## Mathematics

Master Degree in Mathematics
Сipher: 6B060100
Name: Mathematics

Purpose To provide students the fundamental knowledge of mathematics to the study of new quantitative relations and spatial forms of the real world in accordance with the requirements of technology and natural science. Specialist areas are mathematical analysis and function theory, mathematical logic and algebra, stochastic analysis and actuarial mathematics, differential equations, optimization and optimal control, equations of mathematical physics.
Disciplines
 State Compulsory Module History and Philosophy of ScienceForeign language (Professional)PedagogicsPsychology Basic Professional Modules: Required Core Modules : Methods of teaching mathematics Module 1:Methods of teaching in mathematics Organization and Planning of Scientific Research Methods of Teaching Higher Education Mathematics Mathematical and stochastic analysis Module 2: Mathematical and stochastic analysis Mathematical analysis on metric spaces and stochastic analysis Mathematical analysis on Manifolds Algebraic structures Module 3: Algebraic structures Algebraic structures Track 1: Differential equations and mathematical physics Module 1: Qualitative theory of differential equations Dynamical systems theory Qualitative theory of differential equations Module 2: Boundary-value problems for differential equations Boundary-value problems for ordinary differential equations Boundary-value problems for partial differential equations Module 3: Equations of mathematical physics Inverse problems of mathematical physics Additional chapters on differential equations and mathematical physics Module 4: Integro-differential and Navier–Stokes equations Singularly perturbed integro-differential equations  Navier–Stokes equations Track 2: Control theory Module 1: Control theory of deterministic and stochastic systems Constructive theory of boundary value optimal control problems Inverse problems forstochastic differential equations Module 2: Boundary-value problems and control systems Methods for solving boundary value problems Module 3: Theory of dynamical systems  Stability theory for dynamical systems  Theory of phase systems Module 4: Optimal control theory Boundary value optimal control problems Differential games Track 3: Mathematical analysis and function theory Module 1: Mathematical and Complex analysis Multidimensional Complex Analysis Number-theoretic methods in approximate analysis and their applications Module 2: Elements of approximation theory Approximations for functions of several variables Application of approximate calculations to the problems about eigenvalues Module 3: Analytic functions theory Nikolsky-Besov spaces and their applications to boundary value problems for generalized analytic functions Boundary value problems and their spectral properties for equations of mixed parabolic-hyperbolic type Module 4: Some actual problems of the theory of differential operators Theory of identification of the boundary conditions and its applications A nonlinear boundary value problem for an ordinary differential equation Track 4: Actuarial mathematics Module 1: Analysis of time series and regression Theory of time series and forecasting Nonparametric regression Module 2: Application of the theory of random processes Martingales and their applications in finance Distributions theory and their application in insurance Module 3: Actuarial mathematics-1 Pension plans Reliabilitytheoryand its applicationin insurance Module 4: Actuarial mathematics-2 Methods for calculating the allowance for losses Theory of   investment Track 5: Stochastic analysis Module 1: Foundations of Probability theory Mathematical Foundations of Probability theory Sums of independent random variables Module 2: Statistical inferences      Theory of statistical inferences Applied Statistics and Econometrics Module 3: Random equations and diffusion processes Stochastic equations Diffusion Processes and Their Applications Module 4: Random processes Introduction to the theory of martingales Statistics of Random Processes Track 6: Mathematical logic and algebra Module 1: Algebraic structures Theory of finite fields The Galois theory Module 2: Computability theory Computable functions Automata and formal languages theory Module 3: Model theory Elimination of quantifiers Structures and methods of model theory Module 4: Actual problems of cryptography Algebraic foundations of cryptography Modern cryptographic systems Internship Professional Teaching Thesis
Employability Graduates of the master can work in research institutes, in government or private educational institution as a lecturer, in area of communication and banks, actuarial activities.
Further studies Further studies. Master graduates can continue their education at the PhD level in the profession or professions related to this one. They can also continue to research activities in that economy grown back, which will directly apply their knowledge. Access to teacher training (educa­tion track); to Research and Pedagogic training
Programme learning outcomes

A graduate of the Mathematics (MSc) programme must possess the following knowledge, skills, and competences:

• have general knowledge about historic stages of the development of Mathematics as a science as well as current world trends, main directions and scientific schools of Mathematics; topical problems of Mathematics and modern methods of their solution; current state of mathematical education and science;

• know current problems of mathematical analysis, algebra and geometry, computational mathematics, differential equations and mathematical physics, probability theory and mathematical statistics and other fields of Mathematics;

• know analytical, qualitative and numerical research methods necessary for independent research work; methods of creation and use of mathematical models to describe and forecast various physical-chemical and natural processes and phenomena;

• be able to apply the theoretical knowledge received in the fundamental fields of Mathematics to solve theoretical, scientific practical and information search-related mathematical tasks; solve complex interdisciplinary problems; carry out qualitative and quantitative analysis of obtained results; construct standard mathematical and computer models of physical, technical, economic and other processes; teaching general mathematical modules; computer-based search, collection, preparation and processing of information used in his/her professional activity;

• have skills of properly formulating goals and objectives of scientific research as well as the concept of scientific search; performing a patent search and applying for an invention; presenting results of studies (in the form of articles, reports, papers etc.); organising work on a scientific basis; acquiring new knowledge using modern information technologies;

• know history and philosophy of science; research methods used in modern science and links between theory and practice; basics of bibliographic research; foundations of higher-education psychology and pedagogy, and methods of teaching mathematical modules; the methodology of labour and personnel management;

• be competent in a foreign language to be able to conduct professional discussions, read and translate specialized literature.

Mathematical analysis and function theory:

• ability to develop theoretical problems of mathematical analysis and function theory;

• ability to assign new problems and solve using methods of mathematical analysis and function theory;

• ability to apply your knowledge of functional spaces to the solving practical problems;

Actuarial mathematics:

• ability to develop the skills needed to do empirical research in fields operating with random processes data sets;

• have a good understanding of the basic concept of integration with respect to a probability measure and the basic properties of fair games;

• ability to answer simple questions on martingales; experience applications of stochastic processes in discrete financial models.

Stochastic analysis:

• ability to illustrate the basic concepts and important fundamental results of general theory stochastic processes;

• ability to distinguish important classes of the theory of stochastic processes;

• ability to prove the main theorems of the theory of stochastic processes;

Mathematical logic and algebra:

• ability to solve standard problems of mathematical logic;

• ability to explain and manipulate the different concepts in automata theory and formal languages such as formal proofs, non-deterministic automata, regular expressions, regular languages, context-free grammars, context-free languages, Turing machines;

• ability to explain the power and the limitations of regular languages and context-free languages;

• ability to solve problems in elementary number theory, apply elementary number theory to cryptography, develop a deeper conceptual understanding of the theoretical basis of algebra and cryptography.