Магистр наук в области прикладной математики
Overview
The Master of Science in Applied Mathematics is a two-year, 120 ECTS program designed to prepare students to become professional applied mathematicians with advanced theoretical, computational, and research skills.
The program combines advanced mathematics with scientific computing, numerical simulation, statistical learning, data analysis, and mathematical modelling, enabling students to apply mathematical methods to complex problems in science, engineering, technology, finance, and other quantitative fields.
Students build a strong foundation through core courses in Measure Theory and Applications, Scientific Computing, Statistical Learning, and Research Seminar, while a broad selection of electives allows them to develop expertise in areas such as optimization, operations research, probability, numerical methods, mathematical biology, stochastic analysis, partial differential equations, and advanced analysis.
The program places a strong emphasis on research. Students develop a thesis proposal, undertake independent research, and complete a 30 ECTS Master's thesis, preparing them for research-intensive careers and further doctoral study.
The program combines advanced mathematics with scientific computing, numerical simulation, statistical learning, data analysis, and mathematical modelling, enabling students to apply mathematical methods to complex problems in science, engineering, technology, finance, and other quantitative fields.
Students build a strong foundation through core courses in Measure Theory and Applications, Scientific Computing, Statistical Learning, and Research Seminar, while a broad selection of electives allows them to develop expertise in areas such as optimization, operations research, probability, numerical methods, mathematical biology, stochastic analysis, partial differential equations, and advanced analysis.
The program places a strong emphasis on research. Students develop a thesis proposal, undertake independent research, and complete a 30 ECTS Master's thesis, preparing them for research-intensive careers and further doctoral study.
General information
- Campus: Astana, Kazakhstan
- Language: English
- Delivery mode: Full-time, on-campus
- Duration: 2 years
- Total ECTS credits: 120
Program Aims
- To empower academics, researchers and tertiary sector educators in applied mathematics to be the drivers of industrial and educational reform in Kazakhstan.
- To facilitate research activities, increase research productivity and impact, and, hence, promote scholarship of mathematics faculty, with the support from graduate students.
- To foster interdisciplinary research within the Mathematics Department, School of Science and Technology, Nazarbayev University, and beyond.
- To bridge the gaps between mathematical sciences, engineering, and other areas.
- To provide instructions on more focused areas required by advanced mathematicians and by non-mathematical graduate programs.
- To prepare future academics and researchers for careers in research medical, and governmental research and industrial organizations and agencies in Kazakhstan and other countries.
- To prepare researchers capable of conducting high quality research using a wide variety of data and numerical simulation program.
- To prepare competent graduate capable of progressing towards PhD in applied mathematics.
Key Advantages
- Advanced Mathematical FoundationBuild advanced theoretical knowledge that supports modern applied mathematics, including measure theory, probability, differential equations, functional analysis, complex analysis, and stochastic methods.1
- Scientific Computing & Computational SkillsDevelop practical computational expertise through scientific computing, numerical algorithms, simulation, and programming.
Students learn to implement numerical methods and use computational tools to solve real-world mathematical problems.
Scientific Computing includes MATLAB/Octave, preparation for Python-based work, numerical solutions of equations and differential equations, optimization, Monte Carlo simulation, and data analysis.2 - Data, Statistics & Machine LearningGain advanced skills in statistical analysis, statistical learning, data mining, and computational statistics.
Students explore regression, classification, clustering, neural networks, principal component analysis, optimization, and other methods used to extract insight from complex data.3 - Flexible & Specialized CurriculumShape your studies around your academic interests and career goals through a broad portfolio of advanced electives.
Students can explore areas including:
Optimization & Operations Research
Probability & Stochastic Analysis
Data Analysis & Statistical Programming
Scientific Modelling & Simulation
Numerical Methods & Computational Mechanics
Partial Differential Equations
Mathematical Biology
Complex & Functional Analysis
Analytic Number Theory
The curriculum includes 54 ECTS of elective coursework, giving students considerable flexibility to develop an individual academic direction.4 - Mathematics for Real-World ProblemsLearn to translate scientific and engineering challenges into mathematical models and computational solutions.
Students develop the ability to use mathematical modelling, numerical simulation, statistical techniques, and computing in scientific, engineering, governmental, and commercial applications.5 - Research-Intensive EducationDevelop advanced research skills through the Research Seminar, Thesis Proposal, Thesis Research Preparation, and Master's Thesis.
Students learn to analyze contemporary research, formulate research questions, select appropriate analytical methods, conduct independent research, and present their findings.
The MSc research project is expected to demonstrate sufficient creativity or novelty to merit research dissemination at conference-paper level, although publication itself is not a graduation requirement.6
Learning Outcomes
Upon successful completion of this program, students will be able to:
- Demonstrate advanced knowledge of the fundamentals of scientific computing, statistical analysis, and data mining, and software in a synergistic framework.
- Demonstrate an in-depth understanding of the contemporary research literature in their field of study.
- Apply mathematics in real-life applications.
- Take part and initiative in the design of a research project.
- Use simulation and computing requirements for scientific, engineering, government, and commercial applications.
- Select and provide a rationale for the selection of particular paradigms and specialist analytical techniques.
- Demonstrate the ability to explain scientific concepts and research findings, using various modalities of communication, with particular emphasis on tertiary education instruction.
What Will You Learn?
- Advanced Mathematical MethodsDevelop an advanced understanding of mathematical concepts used in applied research, including measure theory, probability, differential equations, complex analysis, functional analysis, and stochastic methods.
Learn to select appropriate mathematical and analytical techniques for challenging quantitative problems. - Scientific Computing & Numerical MethodsLearn how to design and implement computational approaches for mathematical problems using numerical algorithms, scientific computing, numerical linear algebra, differential equation solvers, and simulation techniques.
Students work with computational tools and programming environments such as MATLAB/Octave, with preparation for Python-based scientific computing. - Statistical Learning & Data AnalysisLearn how mathematical and statistical methods can be used to understand complex data.
Topics include regression, classification, clustering, statistical modelling, neural networks, principal component analysis, model selection, and machine learning algorithms. - Mathematical Modelling & SimulationLearn to formulate mathematical models of real-world systems in science, engineering, biology, and other fields.
Use differential equations, numerical analysis, scientific computing, finite element and finite difference methods to analyse models and compute solutions. - Optimization & Decision-MakingExplore mathematical approaches for finding optimal solutions and supporting complex decision-making.
Depending on elective choices, students can study convex optimization, dynamic programming, stochastic optimization, optimal control, operations research, and applications in engineering, finance, and management. - Research & Scientific CommunicationLearn to formulate research questions, review scientific literature, design research projects, analyse results, write scientific papers and reports, and communicate research findings.
Through research-oriented coursework and the Master's thesis, students develop the skills required for independent research, doctoral study, and research-intensive professional roles.
Shape Your Mathematical Focus
Scientific Computing & Numerical Mathematics
Explore scientific computing, advanced numerical methods, numerical PDEs, finite element analysis, computational mechanics, and simulation.
Data, Statistics & Machine Learning
Develop expertise in statistical learning, data analysis, statistical programming, probability, machine learning, and computational statistics.
Optimization & Operations Research
Study optimization methods, operations research, dynamic programming, stochastic optimization, optimal control, and quantitative decision-making.
Mathematical Modelling & Applied Analysis
Explore mathematical modelling, differential equations, stochastic analysis, mathematical biology, fluid modelling, functional analysis, and advanced applied mathematics.
Build your own academic focus through advanced electives and research in scientific computing, data and statistics, optimization, mathematical modelling, and applied analysis.
Curriculum
Year 1: Fall Semester and Spring Semester (32 ECTS)
Year 1: Summer (0 ECTS)
Year 2: Fall Semester (32 ECTS) and Spring (30 ECTS) Semester
Electives
-
- MATH 512 - Optimization Methods and Techniques
- MATH 514 - Operations Research
- MATH 515 - Theory of Probability
- MATH 517 - Scientific Modeling and Simulation with Mathematics
- MATH 518 - Applied Finite Element Analysis
- MATH 541 - Data Analysis and Statistical Learning
- MATH 542 - Statistical Programming
- MATH 551 - Advanced Numerical Methods
- MATH 555 - Stochastic Analysis
- MATH 571 - Advanced Nonlinear Differential Equations
-
- MATH 576 - Numerical Methods for Partial Differential Equations
- MATH 577 - Modeling and Numerical Analysis for Incompressible Fluids
- MATH 580 - Advanced Complex Analysis
- MATH 582 - Mathematical Biology
- MATH 601 - Analytic Number Theory
- MATH 620 - Asymptotic Analysis
- MATH 676 - Advanced Partial Differential Equations with Applications
- MATH 677 - Introduction to Nonlinear Dispersive and Wave Equations
- MATH 680 - Potential Theory and Polynomial Approximation
- MATH 682 - Applied Functional Analysis
Core modules
The list of core modules
This course is an introduction to the subject of measure theory and Lebesgue integration, and its applications in probability and differential equations. The course begins with a discussion of the extended real number system, measurable sets and measureable 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, in which some of the major theorems of probability theorem will be proved.
Course learning outcomes
1. Prove basic propositions about measures and measure spaces.
2. Prove basic propositions about integrals with respect to a measure.
3. Prove basic propositions involving probabilities.
Course learning outcomes
1. Prove basic propositions about measures and measure spaces.
2. Prove basic propositions about integrals with respect to a measure.
3. Prove basic propositions involving probabilities.
Topics: This course will make students familiar with algorithm design motivated by problems in physics, engineering, epidemiology, chemistry, and biology. It will use scientific principles in combination with mathematical methods to implement MATLAB algorithms that are grounded in sound principles of scientific computing and in the understanding of machine arithmetic and memory management. The course is designed to teach students programming skills for carrying out computational tasks. Numerical computing will be used for numerical solution of linear and nonlinear equations, differential equations, optimization problems and Monte Carlo simulations as well as for analyzing data. Students will also learn elements of shell programming under Unix/Linux, as well as use of modern software packages AUTO/BVPSuit and preparing scientific reports in Latex.
Course learning outcomes
1. Implement in MATLAB/Octave algorithms and use modern software packages to solve numerically real-life problems.
2. Write high-quality codes in MATLAB/Octave and be prepared for writing codes in Python.
3. Use modern software development tools and systems.
4. Write reports in scientific style.
Course learning outcomes
1. Implement in MATLAB/Octave algorithms and use modern software packages to solve numerically real-life problems.
2. Write high-quality codes in MATLAB/Octave and be prepared for writing codes in Python.
3. Use modern software development tools and systems.
4. Write reports in scientific style.
This graduate level course combines the application of rhetorical analysis to stylistic conventions of writing in the sciences and math with a focus on clarity, conciseness, and coherence. Students will write in their area of expertise in accordance with ethical and professional standards, prepare coherent reports of research findings, develop skills to author research publications, and present posters and/or oral presentations in conferences. They will learn to skillfully communicate scientific concepts and research findings using various modalities of communication with expert and lay audiences. This course trains students to deliver effective and appealing professional and scientific presentations with attention to best practices in the use of technical English and oral communication.
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.
Course learning outcomes
1. Derive mathematical expressions for a few simple algorithms like ordinary least squares regression, ridge regression, k-means and PCA.
2. Implement clustering, regression and classification algorithms using modern programming languages.
3. Apply gradient descent to solve simple convex optimization problems, and stochastic gradient descent to solve nonconvex optimization problems.
4. Assess whether the solution to a real-world machine learning problem might involve one or more of clustering, regression or classification.
5. Prepare reports and presentations with reproducible code.
Course learning outcomes
1. Derive mathematical expressions for a few simple algorithms like ordinary least squares regression, ridge regression, k-means and PCA.
2. Implement clustering, regression and classification algorithms using modern programming languages.
3. Apply gradient descent to solve simple convex optimization problems, and stochastic gradient descent to solve nonconvex optimization problems.
4. Assess whether the solution to a real-world machine learning problem might involve one or more of clustering, regression or classification.
5. Prepare reports and presentations with reproducible code.
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.
Course learning outcomes
1. Analyze, understand, discuss, and summarize research results.
2. Build arguments and points of view, affirm their thoughts, and establish their own hypotheses on a research question.
Course learning outcomes
1. Analyze, understand, discuss, and summarize research results.
2. Build arguments and points of view, affirm their thoughts, and establish their own hypotheses on a research question.
Research and Thesis Component
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.
Course learning outcomes
1. Design and conduct an original thesis research project.
2. Develop a discipline specific research proposal.
3. Develop a working knowledge of relevant literature and apply it to write background, material and methods, expected results/ preliminary results of the thesis proposal.
4. Defend the research proposal.
Course learning outcomes
1. Design and conduct an original thesis research project.
2. Develop a discipline specific research proposal.
3. Develop a working knowledge of relevant literature and apply it to write background, material and methods, expected results/ preliminary results of the thesis proposal.
4. Defend the research proposal.
This course is designed to facilitate during the Summer Term of first Academic Year the preparation for the Thesis Proposal Research and Thesis Research in the second Academic Year.
The course aims are:
1. to provide students the opportunity to conduct feasibility studies and simulations; 2. to practice methodologies required for their research; 3. to collect data required for their research.
Course learning outcomes1. Confidently apply methodologies required for their research.
2. To develop based on data accumulated a thesis research proposal in the following semester.
The course aims are:
1. to provide students the opportunity to conduct feasibility studies and simulations; 2. to practice methodologies required for their research; 3. to collect data required for their research.
Course learning outcomes1. Confidently apply methodologies required for their research.
2. To develop based on data accumulated a thesis research proposal in the following semester.
Student will conduct independent work under the direction of a supervisor on a research problem in the student's designated area of research. The student will prepare and defend the thesis.
Course learning outcomes
1. Conduct an original Thesis research project.
2. Develop a working knowledge of relevant literature and apply it to write results of the Thesis.
3. Defend the Thesis.
Course learning outcomes
1. Conduct an original Thesis research project.
2. Develop a working knowledge of relevant literature and apply it to write results of the Thesis.
3. Defend the Thesis.
Elective Courses
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.
Course learning outcomes
1. Theory of Convex Analysis.
2. Lagrangian Duality.
3. Fenchel-Rockafellar Theorem.
4. Karush-Kuhn-Tucker Theorem.
5. Deterministic Dynamic Programming.
6. Stochastic Dynamic Programming.
7. Pontryagin Maximum Principle.
Course learning outcomes
1. Theory of Convex Analysis.
2. Lagrangian Duality.
3. Fenchel-Rockafellar Theorem.
4. Karush-Kuhn-Tucker Theorem.
5. Deterministic Dynamic Programming.
6. Stochastic Dynamic Programming.
7. Pontryagin Maximum Principle.
Operations research helps in solving problems in different environments that needs 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.
Course learning outcomes
1. Learn dynamic programming paradigm.
2. Learn Pontryagin maximum principle.
3. Use optimisation theory in finance and engineering.
Course learning outcomes
1. Learn dynamic programming paradigm.
2. Learn Pontryagin maximum principle.
3. Use optimisation theory in finance and engineering.
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.
Course learning outcomes
1. Use sigma-algebras of events of probability.
2. Use the corresponding building objects of probability theory in an abstract setting.
3. Use Gaussian vectors.
4. Use various modes of convergence of random variables.
Course learning outcomes
1. Use sigma-algebras of events of probability.
2. Use the corresponding building objects of probability theory in an abstract setting.
3. Use Gaussian vectors.
4. Use various modes of convergence of random variables.
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. Numerical analysis and scientific computing methods will be used to develop algorithms to compute solutions to the problems.
Course learning outcomes
1. Understanding how to model engineering/physics, biological problems using differential with conservation laws.
2. Understanding mathematics useful for modeling conservation laws with applications.
3. Be able to understand the concept of well-posedness of the mathematical models.
4. Understanding basic analytic methods for solving solutions of the math models.
5. Understanding basic numerical methods including finite element/ finite difference approaches.
6. Understanding basic numerical methods including finite element/ finite difference approaches.
Course learning outcomes
1. Understanding how to model engineering/physics, biological problems using differential with conservation laws.
2. Understanding mathematics useful for modeling conservation laws with applications.
3. Be able to understand the concept of well-posedness of the mathematical models.
4. Understanding basic analytic methods for solving solutions of the math models.
5. Understanding basic numerical methods including finite element/ finite difference approaches.
6. Understanding basic numerical methods including finite element/ finite difference approaches.
Topics: 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.
Course learning outcomes
1. Understand the finite element technology for 1d and 2d boundary value problems.
2. Construct finite element spaces for solving standard boundary/initial value problems in 1d and 2d spatial dimensions.
3. Analyze finite element errors for simple linear boundary value problems.
4. Implement in MATLAB/Octave finite element solvers for standard 1d problems in engineering and science.
5. Use modern software packages to solve numerically boundary value problems in engineering.
Course learning outcomes
1. Understand the finite element technology for 1d and 2d boundary value problems.
2. Construct finite element spaces for solving standard boundary/initial value problems in 1d and 2d spatial dimensions.
3. Analyze finite element errors for simple linear boundary value problems.
4. Implement in MATLAB/Octave finite element solvers for standard 1d problems in engineering and science.
5. Use modern software packages to solve numerically boundary value problems in engineering.
Topics of the course:
Course learning outcomes
1. Select appropriate set of predictors.
2. Model numerical (in the case of regression) and logical (in the case classification) response using a single or multiple explanatory variables.
3. To measure a relative quality of models via information criteria and identify the case when the model is over-fit.
4. Prepare reports and presentations using SAS and SVM-light.
- The supervised learning problem. The statistical models and the maximum likelihood principle.
- Linear regression and logistic regression.
- The model assessment and the model selection.
- Tikhonov regularization.
- The kernel trick.
- Structure of neural networks.
- Vapnik – Chervonenkis theory.
- Principal Component Analysis.
Course learning outcomes
1. Select appropriate set of predictors.
2. Model numerical (in the case of regression) and logical (in the case classification) response using a single or multiple explanatory variables.
3. To measure a relative quality of models via information criteria and identify the case when the model is over-fit.
4. Prepare reports and presentations using SAS and SVM-light.
Introduction to the basics of programming and algorithms in statistics. Exploration data of storage, manipulation, plotting, and analysis. Modularization, conditional execution, looping, and function construction are covered along with program debugging techniques.
Course learning outcomes
1. Apply statistical principles to experiments in various scenarios and understand how to make sense out of the data that is generated.
2. Interpret and present data to a wide audience and turn insight into decisions.
3. Conduct analyses which incorporate data and machine learning techniques to generate more complex understandings of data.
Course learning outcomes
1. Apply statistical principles to experiments in various scenarios and understand how to make sense out of the data that is generated.
2. Interpret and present data to a wide audience and turn insight into decisions.
3. Conduct analyses which incorporate data and machine learning techniques to generate more complex understandings of data.
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.
Course learning outcomes
1. Experiments with Krylov subspaces to solve linear systems and eigenvalue problems, with a preconditioner, and to use them as a tool to analyze different methods.
2. Apply and analyze standard algorithm for small/dense, and large/structured eigenvalue problems.
3. Use and examine advanced methods for initial value problems.
4. Compose numerical algorithms using a programming language, for instance Matlab or Octave.
5. Compare different algorithms based on their convergence speed, accuracy, and stability.
Course learning outcomes
1. Experiments with Krylov subspaces to solve linear systems and eigenvalue problems, with a preconditioner, and to use them as a tool to analyze different methods.
2. Apply and analyze standard algorithm for small/dense, and large/structured eigenvalue problems.
3. Use and examine advanced methods for initial value problems.
4. Compose numerical algorithms using a programming language, for instance Matlab or Octave.
5. Compare different algorithms based on their convergence speed, accuracy, and stability.
Stochastic analysis covers stochastic integration, stochastic differential equations, and related topics. Students are required to have a background in measure theory. Topics include but are not limited to: Ito integrals, Martingales, stochastic differential equations, and some applications in finance.
Course learning outcomes
1. Demonstrate understanding of the basic concepts underlying stochastic analysis.
2. Demonstrate familiarity with a range of examples of these concepts.
3. Prove basic results in this course.
4. Apply the methods introduced in this course to solve various problems in other fields.
Course learning outcomes
1. Demonstrate understanding of the basic concepts underlying stochastic analysis.
2. Demonstrate familiarity with a range of examples of these concepts.
3. Prove basic results in this course.
4. Apply the methods introduced in this course to solve various problems in other fields.
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. Applications to classical mechanics, solid state physics, nonlinear optics, water wave propagation, population dynamics and time evolution of epidemiological models are also introduced.
Course learning outcomes
1. Draw and interpret phase diagrams.
2. Distinguish between stable and unstable equilibrium points.
3. Distinguish types of bifurcation.
4. Find limit cycles and describe their stability.
5. Apply Poincare-Bendixon theorem.
6. Understand and apply the concept of Lyapunov stability.
Course learning outcomes
1. Draw and interpret phase diagrams.
2. Distinguish between stable and unstable equilibrium points.
3. Distinguish types of bifurcation.
4. Find limit cycles and describe their stability.
5. Apply Poincare-Bendixon theorem.
6. Understand and apply the concept of Lyapunov stability.
The course is a blend of conceptual and practical aspects of numerical solutions of partial differential equations. Topics include finite difference, volume, and element methods, and explicit/implicit methods for time integration, convergence, consistency, stability, Courant-Friedrichs-Lewy criteria, Lax equivalence theorem, error analysis, Fourier-von Neumann stability analysis. Applications include the Poisson equation, heat equation, wave equation, advection-diffusion equation, and Stokes equation.
Course learning outcomes
1. Understand several major numerical methods for solving initial/boundary value problems of second and fourth order differential equations.
2. Be able to perform several major discretization procedures to reduce initial/boundary value problems into systems of algebraic equations and to be able to solve such systems numerically.
3. Understand various applications of the numerical methods for initial/ boundary value problems in some science and engineering fields.
4. Be able to discretize some nonlinear problems by the numerical methods and solve the corresponding nonlinear algebraic equations.
5. Be able to develop and implement major numerical procedures for solutions of some project problem approved by the instructor.
6. Be able to prove some theorems related to existence/unique and stability/convergence of the problems.
7. Be able to write a term paper in mathematical formats using Latex or Word-Equations and present the results to peers and the instructor in class.
Course learning outcomes
1. Understand several major numerical methods for solving initial/boundary value problems of second and fourth order differential equations.
2. Be able to perform several major discretization procedures to reduce initial/boundary value problems into systems of algebraic equations and to be able to solve such systems numerically.
3. Understand various applications of the numerical methods for initial/ boundary value problems in some science and engineering fields.
4. Be able to discretize some nonlinear problems by the numerical methods and solve the corresponding nonlinear algebraic equations.
5. Be able to develop and implement major numerical procedures for solutions of some project problem approved by the instructor.
6. Be able to prove some theorems related to existence/unique and stability/convergence of the problems.
7. Be able to write a term paper in mathematical formats using Latex or Word-Equations and present the results to peers and the instructor in class.
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.
Course learning outcomes
1. Learn how to derive the model equations for incompressible fluids.
2. Learn functional analytical tools for CFD.
3. Learn stable finite element discretizations for CFD.
4. Implement in MATLAB/Octave algorithms and use modern multiphysics software package Comsol to solve incompressible fluid flow problems.
Course learning outcomes
1. Learn how to derive the model equations for incompressible fluids.
2. Learn functional analytical tools for CFD.
3. Learn stable finite element discretizations for CFD.
4. Implement in MATLAB/Octave algorithms and use modern multiphysics software package Comsol to solve incompressible fluid flow problems.
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.
Course learning outcomes
1. Demonstrate understanding of the basic concepts underlying complex analysis.
2. Demonstrate familiarity with a range of examples of these concepts.
3. Prove basic results in this course.
4. Apply the methods introduced in this course to evaluate definite integrals and infinite series.
Course learning outcomes
1. Demonstrate understanding of the basic concepts underlying complex analysis.
2. Demonstrate familiarity with a range of examples of these concepts.
3. Prove basic results in this course.
4. Apply the methods introduced in this course to evaluate definite integrals and infinite series.
This course examines the application of ordinary and partial differential equations in the biological sciences. Biological models include single-species and interacting population dynamics, infectious and dynamic disease modelling, regulation of cellular function, molecular interactions, and neural and biological oscillators.
The course also covers mathematical tools used to analyse and interpret biological models, including phase portraits, bifurcation diagrams, and parameter estimation methods.
Course learning outcomes
1. Formulate and solve mathematical models of evolution.
2. Use techniques from difference, ordinary and partial differential equations to describe spread of disease and other biological material.
3. Explain how these techniques are applied in scientific studies and applied in ecology and epidemiology.
4. Find the equilibria of a single-population model and their stability.
5. Use techniques from nonlinear differential equations to explain the oscillation in population dynamics and chemical systems.
The course also covers mathematical tools used to analyse and interpret biological models, including phase portraits, bifurcation diagrams, and parameter estimation methods.
Course learning outcomes
1. Formulate and solve mathematical models of evolution.
2. Use techniques from difference, ordinary and partial differential equations to describe spread of disease and other biological material.
3. Explain how these techniques are applied in scientific studies and applied in ecology and epidemiology.
4. Find the equilibria of a single-population model and their stability.
5. Use techniques from nonlinear differential equations to explain the oscillation in population dynamics and chemical systems.
This course is a first foray into the analytic theory of numbers, or how real and complex function theory (in one variable), primarily developed 150 years ago, can help us understand the properties of numbers. Students will also learn standard techniques in the field, that can be used in a more advanced course: partial summation, Euler-Maclaurin summation formula, the general theory of Dirichlet series, entire functions of finite order. The most remarkable achievements of the 19th century will be presented: the Prime Number Theorem, Dirichlet’s theorem of primes in an arithmetic progression, the transcendence of e and pi. Emphasis will be given in understanding the techniques rather than getting the best possible results.
Course learning outcomes
1. Translate arithmetic into analysis: Name the main arithmetical functions, find their generating functions and calculate their average values. Learn to use simple techniques, like partial summation or the Euler-Maclaurin formula. Understand equivalent formulations of the prime number theorem (PNT). Generalization to primes in arithmetic progressions.
2. Manipulate important analytic functions relevant to number theory: Learn a complex analytic proof of the PNT and how a precise estimation of the prime number remainder is related to the location of the zeros of the Riemann zeta function. Study more properties of the Riemann zeta function. Extend some of these properties to Dirichlet L-functions.
3. View examples of the circle method: the Hardy-Ramanujan-Rademacher theorem, Waring’s or Goldbach’s problems. Analyze in detail one of these.
4. Diophantine approximations: how they are related to questions of irrationality and transcendence.
Course learning outcomes
1. Translate arithmetic into analysis: Name the main arithmetical functions, find their generating functions and calculate their average values. Learn to use simple techniques, like partial summation or the Euler-Maclaurin formula. Understand equivalent formulations of the prime number theorem (PNT). Generalization to primes in arithmetic progressions.
2. Manipulate important analytic functions relevant to number theory: Learn a complex analytic proof of the PNT and how a precise estimation of the prime number remainder is related to the location of the zeros of the Riemann zeta function. Study more properties of the Riemann zeta function. Extend some of these properties to Dirichlet L-functions.
3. View examples of the circle method: the Hardy-Ramanujan-Rademacher theorem, Waring’s or Goldbach’s problems. Analyze in detail one of these.
4. Diophantine approximations: how they are related to questions of irrationality and transcendence.
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.
Course learning outcomes
1. Demonstrate understanding of the basic concepts underlying asymptotic analysis.
2. Demonstrate familiarity with a range of examples of these concepts.
3. Prove basic results in this course.
4. Apply the methods introduced in this course to solve various problems in other fields.
Course learning outcomes
1. Demonstrate understanding of the basic concepts underlying asymptotic analysis.
2. Demonstrate familiarity with a range of examples of these concepts.
3. Prove basic results in this course.
4. Apply the methods introduced in this course to solve various problems in other fields.
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.
Course learning outcomes
1. Solve transport eq.
2. Compute solutions of Laplace and heat eq. using Green’s function.
3. Calculate solutions of the wave eq. by spherical means, Kirchhoff’s and Poisson’s formula.
4. Analyze uniqueness of solutions of main PDEs by energy methods.
5. Examine regularity and properties of solutions of main PDEs.
6. Model easy physical phenomena using PDEs.
7. Understand distribution theory and characterize Sobolev spaces.
8. Explain some open problems in dispersive PDEs.
Course learning outcomes
1. Solve transport eq.
2. Compute solutions of Laplace and heat eq. using Green’s function.
3. Calculate solutions of the wave eq. by spherical means, Kirchhoff’s and Poisson’s formula.
4. Analyze uniqueness of solutions of main PDEs by energy methods.
5. Examine regularity and properties of solutions of main PDEs.
6. Model easy physical phenomena using PDEs.
7. Understand distribution theory and characterize Sobolev spaces.
8. Explain some open problems in dispersive PDEs.
The aim of this course is to provide students an introduction to nonlinear dispersive and wave equations, and the methods used to study the qualitative properties of their solutions. These equations arise as fundamental models of a wide variety of physical systems, including the propagation of waves and the study of nonlinear optics, and are closely related to models of fluids, as well as a number of aspects of statistical and quantum mechanics.
Course learning outcomes
1. Solve linear disperisive equations & wave equations, and understand the properties of their solutions.
2. Prove well-posedness nonlinear dispersive equations in low regularity Sobolev spaces.
3. Prove well-posedness nonlinear wave equations in low regularity Sobolev spaces.
4. Identify functional and harmonic analytic tools that are important to study qualitative properties of solutions to dispersive and wave equations.
Course learning outcomes
1. Solve linear disperisive equations & wave equations, and understand the properties of their solutions.
2. Prove well-posedness nonlinear dispersive equations in low regularity Sobolev spaces.
3. Prove well-posedness nonlinear wave equations in low regularity Sobolev spaces.
4. Identify functional and harmonic analytic tools that are important to study qualitative properties of solutions to dispersive and wave equations.
Numerical approximation with polynomials, both theory and practice. The emphasis is on Chebyshev polynomials on an interval, but some other topics will be covered. Expertise in numerical and theoretical sides of the topic will be expected.
Course learning outcomes
1. Write explicit formulas for polynomial approximation of functions.
2. Implement numerical algorithms for polynomial approximation.
3. Compute Fekete points and logarithmic potentials in basic examples.
4. Analyze the regularity of approximation for smooth and analytic functions.
5. Quantify the Gibbs phenomenon, numerically and analytically.
6. Work with best approximations, and near best approximations.
7. Compute Bernstein-Walsh and Bernstein-Markov constants for polygons.
Course learning outcomes
1. Write explicit formulas for polynomial approximation of functions.
2. Implement numerical algorithms for polynomial approximation.
3. Compute Fekete points and logarithmic potentials in basic examples.
4. Analyze the regularity of approximation for smooth and analytic functions.
5. Quantify the Gibbs phenomenon, numerically and analytically.
6. Work with best approximations, and near best approximations.
7. Compute Bernstein-Walsh and Bernstein-Markov constants for polygons.
This course will make students familiar with mathematical tools :
Course learning outcomes
1. Learn the main concepts of metric, Banach, Hilbert, linear and compact operators.
2. Learn how to generalize the known results from Linear Algebra to infinite dimensional spaces.
3. Learn function spaces needed for solving boundary value problems in physics and Engineering.
4. Learn how to approximate solutions to operator equations.
- 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.
Course learning outcomes
1. Learn the main concepts of metric, Banach, Hilbert, linear and compact operators.
2. Learn how to generalize the known results from Linear Algebra to infinite dimensional spaces.
3. Learn function spaces needed for solving boundary value problems in physics and Engineering.
4. Learn how to approximate solutions to operator equations.
Where do our graduates work?
- Career opportunities
- Applied Mathematician
- Data Scientist
- Data / Statistical Analyst
- Quantitative Analyst
- Operations Research / Optimization Specialist
- Mathematical Modelling Specialist
- Scientific Computing Specialist
- Computational / Research Scientist
- Financial / Risk Analyst
- Quantitative Researcher
- Numerical Simulation Specialist
- Academic Researcher / University Lecturer
- Industries & Sectors
- Technology, Data & Analytics
- Research & Development
- Universities & Research Institutions
- Engineering & Scientific Computing
- Banking, Finance & Insurance
- Government & Public Sector
- Healthcare & Biomedical Research
- Industry & Manufacturing
- Education
- Quantitative & Analytical Consulting
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53 Kabanbay Batyr Ave
Astana city, Republic of Kazakhstan
Astana city, Republic of Kazakhstan