Minimum curriculum requirements for Magister Programmes
in MATHEMATICS

1. GENERAL REQUIREMENTS

Magister programmes in Mathematics last five years (ten terms). It is assumed that the total load is approx. 3000 hours, including 1530 hours covered by the minimum curriculum requirements. This minimum curriculum covers only general, basic, and major subjects shared by all specializations. The recommended fundamental and major subjects may be offered in the form of several separate or combined classes.

2. PROFILE OF THE GRADUATE

Graduates from the magister programmes in Mathematics should be equipped with general knowledge of mathematics, comprehensive enough to allow them to independently broaden their education and practice the profession of mathematician at occupying various positions, including that of mathematics teacher (subject to his/her complying with additional requirements for teacher training studies).

3. COURSE GROUPS AND MINIMUM HOUR LOAD

GENERAL EDUCATION COURSES	270 hours
BASIC AND MAJOR COURSES	1260 hours
Total:	1530 hours

4. COURSES BY GROUPS AND MINIMUM HOUR LOAD

GENERAL EDUCATION COURSES	270 hours
Philosophy	60 hours
Courses in the Humanities (facultative)	30 hours
Foreign Language (English)	120 hours
Physical Education	60 hours

BASIC AND MAJOR COURSES	1260 hours
Introduction to Mathematics	60 hours
Mathematical Analysis	360 hours
Differential Equations	60 hours
Linear Algebra and Geometry	180 hours
Algebra	120 hours
Topology	60 hours
Probability Theory	120 hours
Complex Analysis	60 hours
Functional Analysis	60 hours
Computer Science	120 hours
Physics	60 hours

5. CURRICULUM CONTENTS

2. BASIC AND MAJOR COURSES

1. INTRODUCTION TO MATHEMATICS

Propositional calculus. Functional calculus. Set algebra. Natural numbers, Peano’s axioms, induction. Set product, relations, equivalence relation, principle of abstraction. Functions, images, and reverse images. Power, sets equipotentiality. Finite and infinite sets. Countably infinite sets and sets of power of the continuum. Selection axiom. Kuratowski-Zorn’s lemma. Cantor-Bernstein theorem. Power set. Cantor theorem. Partially and linearly ordered sets and order types. Well-ordered sets, ordinal numbers.

2. MATHEMATICAL ANALYSIS

Real numbers. Axioms of real numbers. Limit of a number sequence. The lower and upper limits of the number sequence and of the real function in point. Continuous mappings and their properties. Compactness, connectivity, and completeness of subsets of the Euclidean space.

Basic elementary functions in the real domain, their continuity and associated limits.

Differential calculus of a real variable function. Physical and geometric interpretation of the derivative. Operations on functions and the derivative. The mean value theorem. Taylor’ formula and its applications.

Tests for series convergence. Absolute and unconditional convergence. Multiplication of series. Point and uniform convergence of functional sequences and series. Weierstrass theorem for the interval. Uniform convergence tests for functional series. Continuity and differentiation of a functional sequence limit and the sum of a functional series. Tylor series and the concept of analytic function of a real variable. Taylor series expansion of basic elementary functions.

Integral calculus for real variable function. Indefinite integral. Elementary integration. Definite integral. Improper integrals. Integral test for series convergence. Geometric and physical application of the integral.

Elements of differential geometry. The tangent line and normal to the curve. Curvature. Natural equation of the curve.

Differential calculus (mappings from R^k to Rⁿ). The derivative and its geometric interpretation. Partial and directional derivatives and function differentiability. Jacobian matrix, jacobian and gradient. Operations on mappings and derivatives. Higher order derivatives. The mean value theorem. Tylor formula – application in studying local extremes. Implicit mapping theorems. Theorem of local reversibility of C class mappings. Local conditional extremes.

Lebesgue measure on R and R^k . Properties of the Lebesgue measure. Measurable functions. The integral and the measure. The necessary and sufficient conditions of integrality in Riemann’s sense. Fubini theorem and substitution theorem. Theorem of going over to a limit under the integral sign. Curvilinear and surface integrals. Differential forms and Stokes theorem.

3. DIFFERENTIAL EQUATIONS

Ordinary differential equations. Introductory concepts: equation, solution, their types, initial value problems, and geometric interpretation. Elementary integrable equations. Equations with separated variables, complete equations and equations reducible to them. Linear equations with fixed coefficients.

Basic theorems on existence and on uniqueness of solutions of initial value problems for the first - and higher -order differential equation systems. Theorem of permanent and smooth dependence of solutions on initial values and parameters.

Basic properties of solutions of linear first order differential equation systems: linear space of solutions of a homogenous system, its dimensions, the basis – fundamental system, fundamental matrix, and Liouville theorem. Form of general solution of a non-homogenous system. Properties of solutions of linear equations of the n^th – order.

Fixed coefficient linear equations systems and algebraic methods used in solving them. Determination of a fundamental system, fundamental matrix and general solution of a non-homogenous system.

Solution stability of a differential equation in Lapunov’s view, stability criteria.

Information on boundary value problems for second-order equations.

Partial differential equations. Introductory information, classification of partial differential equations. Basic limit, initial value, boundary value, and mixed problems; concept of a properly set problem.

The first order partial equations and their relation with ordinary equations, the first integrals.

Approximate solving of differential equations.

4. LINEAR ALGEBRA AND GEOMETRY

Linear spaces, the basis, dimension, coordinates. Linear transformations, matrices, determinants. Linear equation systems. Eigenvalues and eigenvectors. Bilinear symmetrical transformations, square forms and their matrices. Euclidean space and basic concept of Euclidean geometry. Orthogonal and conjugate transformations. Curves and conicoids. The plane isometry group, similarities group. Selected theorems of elementary geometry. Geometry of a triangle. Information about non-Euclidean geometries

5. ALGEBRA

Groups and their homomorphisms, Cayley theorem, normal divisors, quotient groups. Transformation and symmetric groups. Cyclic, abelian, solvable, and simple groups. Theorem of the structure of finitely generated abelian groups. Rings and their homomorphisms, ideals and quotient rings. Polynominal rings, rings of matrices, and power rings; rational number field, real number field, complex number field; finite fields; symmetrical polynominals. Algebraic number field. Fraction field. Algebraic expansion of fields. The field of constructible numbers and geometric structures. Elements of number theory ( Fermat small theorem, examples of applying the ring theory with a unique distribution to solving diophantine equations, whole number rings in square extensions of the rational number field). The fundamental algebraic theorem.

6. TOPOLOGY

Matric spaces: basic examples (Euclidean spaces, functional spaces), convergence of sequences, environments of points, open sets, closed sets, focal points; the interior, boundary and closure of sets, Borel sets. Continuous transformations, homomorphisms, uniform continuity. Complete spaces: Cauchy sequences, Cantor theorem, Baire theorem. Compact spaces; the absolute limit and completeness of compact spaces, Borel-Lebesgue axiom, Cantor set. Finite and countable Cartesian products of metric spaces. Connectivity and areas in Euclidean spaces. Separable metric space and Lindelöf axiom. The concept of general topological space and continuos transformation, Cartesian product of any topological space family. Hausdorff compact spaces and Tichonov theorem. Separation axioms.

7. PROBABILITY MATHEMATICS

Introduction, the intuitive ( frequency) concept of probability. The space of probability (from elementary examples to the general concept). Conditional probability, compete probability formula, Bayes formula, and ballot-box schemes.

Independence of events and sigma-fields, product spaces as probabilistic spaces in a series of independent trials, and Bernoulli schemes.

Random variables and their distributions. One- and multi-dimensional random variables, distribution functions, measure determination using distribution function, discrete distributions, continuous distributions, independence of random variables.

Expected value and other parameters and other parameters of the random variable distribution (variance, covariance matrix). Tschebyschev inequality.

Sequences of random variables and their distribution. Types of convergence in the theory of probability and the relations between them. The weak law of large numbers and the strong laws of large numbers (Borel and Kolmogorov). “Frequency” interpretation of the measure definition of probability.

Adding independent random variables. Transformation method. Generating functions and the branching process. General concept of measure convolution and Fourier transformation. Central limit theorems.

The concept of stochastic process. Kolmogorov theorem of compatible measure. Examples of stochastic processes. The concept of conditional expected value and that of martingale.

Information about elements of statistical inference: estimation problems and hypothesis testing.

8. COMPLEX ANALYSIS

Introductory concepts: complex numbers, closed plane, compact and connected sets, number sequences and series.

Complex functions: complex functions of a complex variable; continuity, derivative, Cauchy-Riemann conditions, elementary functions, logarithm and power; a branch of argument, logarithm and power ; homography, function sequences and series.

Curvilinear integral; complex functions of a real variable; curves, curvilinear integral and the anti-derivative of a function.

Holomorphic functions: holomorphic functions, Cauchy theorem and integral formula for a rectangle, differentiating an integral against a parameter, Morery theorem, Weierstrass theorem of sequences of holomorphic functions, power and Laurent series.

Isolated singular points: expansion in a Laurent series in the neighbourhood of a point; expansion in a power series in the vicinity of a point, isolated singular points, meromorphic functions, Casorati-Weierstrass theorem, identity theorem.

Integration in the complex domain: point index in relation to a curve, cycles, Cauchy theorem, Cauchy integral formula, the residue theorem for any open set, conclusions for sets which do not cut the plane.

9. FUNCTIONAL ANALYSIS

Normed spaces. Banach space. Banach classic sequence space and function space. Hölder and Minkowski inequalities. Series in normed spaces. Finite dimensional normed spaces. Supplementing normed spaces.

Unitary spaces. Hibert spaces. Supplementing unitary spaces. Schwarz inequality. Pythagorean theorem. Orthogonal projection theorem. Riesz theorem of the form of the linear functional. Orthogonalization and orthonormalization of a vector system. Orthogonal and orthonormal systems. Complete orthonormal systems. Fourier series. Bassel inequality. Parseval identity. Riesz- Fischer theorem. The trigonometric system (real and complex form). Fourier system with respect to the trigonometric system. Rademacher system.

Linear operators and functionals in normed spaces. Limits and continuity. The norm for a linear operator and functional. The space of linear limited operators. The conjugate space. Banach-Steinhaus theorem. Banach theorem of open mapping and reverse operator. The closed graph theorem. Hahn-Banach theorem.

10. COMPUTER SCIENCE

Introductory concepts. Review of the use of computer science in a mathematician’s work. Computer assisted mathematical problems solving.

Computer tools: hardware and software. The impact of computer configuration on its capabilities. Peripherals. An operating system as the part of software which coordinates the operation and division of computer resources. Multi task and parallel processing. Computer networks. Network services. E-mail, file transfer, searching and use of remote resources. Group work organization.

Programming. Selection of the algorithm as the result of task analysis. Presentation of the algorithm in the computer readable form. Comments on problem specification. Concepts of the program, module, and procedure. Instructions and declarations. Data and their structures. The global and local nature of data. Programming languages. Methods of describing the content: BNF notation. Elements of a selected high-level programming language (Java, C, C++). The programming environment. Translating programs.

Calculations. Elements of numerical analysis and examples of methods used for solving basic programs.

Application software. Word processors and database systems. Mathematical applications (Mathematica, Maple). TeX.

11. PHYSICS

The role of mathematics in the development and current state of physics. The basic laws of mechanics. Conservation laws. Elements of the electromagnetic field theory.

The principles of quantum physics. Relativistic physics issues.