The Faculty of Mathematics and Mechanics of St.Petersburg University together with its scientific partitions: the Institute of Mathematics and Mechanics, the Computing Center, the IT Research Institute, the Astronomical Institute - is one of foremost scholastic and scientific centers in Russia. It draws the most serious and able entrants being care for mathematics and gives to them the first-class education. |

### About our Faculty

At the present time (October 1-st 2008) there are on the faculty about 1587 students and 299 postgraduates and improvers. On the teaching staff (364 in total), there are 111 professors and 172 assistant professors. Lecturers and other staff of the Faculty works on fundamental and applied problems in all regions of modern mathematics, mechanics, computer science and astronomy. The Faculty of Mathematics and Mechanics has tight connections with institutes of the Russian Academy of Sciences, with the russian and foreign universities. The rating of St.Petersburg University is the next after the Moscow University.

At the present time the interest to the university education is growing. The cause of it is very simple: the prestige of such education is very high. For this purpose many previous education institutes are named now as universities. But the change of the name only means nothing. The authentic education of university has to be based on catholicity and depth, on the many years' experience and famous history, on the glorious teachers and world-wide fame. And although there are many universities in St.Petersburg today, the bona fide University is as previously only one. The main peculiarity of the teaching in university is the combination of the catholicity, width and depth of education with the detailed study of certain more narrow regions, where students obtain an initial experience in the scientific work.

Already during the first semesters our students have the possibility to take part in scientific researches under the supervision of world-class scientists. The catholicity of education is guaranteed by the fundamental courses on mathematics, computer science and programming, theoretical mechanics, on elements of economics and social sciences which are given in 1-4th semesters to students of all specialties. It must be noted that in the whole world there are not very much universities, where analogous courses are set so capitally.

Our best students who are abroad for traineeships, know this by their own experience. So the students that graduates the Faculty of Mathematics and Mechanics by specialties of Astronomy or Mechanics have the same fundamental mathematical education as specialists on mathematics. Much attention on our faculty is devoted to the study of foreign languages, in the first place to the english language. The modern science is international, and a real researcher must not only read papers on his narrow speciality but converse freely with foreign fellows. Therefore the foreign language program together with the compulsory course of english contains optional ones for studying the second language, and some special courses are given in foreign languages.

The basic program on computer science for students of all specialties includes the study of technology of programming, of algorithmical languages and of information systems. Lectures on computer science and the original students' work are performed in up-to-date computer classes. All workplaces in such classes form local computer nets. All local nets are included into the all-university network, which has a direct entry in Internet. Such an organization gives the possibility to students studying information technology in the computer classes to be members of worldwide information community.

After the fourth semester all students are assigned to different departments and take part in research works under guidance of assistant professors and professors. At the end of each school year they defend their research or diploma works.

The best students are send for practice to Universities in USA, Germany, France, Finland, Israel at al. Many of them are invited further to work in these Universities. It is not easy to find to-day some large scientific center where there are no our graduates. They have no difficulties to find the fork. The best graduates go to postgraduate school. Many graduates are invited into known scientific institutes such as St.Petersburg Department of Mathematical Institute in name of V.A. Steklov, Physic-Technics Institute in name of A.F. Ioffe, Pulkov Astronomy Observatory, and other Institutes of Russian Academy of Sciences, into Scientific and Technical companies, Insurances companies and banks.

There is an inter-department specialization on financial mathematics, where specialists on solving of economical problems are prepared. In the basis of this specialization lay the problems of probability theory and of mathematical statistics, which find their application in banks and insurance business. This specialization is created by the departments of Probability theory and Mathematical Statistics, Statistical Simulation, Operations Research, and Theoretical Cybernetics.

Say some words on speciality 511300 of **mechanics and applied mathematics**. It is designed according two-level education system: four years for degree of the bachelor, which is completed by the defence of a graduation work and obtaining the diploma of bachelor, and the following two-year magistracy, which is completed by the defence of a dissertation and obtaining the master's degree. Unlike the traditional system of high education, the two-level system is more flexible. A graduate with the bachelor diploma may choose the further direction of education among several master's programs. It is especially important at the present time when the labor market is characterized by the fast change of economic conditions.

Entrants ask often what is the difference between our applied mathematicians and mechanics and the graduates of analogical specialties of other institutes.The difference is the principal one. Technical education institutes are interested as a rule in some concrete applications of one or another mathematical method to concrete industry problems appearing in the framework of specialization of a given institute. In other words, they study the available algorithms and methods solving the concrete problems. Unlike this, our students are taught to invent the methods, to create new mathematical apparatuses for new problems, which appear in different branches of industry. It is clear that such an activity is impossible without wide mathematical training. One may say that applied mathematicians in industrial institute study the application of the mathematics, applied mathematician in the university study the mathematics of applications.

In order to be a student of our faculty it is necessary to cross the serious barrier of entrance examinations. The main of them, where the preparation of an entrance is tested, is the written examination on mathematics. Problems, which are given to entrants of mathematical faculty, do not need for their solving the experience being out of limits of school program.

All nonresident students (so as nonresident entrants during examinations) are living in the comfortable hostel near the university building. In the last years the places in this hostel have also the students from St.Petersburg that are living in distant regions of the town.

Let us list the departments of the Faculty of Mathematics and Mechanics and their specializations:

the Department of Mathematical Analysis (its chief is prof. V.P. Khavin) with specializations:

functional analysis, constructive theory of functions, linear and complex analysis;

the Department of Higher Algebra and Numbers Theory (prof. A.V. Yakovlev) with specializations:

theory of numbers, theory of groups, algebraic K-theory, Galois theory;

the Department of Higher Geometry (prof.N.Yu. Netsvetaev) with specializations:

algebraic and differential topology, topology of real and complex manifolds, Riemann geometry;

the Department of Differential Equations (prof. V.A. Pliss) with specializations:

theory of stability, theory of smooth dynamic systems, qualitative theory of differential equations;

the Department of Mathematical Physics (prof. N.N. Ural'tseva) with specializations:

nonlinear equations with partial derivatives, spectral theory of differential operators, asymptotic methods;

the Department of Probability Theory (prof. I.A. Ibragimov) with specializations:

limiting theorems, random processes, mathematical statistics;

the Department of General Mathematics and Computer Science (prof. M.A. Narbut);

this department ensures the teaching of mathematics and computer science for all other university faculties (excluding the Faculty of Physics and the Faculty of Applied Mathematics & Control Processes);

the Department of Computational Mathematics (prof. V.M. Ryabov) with specializations:

computational methods for solving of differential and integral equations, theory of approximations;

the Department of Parallell Algorithms (prof. Yu.K Dem'janovich) - new department.

the Department of Statistical Simulation (prof. S.M. Ermakov) with specializations:

imitative and statistic simulation, applied statistics, theory of optimization and design of experiments;

the Department of Theoretical Cybernetics (prof. B.A. Yakubovich) with specializations:

theory of learning and identification systems, theoretical robotechnics, theory of automata, mathematical methods for investigation of nonlinear systems;

the Department of Operations Research (prof. N.N. Petrov) with specializations:

discrete, linear, nonlinear programming and approximation, calculus of variations, theory of optimal control, theory of differential games, combinatorial problems;

the Department of Computer Science (prof. N.K. Kosovskii) with specializations:

functional and logical programming, theoretical fundamentals of computer science, algorithmical languages and methods of compilers constructing, automatic proof of theorems and constructing of effective algorithms, languages of artificial intelligence, information and expert systems;

the Department of Software Engineering (prof. A.N. Terekhov) with specializations:

system programming, realization of operating systems and compilers, technology of software for built-in real time systems, development of special chips, software for communication systems;

the Department of Theoretical and Applied Mechanics (prof. P.E. Tovstik) with specializations: analytical mechanics, theory of shells, theory of automatic regulation and control, electromechanics, theory of vibrations, mechanics of robots;

The Department of Hydroaeromechanics (prof. S.K. Matveev) with specializations:

hydroaeromechanics, gas dynamics and theory of shock waves, theory of carrying surfaces, physical-chemical aeromechanics, aerodynamics of rarefied gases;

the Department of Hydroelasticity (prof. B.A. Ershov) with specialization: dynamics of flexible shells in continuum;

the Department of Elasticity Theory (prof. N.F. Morozov) with specializations: problems on strength and fracture of materials, theory of cracks, problems on mechanics of solid deformable bodies;

the Department of Physical Mechanics (assosiate prof. V.A. Morozov) with specializations: physical foundations of mechanics of solid bodies, fluids, and gases, dynamics of plasma, biomechanics;

the Department of Astrophysics (prof. V.V. Ivanov) with specializations: theoretical astrophysics, observation astrophysics, radar astronomy;

the Department of Celestial Mechanics (prov. K.V. Kholshevnikov) with specializations:

gravitational fields of celestial bodies, star's astronomy, evolution of Solar system, evolution of triple star systems;

the Department of Astronomy (prof. V.V. Vityazev) with specializations: cosmic astrometry, radar astronomy.

#### St.Petersburg Mathematical School: inheritance of scientific traditions.

At the creation of St.Petersburg University on the mathematical faculty four departments were open: pure mathematics, applied mathematics (under which the mechanics was assumed), astronomy, and physics. With such a structure (but with some more departments) the university existed more than hundred years, and only in 20th of our century the faculty of physics and mathematics was reformed in two faculties: of physics and of mathematics and mechanics. It is necessary to note the succession of scientific traditions, which were inherited by the faculty from the celebrated St.Petersburg Mathematical School. The conceptual predecessor of this school was Leonardo Euler who worked during first half of 18th century in St.Petersburg Academy of Sciences and taught in the Academy University, which was the predecessor of our university.The school in question was created by the genial mathematician P.L. Chebyshev (1821-1894) who was the father of many new directions in theoretic and applied mathematics. His works on the theory of numbers were the beginning of investigations continued by his progenies being also the professors of the university academicians E.I.Zolotarev, A.A. Markov, I.M. Vinogradov, Yu.V. Linnik, associate members B.N. Delone and D.K. Faddeev, professors B.A. Venkov and Z.I. Borevich.

Due to works of P.L. Chebyshev and his successors the probability theory has became a strong mathematical science. Chebyshev was the author of shining works on the theory of moments, limiting theorems, the law of large numbers. These works were continued by his pupils academicians A.A. Markov, A.M. Lyapunov, S.N. Bernstein and then by Yu.V. Linnik whose pupils I.A. Ibragimov and V.V. Petrov successfully work now in the same field. The rise in the university of investigations in the domain of differential equations and mathematical physics is due to academicians A.M. Lyapunov, B.A. Steklov (the pupil of A.M. Lyapunov), associate member H.M. Gunter (the pupil of A.A. Markov), associate member I.A. Lappo-Danilevskii, academicians V.I. Smirnov (the pupil of V.A.Cteklov), N.E. Kochina, N.P. Erugin, S.M. Losinskii. N.M. G'unter was the creator of the Department of differential equations. Academician V.I. Smirnov has created the Department of mathematical physics, where his pupils are working successfully at the present time.

In 1920th investigations of professor G.M. Fichtengolts on the theory of functions of real variable have appeared. They were followed by the works on functional analysis of his pupils academician L.V. Kantorovich and professor B.Z. Vulikh. At the same time due to professor B.N. Delone the work on algebra and geometry was enlivened. Investigations on algebra of the associate member D.K. Faddeev and on geometry of academician A.D. Aleksandrov and his pupil Yu.A.Volkov have appeared. From mid 1960th investigations on topology under guidance of professor V.A. Rokhlin were intensively developing.

At the beginning of 1950th on the faculty the Department of computational mathematics was opened. His first chief was professor V.I. Krylov, then this place was taken by professor M.K. Gavurin. The teaching to programming and investigations on computer science were initiated on the faculty together with the appearance of the first computer in 1957. Further, in the connection with the fast increase of investigations on applied mathematics, the Department of computational mathematics was divided into four departments. In 1970 the Department of computers software (which is renamed now as the Department of computer science) and the Department of operations research were created. During the following years the Departments of theoretical cybernetics and statistical simulation were formed.

The first investigations on mechanics in the university (in the previous century) are due to academicians I.I. Somov and P.L. Chebyshev and associate member D.K. Bobylev. With the name of I.I. Somov the introduction into mechanics methods of vector geometry and vector analysis is connected, and with the name of P.L. Chebyshev --- the science on synthesis of mechanisms. D.K. Bobylev was in Russia one of creators of analytic direction in mechanics, which runs back to Lagrange and Ostrogradskii.

The first chief of the Department of **theoretical mechanics** was professor N.V. Rose known by his works on magnitology and textbooks on theoretical mechanics. During following years this Department was lead by associate member Yu.A. Krutkov and professor N.N. Polyakhov. The chief of Department of elasticity was associate member G.V. Kolosov well known in Russia and abroad as one of creators of effective solving methods for plane problem of elasticity theory. After him this department was led by Honored scientist of Russia E.L. Nikolai and academician V.I. Smirnov. For a long time on this department academician V.I. Novozhilov worked, who is the chief of St.Petersburg School of shell's theory, nonlinear elasticity, theory of strength and fracture.

The chiefs of the Department of **hydroaeromechanics** were associate member A.A. Satkevich, being the author of well known manuals on the theory of hydraulic turbines and aerodynamics, and after him associate member S.V. Vallander, the creator of dynamic theory of rarefied gas.

The chiefs of Department of **astronomy** were academicians V.K. Vishnevskii, A.N. Savich, and Honorary academician S.P. Glasenap who in 1881 has created the university's observatory, which exists till the present time. The first interests of astronomers lay in the domain of astrometry, then the investigation of associate member A.A. Ivanov on celestial mechanics were appeared. In 1920th the interests were turn to the astrophysics. It was is connected with the names of academicians V.A. Ambartsumyan and V.V. Sobolev who have created the well known school of theoretic astrophysics.

The distinguished feature of teaching and investigations, which had glorified in the whole world the St.Petersburg Mathematical School, is the aspiration to set oneself firstly for solving of the most important for the time being practical problems and solve them on a high mathematical level notwithstanding any difficulties. A shining example of such an approach is the work of two graduates of our university academicians L.V. Kantorovich and G.I. Marchuk. L.V. Kantorovich was a specialist on very fine and abstract questions of the functional analysis. In 1938 according to a task set by Veneer trust he addressed himself to a problem on the most beneficial use of machines. The method created by him was laid in the basis of the linear programming being at the present time one of most wide spread apparatus to solve extremal problems. The investigations of L.V. Kantorovich are marked by Lenin's and State prizes, in 1975 Nobel's prize on economics was awarded to him.

G.I. Marchuk, a graduate of the mechanic department of the faculty, has became a greatest specialist in the region of computational mathematics and mathematical simulation of nuclear reactors, of dynamics of atmosphere and ocean, of ecological processes. He is a laureate of Lenin's and State prizes, for a long time he was the chief of State Committee on Science and Technics, and in 1985 was chosen as the president of Academy of Sciences.

About the work of the lecturers and scientists of the faculty and its graduates it is possible to say and write very much. The main feature of the faculty is the traditional priority of fundamental research, which guarantees the qualitative specialist training and the connection with modern applications. For example, the fundamental investigations on astronomy have given the possibility to create together with departments of Academy of sciences the industrial basis in astronomic observatory of St.Petersburg university for constructing and building modern telescopes. Another example may by the investigations on computer's software, which there initiated in the laboratory of mathematical linguistics of the Institute of mathematics and mechanics and then were continued on the Department of computer's software together with Computing center. These investigations are of great interest for many industrial organizations. There are also many other fundamental developments on mathematics and mechanics, which may be used in many implementations.

From 1960 in St.Petersburg there exists a system of study groups for schoolboys and schoolgirls, which is attached to the faculty of mathematics and mechanics. It is a so called Youthful Mathematical School (YMS). In these groups teaching is performed by students and graduates under supervision of faculty's lecturers. Many of present-day professors and assistant professors were at some time pupils of the YMS.

In the study groups the pupils solve nonstandard ("olympiad's") problems, play mathematical games, take part in competitions. There they study "nonschool" parts of elementary mathematics, for example, some elements of graph theory, combinatorics, mathematical logic, projective geometry. The groups of YMS are oriented to children from different districts of the town having different levels of training. YMS is always open for schoolboys and girls of 5-7 classes. Many of them enter YMS in autumn, after they took the part in one of olympiades of YMS, whose terms are sent to all St.Petersburg schools. Each study group has a program for several years. Many members of such groups enter special town schools with prevalent study of mathematics and physics, continuing at the same time session in YMS.

#### On the department of mathematics

From the all three sciences being studied on the faculty of mathematics and mechanics You are, certainly, the best of all acquainted with the mathematics. In the school course You learned firstly the arithmetic, then algebra and geometry, elements of analysis. From this course You have known properties of points and straight lines, the rules for solving linear and quadratic equations, You were taught to draw the diagrams of simple functions, to find their derivatives and primitives.From the courses of modern mathematics You will have known that the geometrical figures, algebraic equations, and functions are the simplest variants of mathematical objects such that the necessity of their study is demanded by the complexity of world around us. You shall study the multidimensional spaces and even spaces having infinity of dimensions whose elements in its tern may have a very complex structure, solve differential and integral equations whose unknowns are not numbers but functions. You shall learn the laws, which rule languages for communication between human beings and computers or random walk of particles in microcosm.

The role of mathematics in the human society is defined first of all by the wide possibilities of its applications. By this cause the specialty of a mathematician can be obtained in two departments, "pure" and "applied" ones.

Persons working in "pure" mathematics are attracted by mathematical problem themselves , apart from their usefulness. The study of mathematics, the selfsufficient mathematical creative work is a gripping, engrossing affair. Often mathematicians tend to investigate some intrinsic mathematical problems, which from ancients times are of interest for them.

In our days mathematics is a fundamental science having the widest applications. The most important parts of mathematics are analysis, algebra, and geometry. Despite their long history, they are in the center of mathematicians interests also in our days.

The Department of **mathematical analysis** was created at the beginning of 1930th. Till the 1952 its chief was one of pioneers of the theory of functions of the real variable professor G.M. Fihtengolts. After him professors S.M. Losinskii, I.P. Natanson, B.Z. Vulikh, B.S. Pavlov were at the head of the department. At the time of its creation on the department the investigations of the fast developing during the following years theory of functions of the real variable were initiated. The substantial contribution to this discipline was made by professors G.M. Fihtengolts, L.V. Kantorovich, and I.P. Natanson who was the author of the first domestic textbook on this discipline.

In 1934 academician V.I. Smirnov has created the seminar on functional analysis. In the works of L.V. Kantorivich and other members of this seminar the basis of a new direction in the functional analysis was laid, which now is known as the theory of partially ordered spaces. In other works of L.V. Kantorovich (written before 1941) was formulated yet one important direction of the functional analysis. It was the theory of generalized functions (distributions), whose fast development began in 1950th.

The works of professors G.M. Golusin and N.A. Lebedev were large contributions to the geometric theory of functions. At the present time on the department different directions of theory of functions and functional analysis are represented. Investigations on theory of approximation are traditional for the department. They there initiated in the last century by the great Russian mathematician P.L. Chebyshev. Now this direction is headed by professors V.V. Zhuk and G.I. Natanson. The seminar under supervision of G.I. Natanson more than 30 years associates St.Petersburg's mathematicians working in this domain.

Another traditional and important direction is the theory of analytic functions and operators. This domain borders on as classical function theory so as functional analysis. Here professors V.P. Khavin (a pupil of V.I. Smirnov), A.B. Aleksandrov, S.A. Vinogradov, and N.A. Shirokov are working. Their works and works of their pupils are well known to specialists. The seminar of professors B.P. Khavin and N.K. Nikolskii is an international center of investigations in this domain of mathematics. During last 15 years four graduates of the department who took part in this seminar have got the prestige international prize of Salem for works of young mathematicians in the theory of functions.

Yet one direction of department's scientific work is the theory of groups representations, the ergodic theory, and the geometry of Banach spaces. With this direction the works of professors A.M. Vershik, S.V. Kerov, B.M. Makarov are connected. A.M. Vershik is the head of seminar on the theory of representations and ergodic theory. The department is connected tightly with the laboratories of mathematical analysis and of complexity of algorithms, which enter the St.Petersburg department of the institute of RAS in name of V.A. Steklov. Many scientists from the staff of this institute give lectures for students attached to the department and control their diploma works. During last 5 years the members of the department have edited in the Russia and abroad more than 10 monographs and textbooks.

The Department of **higher geometry** has two specializations: topology and geometry. The St.Petersburg topological school was created in mid 1960th by professor V.A. Rokhlin. The main subject of investigations of the Department is the topology of differentiable manifolds, i.e. studying (with the use of methods of modern algebra) the structure of spaces, whose topological structure "in the small" is analogous to that of a multidimensional Euclidean space. The geometric school of the faculty has also more former traditions connected first of all with the name of academician A.D. Aleksandrov. Scientific interests of department's scientists lie in the domain of multidimensional geometry of polyhedrons and convex bodies, in the domain of Riemann geometry, which studies the spaces whose geometry "in the small" is different from the Euclidean one, and the yet more general spaces.

The department has tight connections with the laboratory of geometry and topology of the mathematical institute of RAS in name of V.A. Steklov. The lecturers of the department take part in the work of the town seminars on topology and geometry, the members of laboratory staff professors Yu.D. Burago, S.V. Buyalo, and V.A. Zalgaller conduct general and special courses on the faculty.

The Department of **higher algebra and theory of numbers **was based in 1932. For the time of its existence its workers have got many notable results. The worldwide celebrity have the works on theory of numbers of academician Yu.V. Linnik. In the works of associate member of RAS D.K. Faddeev among of many important results there are initial notions (analogous notions have appeared independently in works of american mathematicians) on gomological algebra being the new and very important branch of modern algebra. Professor A.V. Yakovlev have solved in considerable part the problem on immersion of fields. Professor S.V. Vostokov has formulated the reciprocity law in the most general form.

The founders of scientific school in St.Petersburg on **probability theory **were great Russian mathematicians P.L. Chebyshev, A.M. Lyapunov, and A.A. Markov. Their works have played the defining role in the development of the theory of probability. But the investigations in this domain were initiated in St. Petersburg before them. V.Ya. Bunyakovskii (being the vice president of St.Petersburg's Academy of Sciences for 25 years) has written the first textbook on probability ( F. Gauss learned Russian by this book). In particular, this book contains the first consideration of the problem on statistical inspection of production quality. Academician S.N. Bernstein worked in first decades of our century in St.Petersburg University. But the autonomous Department of the probability theory did not exist till the December 1947, when Yu.V. Linnik was appointed as the chief of **the Department of probability theory and mathematical statistics** according to dictation of rector A.A. Voznesenskii.

During first 5 years the members of department were Yu.V. Linnik, N.A. Sapogov, O.V. Sarmanov (a pupil of S.N. Bernstein), and V.P. Skitovich. Yu.V. Linnik give lectures on mathematical statistics and theory of random processes, on theory of probability for students of mechanic specialties and on the faculty of economics (before Yu.V. Linnik the course on the faculty of economics was conducted by professor A.A. Markov, the sun of the founder of Markovian processes).

In the following years the members of the department became V.V. Petrov, I.A. Ibragimov, Yu.A. Davydov, A.G. Bart, L.V. Osipov, Ya.Yu. Nikitin, S.S. Vallander, V.B. Nevzorov, C.M. Ermakov, V.V. Nekrutkin, V.N. Solntsev (the last three have passed now to the Department of statistical simulation organized by S.M. Ermakov), A.I. Martikainen, S.M. Anan'evskii, O.V. Rusakov, A.N. Frolov. Combine jobs on the department hold N.N. Vorob'ev, V.N. Sudakov, A.M. Kagan, M.S. Nikulin, V.N. Solev, M.I. Gordin, V.A. Egorov, A.N. Borodin, M.A. Lifshits, A.Yu. Zaitsev, L.V. Rozovskii, S.V. Malov, E.V. Polikarpov. Almost all of them were the graduates of the department and then postgraduates or fellow applicants on the department.

Among graduates of the department are professors I.V. Romanovskii, A.A. Zinger, L.B. Klebanov, A.N. Tikhomirov, L.A. Petrosyan, O.N. Bondareva, O.M. Kalinin, Yu.V. Borovskikh, N.V. Smorodina, V.A. Statulavichus (who together with I.P. Kubilyus, the previous aspirant of Yu.V. Linnik, has founded the school on probability in Lithuania), professors of Dresden University R. Rihter and W. Wolf, professor of Geteborg University T. Arak, professor of Groningen University (Holland) T. Mikosh, professors of Meriland University (USA) A.M. Kagan and L.B. Rukhin, professor of Magdeburg University G. Kristof, and professor Chen' Khan'fu (Peking).

On the Department of** differential equations** the ordinary differential equations are studied. Such equations arise in very different domains: in mechanics and physics, biology and chemistry, electronics and economics. In St.Petersburg University the theory of ordinary differential equations was developed by academicians V.A. Steklov, V.I. Smirnov, N.E. Kochin, by associate member I.A. Lappo-Danilevskii, by Honoured Scientist S.M. Lozinskii, and by their pupils. The department was founded in 20th of our century. Its first chief was associate member of Academy of Sciences N.M. G"unter, and after him was professor N.P. Erugin. The last 37 years the department is headed by associate member of RAS V.A. Pliss.

Since many very simple differential equations cannot be integrated exactly (it was proved by Liouville in the 19th century), the main attention on the department is given to qualitative methods, to the art of proving different properties of solutions without solving of equations themselves.

The one of primary problems is the asymptotic behavior of solutions, which is the subject of the stability theory. The mainland contribution to this theory was made by the great Russian scientist A.M. Lyapunov who worked in St.Petersburg about all his life. Till the present time this theory is one of main directions of work for department's scientists. Certainly, the class of problems of this theory is now much wider. It includes hyperbolic sets, invariant manifolds, and the recently opened rather curious the phenomenon called "a strange attractors" (or deterministic chaos). Many results on stability belong to V.A. Pliss, there exists even the known "Pliss' principle of reducing".

The subject of the local qualitative theory is the topological organization of singular points. It is a very difficult problem, and at the present time only the two-dimensional space has been studied (in many ways it is the merit of professor A.F. Andreev).

The theory of smooth dynamic systems studies the global topological structure of dynamic systems. Its methods and problems are very diverse and interesting. For example, professor Yu.N Bibikov is occupied with the bifurcation theory (bifurcation is an abrupt change of system's properties under smooth change of parameters) and professor S.Yu. Pilyugin is occupied with structurally stable systems (geometric properties of such systems do not change under small perturbations). They both are world-wide known specialists.

The Department of** mathematical physics** exists from 1956. The subjects of department' scientific investigations are different, but all problems are inspired by Nature being the best mathematician of all times and generations. The mathematical physics is, in spite of its name, a part of mathematics. But its roots are in physical theories describing the nature. Nature has a "good feel": problems stated by it are often very in-depth and interesting but sometimes very complex. It is possible that the problem on complete investigation of correctness of basic boundary problems connected with the mathematical theory of motion of fluid is compatible by its complexity with the celebrated great theorem of Fermat.

At the present time the members of the department investigate problems, which appear in different regions of modern physics. These are problems of hydrodynamics, mechanics of continuum, the problems of diffusion, the problems of the theory of waveguides and lightguides, of the theory of diffraction, the problems of spectral theory and physical kinetics.

On the department of mathematics there is also the Department of **general mathematics**. This department has no graduates, it provides other faculties of St.Petersburg University with teaching of mathematics and computer science. It was founded in 1936 by the initiative of academician V.I. Smirnov and associate member N.S. Koshlyakov who became its first chief. Academician V.I. Smirnov attended this department. His recommendations on teaching of higher mathematics on faculties of natural sciences were very useful. V.I. Smirnov is the author of very popular five-volume "Course of Higher Mathematics".

After N.S. Koshlyakov the chiefs of the department were: Honoured Scientist professor K.F. Ogorodnikov, associate member R.O. Kuz'min, academician V.I.Krylov, professors D.M. Volkov and M.M. Smirnov, assistant professor N.A. Sakharnikov, professors A.S. Merkur'ev and (at the present time) M.A. Narbut.

Lecturers of the department works in different regions of pure and applied mathematics, such as algebraic geometry and K-theory, artificial intelligence and pattern recognition, nonclassical problems of mathematical physics and so on.

#### The department of applied mathematics

The role of mathematics in the human society is defined by wide possibilities of its applications. The boisterous growth of investigations in the domain of applied mathematics was the cause of creation in the framework of the faculty of a department with the same name. This new department is based on fundamental mathematical education while creating a new mathematical apparatus for new appearing problems in different branches of science.The Department of **computational mathematics**, the oldest one in the department, was created in 1951, in the time of appearance of first native computers, in the connection with the vital necessity of training specialists in the domain of computational mathematics and programming. The founders of the department were academicians L.V. Kantorovich and V.I. Krylov.

The scientific specialization of the department were methods of computations. As it is well known, many practical and theoretical problems from different domains of physics, mechanics, and technics are reduced to a mathematical model (task), which is described by different functional equations (algebraic, differential, integral and so on) or by their systems. It is a very seldom chance to find the precise solution of such task. Therefore the necessity appears to find its "approximate" solution.

At the time of mathematical stating of a problem mathematicians work in the tight contact with customers who is interested in the fast obtaining sufficiently precise (reliable) and inexpensive solution. Under these conditions, which are rather conflicting, a mathematician have to choose or develop an algorithm of solving such that it would not be very laborious as with respect to time spent by the mathematician so as with respect to the time of the algorithm realization by a computer and such that it permits to estimate the error of approximate solution. The development of methods for numerical solving different problems, their investigation and realization is the subject of scientific work of the department.

The theoretical basis of such a work is containing in the monographs of the above mentioned academicians L.V. Kantorovich and V.I. Krylov and professors M.K. Gavurin, S.G. Mikhlin, and I.P. Mysovskikh. In the process of teaching to the computational mathematics and its following use also all mathematical disciplines studying on the faculty are used together with computers and programming languages. Such a wide outlook gives our graduates the possibility to work successfully in computing centers of different organizations and institutes, to conduct the scientific work. The department has prepared many post-graduates from Germany, Vietnam, China, Egypt, Bulgaria, and Syria an made an substantial contribution into the development of computational mathematics in China.

There are many random phenomena in our world. One of methods to study these phenomena is their simulation. This is the subject of the Department of statistical modeling. Though probabilistic models were used widely also in the 19th century, they were at that time some part of probability theory. Only in 50th of the 20th century, when first computers have appeared, the statistical simulation gets an autonomous meaning.

The Department of **statistical modeling** was open under assistance of academicians Yu.V. Linnik and G.I. Marchuk who appreciated perspectives of this new scientific tendency. Indeed, among other applications, apart of theory of probability and mathematical statistics, the statistical modeling has tight connections with the computational mathematics, theory of numbers, mathematical physics, and theoretical cybernetics. Here there are very interesting theoretical problems, which are waiting for talented young investigators. Another directions of department's activity are applied aspects of mathematical statistics, mathematical methods in biology, the theory of complex systems, the theory of mass servicing, market models, the financial mathematics.

The wide spectrum and topicality of investigations facilitated the formation of international connections. For last 5 years the department has organized 4 international scientific conferences. Its graduates have great success on the labor market in Russia and also work abroad in banks, firms, and universities The operations research in the wide sense is a science to take decisions. The process of coming to a decision in a real situation is complicated by a nebulosity of the goal, deficit of information, and the presence of conflict relations, which lead to active contradictions with our designs. These factors can be formalized and be subjects of a mathematical investigation.

The Department of **operations research** trains specialists on the theory of optimal and conflict control, on the mathematical economics, and also on development and usage of software for these and some other domains. Graduates of the department get the experience of solving many extremal problems and are ready for mathematical investigation of concrete control problems under conditions of ambiguity and conflict. They are acquainted with the modern computer technics and with mathematical methods of solving economical and technical problems.

The Department of **theoretic cybernetics** of St.Petersburg University was created in 1971 on the basis of the analogous Laboratory of Scientific Institute of mathematics and Mechanics of Leningrad University. On the department, which is tightly connected with the laboratory mentioned, the scientific collective was formed, which works on problems of mathematical cybernetics such as the analysis and synthesis of systems of automatic control, the optimal and adaptive control, theory of filtration, and on some other problems.

The kernel of department' staff consists of 12 candidates and 4 doctors of physical and mathematical sciences ( the last 4 are: associate member of RAS professor V.A. Yakubovich, professor G.A.Leonov being also dean of the faculty, chief of laboratory professor A.Kh. Gelig, and professor V.N. Fomin). The scientific production of the staff contains hundreds of works, among them are about 20 monographs.

The main task of the department is the training of specialists on mathematical cybernetics. Graduates of the department possess different mathematical disciplines in their interaction. At the present time about 100 postgraduates are prepared, more than 90 of them have defended their thesisses. The department collaborates actively with scientific institutes of Sweden, USA, Germany, France, Australia. Its graduates are working in all known scientific centers of these lands.

The general scientific tendency of the department is connected with solving of mathematical problems appearing in the process of constructing systems, which are able under the minimal help of a human or without such help to make some intellectual work. The subject of the cybernetics is "the investigation of systems of any nature that are able to receive, store, and process information and use it for control and adjustment" (A.N. Kolmogorov).

Among other classic methods of control theory on the department the methods of adaptive control are developed. It means the control under condition when the models of executive bodies and external media are not known completely and the classical control methods are not sufficient (for example, when space robots are constructed).

In the framework of mathematical cybernetics it is not necessary to make a real robot. We mean an information control system, which defines the robot's intelligence. Such a system can be created, for example, as a package of computer programs, and robot's behavior is simulated by a computer. It must be emphasized that such an imitation is a result of large theoretic investigations of corresponding problems, which need the powerful set of tools connected with many mathematical sciences.

#### The department of computer science

The interest to methods of computational mathematics and to creation of devices for automatization of computations has appeared in St.Petersburg University long ago. It is sufficient to note that in 1878 the founder of St.Petersburg Mathematical School P.L. Chebyshev invented an original arithmometr and created methods of approximate computations, which did not lose their meaning till the present time. The computerization of the faculty was started in 1957, when the first computer URAL-1 was obtained. At the same time the teaching of programming and investigations on computer sciences were started. In the following years computers M-20, BESM-3M, M-222, and ODRA-1204 were sequentially obtained.In 1970 the learning of programming was organized for all specialties of the day department. The assimilation of the first native computers, though being imperfect technically, gave the possibility to the faculty to be on the first line of computerization. 1973, the year of obtaining of the first computer ES-1030, was the beginning of era of general computerization, when the display of a computer becomes one of main instruments of a student. At the end of 1980th on the faculty were about 150 displays. The availability of computers ( at different times on the faculty there were up to 9 big computers) permitted, besides of their use for student's training, to perform large works on software development.

Under the guidance of professor A.N. Terekhov the compiler of algorithmic language ALGOL-68 for computers of ES-system was created and under the guidance of professor V.O.Safonov the compiler of the language "Pascal" for Russian supercomputer "Elbrus" was realized. In all these works many students and post-graduates of the faculty took part. Due to the many-year experience of the computers' use in different domains of mathematics a large set of packages of applied programs, about 400, were written by staff of laboratories and departments of the faculty.

From the beginning of 1990th in the teaching process and in the scientific work the role of personal computers becomes more and more significant. For training in computer science there are now several computer classes supplied by personal computers. These computers are different, from an old XT to modern Pentium. But they all are included into the local net and may use all resources of this net (computational, program, and informational). Local nets are included into all-university net and via it have connection with Internet. The students of specialization of computer science, and also of other specializations, have a possibility to study the modern achievement of such world-wide known firms as Intel, Sun, Microsoft, IBM.

On the department of computer science students obtain the speciality "software for computers and automatic control systems". In 1970 the Department of software was created, which in 1995 was renamed as the Department of** computer science**. At the end of 1995, according to the resolution of the Scientific Counsel of the faculty, on the base of Laboratory of System programming the Department of System Programming was created.

The department of software engineering trains the specialists on the very fast developing speciality called informatics. Here students study the architecture of modern computers, algorithmical languages and systems of programming, methods of compiler's constructing, operational systems, and computer's nets, which are the traditional disciplines of informatics. Besides them, there are on the Department new directions such as expert systems, systems of artificial intelligence, computer algebra and parallel computations, systems of information protection and programming in networks. The specializations on the mathematical logic is possible: automatic proving of theorems, methods of solving of mass problems, constructing of effective algorithms.

The Department of** system programming** is young, but the experience collected and its staff consisting of qualified mathematicians, system programmers and engineers of the Laboratory of system programming can develop the following specialties: system programming, realization of operational systems and compilers, technology of software for build-in systems of real time, software of telephone stations and systems of communication. The training on the department of computer science allows its graduates to work with different modern computer systems, and they have no difficulties with employment. The best of them continue their education as postgraduates.

Beginning from 1996 in St.Petersburg the semifinal of world championship of universities on computer programming takes place. These team competitions have the great authority. In their final 50 teams take part, among them teams from best universities of USA, Europe and Asia present. On the St.Petersburg city competitions, which goes by the same rule as the world championship, the last two years the team of the Faculty of mathematics and mechanics gains the first place. According to results of the semifinal 1996, three teams from Russia went to San-Hose (the capital of Silicone Valley, USA). Among them was the team of St.Petersburg University, which was formed from students of the Faculty of mathematics and mechanics. In the result our team has occupied the eighth place; in November 1997 on the semifinal in St.Petersburg about 40 teams from main technical high schools of Russia and near countries were present, and about 20 teams from Siberian amd Far East took part in the competitions via Internet. The team number 1 of St.Petersburg University, which was brought out by our Faculty, has occupied the first place with a good breakaway, having solved one problem more than the nearest competitors. So, the faculty team had gain the regional competition and got the right to take part in the world championship in Atlanta, USA, in 1998, where it have gained the second place. In 2000 and 2001 our team was a winner of the finals of world championships (in the USA and Canada) of universities on computer programming.

#### The department of mechanics

Mechanics is a science on a simplest form of materia motion being the mechanical motion and on the connected with this motion interactions between bodies. In the school course mechanics is the part of physics. In our university (analogously to many others) it is studied on the faculty of mathematics and mechanics. It is due to several causes.Firstly, from the time of its creation the mechanics was a branch of the applied mathematics. It is sufficient to recall the names of Archimed, Newton, Euler in order to say that mathematics and mechanics were developing in the tight connection of one with another. In many cases the development of mathematics gave possibilities to solve problems of mechanics, which cannot be solved by old methods. But in many cases the mechanics helped mathematicians to solve some problems. Mathematics has obtained from mechanics many problems whose solving generated new domains of contemporary mathematics. It is important that the same process continues at our days.

Secondly, mechanics is studied on mathematical faculties also by the cause that main complexities of mechanics' problems are their mathematical statements and solving. Physical laws such as laws of Newton, Hooke, Boyle-Mariotte, which define the interaction of bodies and must be given in a problem setting, are supposed to be known.

A specialist in mechanics having the university profile is not an engineer. He has no special technical information, which has a graduate on an technical high school, he does not know the technical details of some concrete constructing or calculation. His task is different. Using his basic mathematical and mechanical training, he had to set correctly problems, to develop methods of solving general and complex mathematical tasks, connected with different mechanical phenomena.

The Department of **theoretic and applied mechanics** is the one of oldest departments. Under different names it exists from the days of university foundation. But it would not be compared with an herbarium. On the department and in attached to it laboratory of applied mechanics many scientific works on dynamics of constrained systems and on the theory of control (including electromechanics and robotechnics) are performed. Deformations of complex mirrors of space telescopes and rotating motion of artificial satellites are studied. Approximate methods of calculations of shocks of elastic and elastic-plastic bodies are developed. The students study different parts of theory of stability, theory of linear and nonlinear vibrations of mechanical systems and solid bodies such as bars, plates, shells; methods of asymptotic analysis, methods of optimization, the theory of gyroscopes and the theory of automatic adjustment. On the senior courses much attention is given to the practice in laboratories with the mandatory use of modern computers.

One characteristic feature of the Department of **hydroaeromechanics** is the variety of investigations. Before the Second World War some important results in the domain of supersonic gas flow, theory of carrying surfaces and transient flows in rivers and channels, mechanics of viscous fluid, hydrometeorology, and experimental aerodynamics were obtained. From 1959 by the initiative of professor S.V. Vallander, who from 1952 till 1975 was the chief of the department, the development of mechanics of rarefied gas was initiated. The appearance of this direction was directly connected with the needs of rocket technics and cosmonautics. The rarefied gas is considered not as a continuum but as a collection of randomly moving molecules. In the dynamics of rarefied gases notions and methods of probability theory and statistical physics are essentially used. In 1973 for these works the State prize was awarded to workers of the department.

The department prepares specialists on three specialties: hydroaeromechanics, gas dynamics, and physic-chemical aeromechanics. The first one is connected with problems of shipbuilding and building of turbines, the second --- with avia- and rocketbuilding. The third speciality is connected with the necessity to take into account physical and chemical processes going in highspeed and hightemperature gasdynamics , with the needs of chemical technology and laser technics.

The Department of **theory of elasticity** was created in 1929. The elasticity theory has became a science in times of Renaissance, when near its cradle there were such scientific giants as Leonardo da Vinci and Galilei. It has its roots in very ancient times of first erections and calculations "on strength". Would the airplane cope with the redoubtable phenomenon of flatter, would be the elasticity of vasculature of the brain be sufficient to bear stress phenomena, how to stop the brittle crack running along gasmine, how to crash fast a strata, how to increase the strength of armor - these and many other questions solves the mechanics of deformable rigid body.

The wide diapason of possibilities will be opened for a student who chooses this department: it is possible to perform scientific experiments and investigate such amazing phenomena as the memory of the form, superelasticity, or superplasticity; to invent superstrong composite materials; to X-ray materials and define their dependability by obtained characteristics; to calculate domes, ships, airplanes, space crafts, human hearts; it is possible to study the nature of destruction, simulate by computers processes going in rigid bodies, to study applications of the theory of complex variable or functional analysis to the theory of elasticity , to define new mechanic effects by the end of your pen.

The Department of **physical mechanics** is rather young. Here the physical basis of rigid bodies, fluids, gases, and the dynamics of plasma are studied. The significant increase of traverse speeds and deformations of mechanical systems is accompanied by complex physical and chemical processes. To solve successfully mechanical problems, the knowledge of different fundamental parts of physics and chemistry is necessary. Theoretical investigations are supplemented by experiments on unique devices.

The study of the most complex processes of fluids and gases interaction with solid bodies shows the necessity of concurrent account of characteristics of gases and fluids, of rigid body properties, and to define conditions on the surfaces of their contacts. Therefore in 1987 the Department of **hydroelasticity** was founded. Its first chief was academician N.S. Solomenko. On this Department students study subsonic and supersonic flow of fluids and gases around deformable rigid bodies, the theory of shock waves, their interaction with shells and plates, the theory of carrying surfaces, and many other subjects. At the last time a new direction in mechanics is developing intensively. It is mechanics of biological objects (of human beings and animals), which is called biomechanics. It studies the blood motion in blood-vessels, deformations of certain human organs under the action of static and dynamical efforts ( for example, apparatus of vision, audition, and others) Biomechanics took a deserving place in the work of other departments of the faculty. Laboratories of the faculty have a rich experimental basis. After passing to Old Petergof experimental laboratories were roughly increased. They have the modern equipment satisfying world standards.

#### The departament of astronomy

St.Petersburg is legally considered as important center of domestic astronomy. Here there is the wide known Pulkov Observatory of Russian Academy of Sciences. Another academical centers in St.Petersburg are Institute of Theoretic Astronomy and the recently created Institute of Applied Astronomy. Their staff consists mainly of graduates of astronomic departments of the St.Petersburg University.More than hundred years ago in the university a small astronomic observatory was open. Now it is reorganized into Astronomical Institute having in his staff 10 doctors and more than 30 candidates of sciences performing successfully as observations so as theoretic works. Now there is a blusterous development of astronomy, which brings almost every day new achievements in the study of Universe. To a great extent this success is due to telescopes disposed in the space on space crafts. Our graduates in different astronomical centers accomplish enthralling investigations on the first line of science.

The learning on the astronomic department is not an easy task since before the astronomy students must have learned capitally the mathematics and physics. But the experience of previous generations of students shows that under the appetite to mathematics and sufficient obstinacy all difficulties may be overcome.

The central place in the modern astronomy belongs certainly to **astrophysics**. The analogous position occupies the Department of astrophysics on the astronomic department of the faculty. It is existing from 1933, when it was founded by academician V.A. Ambartsumyan. He and his successor academician V.V. Sobolev are authors of first-class investigations in different domains of theoretical astrophysics. The scientific school of Ambartsumyan-Sobolev is an established world leader in analytical theory of radiation's carry. These investigations are connected with classic objects of astronomy such as stars, nebulas, galaxies. In the domain of investigation of nebulas on the department methods of definition of main characteristics such as temperature and chemical composition of nebulas were created and specified. The construction of sequential theory of star's spectrums formation was a fundamental achievement in the investigation of stars.

Our theorists study actively problems of cosmic gasdynamics, which conquers now an important place in astrophysics. To day astrophysicians-theoretists spend more time near their computers than observers near their telescopes. But despite of this, the basis of astrophysics is the observation. The most part of faculty's astrophysicists are observers. Many programs on observations are performed by university's astrophysicists on the greatest instruments of Russia, including the 6-meter optical telescope and 600-meter radiotelescope of Academy of Sciences.

What are scientific interests of astrophysicists-observers? During many years they investigate the polarization of starlight, which was opened by professor V.A. Dombrovskii and simultaneously by two american scientists. Also the polarization of nebulas' radiation is studied. The polarization of radiation of eminent Crablike nebula was opened also on the our Department of astrophysics. It was a landmark in development of astrophysics, which has converted the hypothesis on important role in Universe of the so called synchrotron mechanism of radiation into a firmly stated fact. The very interesting and cryptic in many aspects celestial objects are quasars and active galaxies. University's astrophysicists-observers also study these amazing objects during many years.

**Celestial mechanics**, it sounds not commonly, does not it? But already Galilei and Newton have shown that there is no principal difference between heaven and earth. The one and the same law rules the motion of celestial and earth bodies. But spite of all this, the specificity of celestial bodies is such that the celestial mechanics is considered to be a separated branch of the astronomy. See yourself: on the heaven we may neglect any friction, take into account forces of gravitation only, predict the motion with precision up to eight place after point!

Classical theories on representation of a gravity field work good for ideally smooth bodies such as ellipsoids. For bodies of angular form this representation is much worse, and this effect is more stronger, more nearer a space craft is to the body's surface. Therefore for cosmic cobbles like Fobos it is necessary to develop some new remedies for description of their gravity fields. By the way, the same is true for Earth. Artificial satellites on low orbits are sensitive for mountains, ocean dimples and other abnormalities of gravity. The relativistic celestial mechanics was developing on the department in last years, when the preciseness of positions' location in astronomy has became such that the theoretic description of celestial bodies motion without taking into account relativistic effects would be differ distinctly from the observable one.

The Solar system is so old that all unstable orbits, excluding very seldom cases, have gone long ago, and we see now the stable motions only. But many stars are yet young, and in triple stars often unstable motions take place such that these systems crash. The stating of laws in triple systems is essential for understanding galaxy's evolution. The integrability problem of equations of celestial mechanics is rather mathematical than astronomical problem. Nevertheless the successes of last years in solving this problem of integrability have also pure astronomic consequence since they tell us on regular or chaotic character of celestial bodies motion.

There is an interesting historic fact: the Department of **astronomy** of our university is older than the university itself. It was opened in Main Pedagogical Institute, which was reorganized in St.Petersburg University in 1819. At the present time on the Department of astronomy and in the laboratory attached to the department the training of specialists on astrometry is performed.

Traditionally astrometry was considered as a "calm" science which solves only its "eternal" problems such as constructing inertial frames of reference and the refinement of astronomic constants. But during the last 10-15 years the astrometry was an arena of dramatic doing. The cause is that instead of classic instruments and methods of observation now the new nontraditional technique appears. First of all these are radiointerferometers that are systems of two or more radiotelescopes disposed at the distance of several thousands kilometers.

No less perspective is also the cosmic astrometry. Astronomers always dreamt of telescopes placed out of atmosphere limits, being free of gravity forces deforming their measuring systems. Now these dreams are realized in space telescopes. The modern astrometrist is an astronomer who knows in details methods of classic astrometry, is a good mathematician and programmer, knows the celestial mechanics and radar astronomy.

Information on all entrance questions is available in the receiving commission of the faculty, numbers of phones **428-69-44, 428-42-10**.

Postal address: **198504, St-Petersburg, Stary Peterhof, Universitetsky prospekt, 28, Mathematics and Mechanics Faculty, St-Petersburg University, Dean's office, room 3540. Admission Board. **

E-mail: **
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