Department of Mathematical Analysis

The Department of Mathematical Analysis was founded in 1934. For forty years (1934-1974) the Department was headed by the Honoured Scientist of Russian Federation, Professor, Doctor of Science Boris M. Gagayev (1897-1975), then (1974-1998) by Anatolii N. Sherstnyov, Honoured Scientist of the Republic of Tatarstan, Professor, Doctor of Science. Now the Department is headed by Semen R. Nasyrov, Professor, Doctor of Science.

Staff

General courses taught at the Department for the whole Institute of Mathematics and Mechanics include "Mathematical Analysis", "Functional Analysis and Integral Equations", "Theory of Probability and Mathematical Statistics". In addition, for students of Mechanics, the general course "Theory of Functions of Complex Variable" is read.

In the above-mentioned courses, the following manuals and guides written by our staff are used:

• L.A.Aksent'yev Problems on Theory of Functions of Complex Variable and Operational Calculus. Kazan University Press, (2nd edition), 1984.
• L.A.Aksent'yev Complex Potential of the Flat Field . Kazan University Press, 1989.
• A.N.Sherstnyov The Guide of Lectures on Mathematical Analysis, (2nd edition), Kazan University Press, 1993.(Sub title of the Committee for Higher School at Ministry of Science and Technology Politics of RF).
• A.N.Sherstnyov Elements of Measure Theory and the Lebesgue Integral , Kazan University Press, 1993.

The Chair graduates students in two specializations: "Functional Analysis" and "Geometrical Theory of Functions".
The students who choose our Chair for their specialization start to be involved in the Chair's scientific problems from very early steps in their university courses. They participate in the seminar "Additional Chapters of Analysis", where they can prepare themselves to their specialization in two years. Afterwards, already within the frameworks of their chosen specialty, they assist in a cycle of interconnected special courses, which lead them to the contemporaneous fundamental problems of the modern Functional Analysis and Theory of Functions. In a number of cases, the Chair forms specialists along personalized individual Curriculum, provided that the main special courses in the Curriculum of a specialization are treated in a specific way.

This specialization was opened in 1974 on the basis of the scientific seminar "Operator Algebras and their Applications" (supervisor - Honoured Scientist of RT, professor A.N.Sherstnyov ). Mathematical problems, which led to this specialization, are related to an intensively developed branch of Functional Analysis, namely "Topological Algebras and Their Representations". Active interest on this domain is based on rich achievements in this area; on the other hand, topological algebras and their representations constitute a powerful tool in contemporaneous Mathematical Physics. One of the central problems of the theory of topological algebras is the construction of an integration theory there. The objective of the specialization is a fundamental preparation of specialists in functional analysis. An active interest to this part of Mathematical Analysis is conditioned, on one hand, by large fruitful achievements in the last few years. On the other hand, it is also due to the importance of topological algebras and their representations in the capacity of tools in the mathematical part of modern mathematical physics. One of central problems of the theory of topological algebras is a construction of a general integration theory in these algebras. The objective of the specialization is a fundamental preparation of specialists in the functional analysis, who must posses a free command in mentioned domain. Members of the Chair, who actively work in this domain, lecture a series of special courses which form the basis of this specialization. Graduates who continue along the same specialization in postgraduate courses as the Chair often work in scientific R&D institutions.

Within the framework of this specialization one can encounter the following special courses:

• Elementary theory of Hilbert spaces (36 hours).
• Vector spaces (special seminar) (36 hours).
• Banach algebras and their representations (34 hours)
• Finite-dimensional operator algebras (special seminar).
• Spectral theorem (54 hours)
• Glisson theorem (special seminar) (36 hours).
• Topological vector spaces (special seminar)(36 hours).
• Elementary theory of von Neumann algebras (68 hours).
• Theory of Tomito-Takesaki (32 hours).

Methodical publications for these courses:

• G.D.Lugovaya, Elementary theory of Hilbert spaces, Kazan University Press, 1985.
• F.F.Sultanbekov, Finite-dimensional operator algebras , Kazan university Press, 1986.
• N.V.Trunov, Spectral theorem, Kazan University Press, 1989.
• A.N.Sherstnyov, Vector spaces, 1994 (electronic version appr. 33KB).
• A.N.Sherstnyov, Banach algebras and their representations, Kazan University Press, 1984.
• A.N.Sherstnyov, Elementary theory of von Neumann algebras, Kazan University Press, 1988.

This specialization was founded early in the 70's on the basis of an existing scientific seminar "Geometric Theory of Functions" (supervisor Honoured scientists of RT, Prof. L.A.Aksent'yev). Problems connected to specialization were based on ideas and results by S.N.Andrianov, F.D.Gakhov, M.T.Nuzhin, and G.G.Tumashev. This specialization combines preparation of students for work on modern theoretic problems (fine problems of conformal mappings, Teichmuller spaces) with application to the theory of boundary value problems in concrete problems of the mechanics of continuous media. Modern theoretical methods for solving applied problems supply to graduates in this specialization an ample field of applications of their knowledge in both scientific research and production. This specialization was opened on Chair in 1978. Members of Chair actively work in the domain of the geometrical theory of functions of complex variable and lecture on a series of special courses which form a basis of this specialization.

Within its framework the following special courses are read:

• Symmetry method (36 hours)
• Application of the symmetry method to conformal mappings and boundary value problems (34 hours)
• Geometrical theory of functions of complex variable (70 hours)
• Boundary value problems for analytic functions (54 hours)
• Riemann surfaces and quasi-conformal mappings (54 hours)
• Inverse boundary value problems (special seminar) (70 hours)
• Univalent resolvability of boundary value problems (special seminar) (34 hours)
• Boundary value problems of the mechanics of continuous media (special seminar) (36 hours)
• Complex potential of a flat field with elements of mathematics modeling (special seminar) (70 hours)

Methodical publications:

• L.A.Aksent'yev, Symmetry Method, Kazan University Press, 1991.
• L.A.Aksent'yev, Application of symmetry method to conformal mappings and to boundary value problems, Kazan University Press, 1993.

Research and Investigation

• Univalent resolvability and well-posedness of inverse boundary value problems with applications to hydromechanics
• Methods of the functional analysis: Stochastic structures of the functional analysis
• Construction of measures and integrals on topological- algebraical and order structures

Scientific seminars

• Operator algebras and their applications.
  
• Geometrical theory of functions.