Department of Mathematics

Graduates are expected to acquire advanced mathematical, statistical, and computational skills as relevant to their subjects of their Master projects. Furthermore, they will develop skills in interdisciplinary numerical simulations, statistical computing, and computer aided design environment and will acquire a command of scientific communication skills. The program is designed to take advantage of the faculty expertise and the computational resources of the School of Science and Technology, Nazarbayev University Research and Innovation System. Graduates of the program will have advanced research skills that will enable them to make contributions in technologically diverse and dynamic economic environments, both in Kazakhstan and abroad.

For inquiries, please contact the Graduate Program Director.

By the successful completion of the M.Sc. in Applied Mathematics program, students should be able to:

1. demonstrate advanced knowledge of the fundamentals of scientific computing, statistical analysis, and data mining, and software in a synergistic framework;

2. demonstrate an in-depth understanding of the contemporary research literature in their field of study;

3. apply mathematics in real-life applications;

4. take part and initiative in the design of a research project;

5. use simulation and computing requirements for scientific, engineering, government, and commercial applications;

6. select and provide a rationale for the selection of particular paradigms and specialist analytical techniques;

7. demonstrate the ability to explain scientific concepts and research findings, using various modalities of communication, with particular emphasis on tertiary education instruction.

**Framework of the Applied Mathematics Master's Program**

MATH 510 Measure Theory with Applications

This course is an introduction to the subject of measure theory and Lebesgue integration, and its applications in probability. The course begins with a discussion of the extended real number system, measurable sets and measurable functions. Measures are then introduced and used to construct the theory of Lebesgue Integration. After introducing the major theorems of Lebesgue integration, the course will conclude with applications of the theory to probability.

MATH 519 Scientific Computing

Algorithm design motivated by problems in physics, engineering, epidemiology, chemistry, and biology; implementation techniques in MATLAB; elements of shell programming under Unix/Linux and use of modern software packages for numerical computations; preparation of reports in the scientific style and maintain scientific software.

MATH 540 Statistical Learning

The course covers theoretical foundations and applications of machine learning models. Topics include supervised methods for regression and classification (linear models, trees, neural networks, ensemble methods, instance-based methods), Bayesian parametric learning, density estimation and clustering, sequential models. Programming projects covering a variety of real-world applications are assigned.

WCS 501 Science Communications

This course provides the graduate students with skills to write their research reports and scientific articles in English with proper use of style and grammar, as required in international conferences and journals. This course also trains the graduate students to deliver effective and appealing professional and scientific presentations, with attention to best practices in the use of technical English and vocal-gestural communication.

MATH 592 Research Seminar

This course provides graduate students with a broader view of the latest research in Mathematics. Seminar speakers are faculty members, PIs of research institutes and visiting academics. Students are required to attend all seminars, to participate in the weekly discussion sessions and to submit via TurnitIN reports on three of the seminars.

MATH 691 Thesis Proposal

This course is designed to monitor progress and develop understandings and skills to plan and conduct independent research at the MSc level. The student will develop under the guidance of the supervisor a research proposal that will state a research problem/question/hypothesis, its background, outline a research strategy and experimental approach, method of data collection, interpretation and validation. The thesis research proposal constitutes a partial fulfillment of the requirements Master's degree, and will be used in the course Thesis Research in part as a basis to monitor the research progress.

MATH 599 Thesis Research Preparation

This course is designed to facilitate during the Summer Term of the first Academic Year the preparation for the Thesis Proposal Research and Thesis Research in the second Academic Year. The course aims are: to provide students the opportunity to conduct feasibility studies and simulations; to practice methodologies required for their research; to collect data required for their research.

MATH 692 Thesis

This course is the last in the series of research courses for master students that leads to successful writing, completion and defense of the MSc thesis. The course main aim is to support the students in writing their thesis and regular monitoring of its progress. Secondly, the facilitation of the communication between course directors, supervisors and students and providing a framework for presenting and disseminate data and knowledge connected to the experimental and/or theoretical part done during the thesis. This is mostly done in form of two short presentations with discussion in the presence of supervisors, program directors, thesis defense committee and department staff. The basis of these presentations represents the basis for the presentation used during the thesis defense.

MATH 512 Optimization Methods and Techniques

The course covers an introduction to optimization methods and techniques. Lectures will discuss the main theoretical results, the most relevant algorithms and show how to use them for solving real case studies. Relevant optimization software will be introduced as well.

MATH 514 Operations Research

Operations research helps in solving problems in different environments that need decisions. This course covers topics that include: mathematical programming, stochastic optimization, fuzzy optimization, dynamic programming, deterministic and stochastic optimal control, multi-criteria decision aid, and application to engineering, finance, management.

MATH 515 Theory of Probability

This is a graduate-level course on measure-theoretic probability theory. The course starts with the probabilistic measure and integration theory and elaborates on conditional expectations from a measure-theoretic perspective. Random variables, distributions, moments, and characteristic functions in an abstract setting are thoroughly covered. Convergence modes of random variables along with Gaussian processes are also studied.

MATH 517 Mathematical Modeling and Simulation

This course will use scientific principles in physics, biology and engineering in combination with mathematical methods to formulate and design mathematical models to study real life problems.

MATH 518 Applied Finite Element Analysis

Initial/boundary value problems (IVP/BVP) in science and engineering with industrial applications, weak formulations for IVP/BVP, Sobolev spaces used in FEM, piecewise polynomial interpolations, basis shape functions in natural coordinates, local and global shape functions in spatial one, two and three dimensions, Galerkin method, Rayleigh-Ritz method, local and global finite element matrices, connectivity and nodal degrees of freedom, numerical integration, non-conforming FE, mixed FEM for Stokes problem, stabilized FEM for convection-dominated flow problems, FE error analysis, Superconvergence, quadrature error, discontinuous Galerkin methods, variational time discretizations and FE error analysis for parabolic problems, industrial applications of finite elements for heat transfer, structural, and fluid flow models, FE software package Comsol and Matlab PDE Toolbox, validation and presentation of simulation results.

MATH 541 Data Analysis and Statistical Learning

The course begins with the general formulation of the supervised learning problem and proceeds with the model assessment/selection issues, the comparison of models. We proceed to covering regularization, kernels, design of neural networks and, finally, based on Vapnik-Chervonenkis theory, we discuss how sample size affects the generalization power.

MATH 542 Statistical Programming

The course deals with data analysis using advanced statistical programming such as R or SAS. Students will learn syntax, how to read fixed and free format data, use built-in and user-written functions, optimization and graphical capabilities. The applied statistical problems considered include linear and nonlinear regression, experimental design, and others.

MATH 551 Advanced Numerical Methods

This course covers advanced numerical methods that are used in large scale scientific and engineering computations and simulations. The topics include Krylov subspace methods and preconditioning, conjugate gradient method and BiCGstab, iterative methods for eigenvalue computation, and advanced numerical methods for initial value problems.

MATH 555 Stochastic Analysis

This course covers various topics in stochastic calculus, providing students with a comprehensive understanding of the subject.

Students are expected to have a solid background in measure theory and elementary probability theory. Topics include conditional expectation, martingales, stochastic integration, stochastic differential equations, and some applications in finance. This course emphasizes rigorous theoretical foundations and explores the practical application of these theories in real-life problems, with a specific focus on their relevance in the field of finance. Any background in finance is not needed but a strong background in analysis and probability is required.

MATH 571 Nonlinear Differential Equations

Advanced topics are presented concerning nonlinear ordinary and partial differential equations and their solutions. After a quick review of first-order (non) autonomous and (non) linear ordinary differential equations, higher order systems of nonlinear ordinary differential equations are examined from the viewpoint of their critical points, phase space trajectories, Poincare maps, periodic and quasiperiodic solutions. The Poincare-Bendixson theorem, Lyapunov stability, stability analysis by linearization, perturbation methods (Poincare-Lindstedt, multiple time scales, averaging method) and bifurcation theory are described. Singularity analysis of integrable and nonintegrable nonlinear differential equations in the complex domain, and applications to classical mechanics, solid state physics, nonlinear optics, water wave propagation, population dynamics and time evolution of epidemiological models are also introduced.

MATH 576 Numerical Methods for Partial Differential Equations

Finite difference, volume, and element methods; explicit/implicit methods for time integration; Convergence, consistency, stability; the Poisson equation, heat equation, wave equation, advection-diffusion equation, and Stokes equation.

MATH 577 Modelling and Numerical Analysis for Incompressible Fluids

This course will make students familiar with mathematical modeling and numerical schemes for problems related to incompressible fluids. The main focus in the numerical analysis will be on stable finite element discretizations for Stokes equations and stabilization methods for convection-dominated problems. The course is designed to teach students numerical analysis for scalar transport equations and Navier-Stokes equations. Students will also learn implementation aspects for presented schemes and modern multiphysics software package Comsol to carry out simulations for incompressible fluid flows.

MATH 580 Advanced Complex Analysis

This course intends to be a rigorous introduction to complex analysis at the graduate level. Complex analysis is essential in many areas of pure and applied mathematics. Students are expected to have passed the undergraduate complex analysis. The topics that will be covered in this course includes Cauchy’s theorem, Runge’s approximation theorem, meromorphic functions, Paley-Wiener theorem, infinite products, Hadamard’s factorization theorem, analytic continuation, etc.

MATH 582 Mathematical Biology

This course provides an introduction to the use of discrete and differential equations as well as PDEs in the biological sciences. Biological models will include single species and interacting population dynamics, modeling infectious and dynamic diseases, regulation of cell function, molecular interactions, neural and biological oscillators. Mathematical tools such as phase portraits, bifurcation diagrams, and parameter estimation techniques that are introduced to analyze and interpret biological models will also be covered.

MATH 601 Analytic Number Theory

This course delves more deeply into the theory of numbers and is a natural follow-up of Math 301 Introduction to Number Theory. Number theory in Z is explored here with the help of functions of a real and complex variable. Both multiplicative and additive theories are introduced, in the form of the Prime Number Theorem, Dirichlet's Theorem of primes in arithmetic progressions, examples of applications of the circle method and of methods of Diophantine approximation.

MATH 620 Asymptotic Analysis

This course aims to introduce a diverse range of topics in asymptotic analysis, a field whose tools find extensive applications across various disciplines, including applied mathematics, physics, and engineering. This course delves into methods for approximating functions defined by integrals or solutions of differential equations. Students are expected to have a background in complex analysis at the undergraduate level.

MATH 676 Advanced Partial Differential Equations with Applications

This course covers: Laplace and heat equations, wave equations, fundamental solutions, maximum principles, Sobolev spaces, embedding theorems, weak solutions, energy methods, regularity of solutions, existence and uniqueness of solutions, general linear elliptic, parabolic and hyperbolic equations, and applications.

MATH 680 Potential Theory and Polynomial Approximation

Numerical approximation with polynomials, both theory and practice. The emphasis is on Chebsyev polynomials on an interval, but some other topics will be covered. Expertise in numerical and theoretical sides of the topic will be expected (at the appropriate level).

MATH 682 Applied Functional Analysis

This course will make students familiar with mathematical tools: complete metric spaces, contraction mapping theorem, Banach spaces, completion, function spaces, linear operators, inverses, the Uniform Boundedness Principle and the Closed Graph Theorem, approximate solution to operator equations, Frechet derivatives, Newton-Kantorovich method, Schauder fixed point theorem, Hilbert spaces, projections, generalized Fourier series, Riesz representation theorem, Ritz method, generalized solutions, Sobolev spaces, Lax-Milgram theorem, Compact operators, Spectral theorem, Self-Adjoint Compact Operators. The course is designed to teach students how to apply functional analysis to solve real-world problems in mathematical physics and engineering.