IM SB RAS
S. L. Sobolev IM SB RAS. 2003 .
 
Main fields of research 


Algebra, theory of numbers, and mathematical logic

This direction of researching was initiated at the Institute of Mathematics in 1960 by one of the most prominent mathematicians of the 20th century — Anatolij Ivanovich Mal'tsev (1909-1967). A. I. Mal'tsev is the author of profound works on application of mathematical logic to algebra and mathematical cybernatics. According to american mathematicians A. Tarskij, L. Henkin and D. Monk, A. I. Mal'tsev pointed out the way from logic to algebra. "Mal'tsev conditions", "Mal'tsev theories", "Mal'tsev theorems", "Mal'tsev algebras" are widely used in modern mathematical language. The theory of axiomatizable classes of algebraic systems, developed by A. I. Mal'tsev and A. Tarskij, led to overall synthesis of ideas of algebra and mathematical logic, and with new strength (now — in 20th) demonstrated the unity and integrity of mathematical science. At present, in the frames of the theory of axiomatizable classes, the theory of varieties, which became the central part of general algebra, received wide development. The theory of quasivarieties and Horn logic are also successfully developed.
Profound investigations are carried on in the theory of models, which became an independent part of mathematical logic. This theory studies the connections between formal languages and their interpretations or models.
At present, investigations in mathematical logic and algorithm theory are headed by Academician Yuri Leonidovich Ershov. Due to intensive development of computers, the notion of algorithm became one of the most important notions in modern mathematics. Along with studying the theory of numberings, which dates back to A. I. Mal'tsev, the problems of elementary classification of algebraic objects and problems of algorithmic decidability or undecidability of elementary theories of various classes of algebraic systems are studied. Problems of generalized theory of computability are investigated with the aim of further development of classical recursion theory.
Investigations in algorithm theory, constructive models and algebras, applied logic are headed by Corresponding Member of RAS Sergej Savost'janovich Goncharov. He and his followers develop the theory of algorithmic dimensions, solve the problems of existing for decidable and constructive models. On this way the problems, founded by A. I. Mal'tsev, M. Morly, K. Ash, Yu. L. Ershov and other famous mathematicians, were solved. Profound investigations are carried on in the fields of nonclassical logics and semantic programming.
Investigations in the general theory of rings were originated by a prominent algebraist Corresponding Member of the USSR Acad. Sci. Anatolij Illarionovich Shirshov (1921-1981). He created a large scientific school on the theory of rings and Lie algebras. In this school both classical leads and new ones, united in the frames of the theory of rings close to assotiative and the algebraic theory of projective planes, are now successfully developing. Some particular researchers are connected with quantum mechanics and geometry.
Corresponding Member of the USSR Acad. Sci. Mikhail Ivanovich Kargapolov (1928-1976) was the direct organizer of investigations in the theory of groups in Mal'tsev's logic-algebraic school. Under his guidance, a large group of professors formed a group of scientists, which successfully works in Siberia just now. At present, investigations in the theory of groups continue in the field of finite groups, decidable and generally decidable groups, matrix groups, lattice-ordered groups and groups of authomorphisms, pro-finite groups. The publication "Unsolved Problems of the Group Theory. Kourovka Notebook" received worldwide fame.
Since 1962, the Siberian Logic-Algebraic School publishes a specialized journal "Algebra and Logic", which is translated into English.

© 2004, Sobolev Institute of Mathematics of the SB RAS, Novosibirsk
     Omsk Branch of the Institute