.G.Chebotarev, V.V.Morozov, who was the head of Department of Algebra from 1947 to 1970, and their students). A.V. Dorodnov, a student of N.G. Chebotarev, received the final solution of one of the classical problems of antiquity - the problem of squarable lune. I.D. Ado in his PhD thesis got an exact finite-dimensional representation of finite-dimensional Lie algebras over the fields of characteristic zero (1935). Due to this result he was awarded the degree of doctor Science immediately, bypassing the candidate degree. V.V. Morozov in his PhD thesis gave an enumeration of all primitive representations of simple Lie groups (1938). Later in his doctoral thesis he received an enumeration of all maximal subgroups of non-semisimple Lie groups (1943). This result, together with the results of the Moscow mathematician E.B. Dynkin, gave a complete solution to the problem of classifying all primitive representations of Lie groups. This problem was stated by S. Lie in the XIX century.

A number of students of N.G. Chebotarev studied his problem on extendibility of polynomials. A polynomial f(x) is called M-extendable, where M is a subset of complex numbers, if by adding some higher-order terms to f (x) it can be obtained by a polynomial whose roots belong to M. The Chebotarev's student A.I. Gavrilov proved that a polynomial is M-extendable if M is a circle of nonzero radius with center at the origin. Another his student N.N. Meiman investigated the case where M is the set of real numbers. In this case the problem of extendability of a polynomial reduces to the verification of an infinite number of inequalities. N.N. Meiman managed to develop an algorithm which in a finite number of steps helps to determine whether those conditions are true. Due to these researches, N.N. Meiman also was awarded the degree of doctor of Science, bypassing the candidate degree.

After the death of N.G. Chebotarev in 1947 his disciple V.V. Morozov headed Department of Algebra. Naturally, the research conducted at the department during the life of N.G Chebotarev had a significant impact after his death as well. VV Morozov continued his research on the theory of Lie Groups and Theory of Resolutions. Another Chebotarev's disciple A.V. Dorodnov (he headed the department after V.V. Morozov retired in 1971 and until 1976) studied subfields of fields of the algebraic functions.

Unfortunately, after the death of Chebotarev the Kazan algebraic school gradually ceased to exist. It's well known that school of N.G. Chebotarev consisted only of his disciples, the most prominent of them were I.D. Ado, A.V. Dorodnov, N.N. Meiman and V.V. Morozov, however they didn't have their own disciples at that time. Soon after the death of Chebotarev, among his disciples only A.V. Dorodnov and V.V. Morozov remained at the department. N.N. Meiman in 1936 moved from Kazan to Kharkov, later in 1938 he moved from Kharkov to Moscow starting researching a new topic. In Moscow he worked in Institute for Physical Problems, as well as in Institute of Theoretical and Experimental Physics of the USSR Academy of Sciences. He worked in a group of L.D. Landau in close cooperation with a group of Ya.B. Zeldovich. He took part in elaboration of the mathematical side of nuclear weapons: led by N.N. Meiman the computing bureau of the Institute for Physical Problems performed calculations of energy allocations of atomic bombs . For this work, in 1953 he was awarded the Stalin Prize (commented as ''for research in the theory of stability of difference schemes'') .

Even earlier, in 1935, I.D. Ado went to work in Kazan Chemical-Technological Institute. However, his creative activity decreased for various reasons in the postwar years. Due to poor health V.V. Morozov gradually withdrew from energetic scientific activity, focusing on pedagogical and scientific-organizational activity.

Since 1976, the Department for 10 years was headed by Associate Professor Yu.B. Ermolaev who was not being a doctor of Science could not longer hold the position of head. At that time the department did not have doctors of Science, moreover, Kazan University didn't have staff suitable for the position of the head of Department of Algebra. After several unsuccessful attempts to invite to head the department of someone from a side, the university administration considered the question about closing the department, more precisely its accession to the Department of Geometry. However, against the prospect of loss Department of Algebra created by N.G. Chebotarev and almost entirely consisting of V.V. Morozov's students, the department's staff rebelled, as well as many of the faculty's Professors. As a result, in January 1989 M.M. Arslanov from Faculty of Computanional Mathematics and Cybernetics was invited to head the Department of Algebra. Shortly before this he became a doctor of Science approving his doctoral thesis at Institute of Mathematics of Siberian Branch of the USSR Academy of Science. The administrations of Kazan University and Faculty of Mathematics and Mechanics tasked him to bring the department out of the crisis .

In the early 90s the main efforts of the department staff were aimed on improving research in the department, on the expansion of its topics involving talented young reserachers, as well as on improving its material and technical base. For this reason, the department organized several research seminars and conducted individual work with students, signed several agreements with enterprises of the city to carry out research and development on their topics.

These efforts eventually allowed to achieve the goals. Gradually, the department began to play a significant role in research and educational activities of the faculty. Expansion of the research subjects, conducted in the department, and attraction of promising and talented young mathematicians allowed to found Department of Algebra and Mathematical Logic in the Chebotarev's Research Institute of Mathematics and Mechanics at KSU in 1993. In fact, this department was a division of Department of Algebra, all its staff were also professors or lecturers at Department of Algebra, and most of the professors of Department of Algebra were cooperating with Department of Algebra and Mathematical Logic.

Soon the first doctors of Sciences grown up at Department of Algebra. In 1995, I.I. Sahaev approved his doctoral thesis (on Theory of Rings and Modules) in Saint-Petersburg State University, and in 2003, S.M.Skryabin approved his doctoral thesis (on Lie Algebras) in the Steklov Institute of Mathematics. In 2009, a doctoral thesis (on Constructive Algebras) was approved by I.Sh. Kalimullin, a young mathematician of the department, in Kazan University. Also in Kazan University in 2011, Associate Professor S.N. Tronin approved his doctoral thesis (on Theory of Categories theory and Operads) and in 2014, A.N. Frolov approved his doctoral thesis (on Computability Theory). In the next few years the department is planning to approve another three doctoral theses (by A.N. Abyzov, S.N. Ilyin, and M.Kh. Faizrakhmanov).

Since M.M. Arslanov became the head of Department of Algebra the topics of the department research were expanded by Mathematical Logic, in particular, by Computability Theory and Computable Algebras. Hence, the department was renamed to Department of Algebra and Mathematical Logic in 2007. Currently, the department conducts research in various areas of the modern Algebra and Mathematical Logic.

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