Department of Mathematics

The Bachelor of Science Program in Mathematics is a 240 ECTS program and is to be completed in 8 regular semesters. The program is designed to train students in the theory of mathematics and to develop mathematical skills and mastery, used in the modern applications of mathematics in science and engineering. Through specialized courses as part of mathematics elective and technical elective requirements, students will have the opportunity to deepen and broaden their mathematical foundations and skills, which will make them more competitive in the mathematics-related job market and/or will prepare them for graduate studies in the area of pure and applied mathematics or statistics. Flexibility in selecting non-mathematics electives will also allow students to complete a minor in a different discipline.

- Identify, formulate, and solve broadly defined technical or scientific problems by applying knowledge of mathematics and science to areas relevant to the discipline.
- Formulate or design a system, process, procedure, or program to meet desired needs.
- Develop and conduct experiments or test hypotheses, analyze and interpret data and use scientific judgment to draw conclusions.
- Communicate effectively with a range of audiences.
- Understand ethical and professional responsibilities and the impact of scientific solutions in global, economic, environmental, and societal contexts.
- Function effectively on teams that establish goals, plan tasks, meet deadlines, and analyze risk and uncertainty.
- Understand theorems and other mathematical statements; reproduce or elaborate their proofs.

The B.Sc. in Mathematics program provides education and training with depth in the core and breadth in electives aims to enable students to:

1. address and solve both research and real-world problems by applying their theoretical knowledge, critical thinking, and practical skills in mathematical and natural sciences;

2. become leaders through seeking collaborative opportunities and interacting professionally with a variety of stakeholders;

3. contribute independently or as a member of a multi-disciplinary organization in industry, academia, or government within Kazakhstan or abroad;

4. recognize both the potential for and the limitations of scientific processes in solving real-world problems, and respond appropriately to values and conflicts within the profession.

MATH 162 Calculus II

MATH 251 Discrete Mathematics

MATH 263 Calculus III

MATH 273 Linear Algebra with Applications

MATH 274 Introduction to Differential Equations

MATH 302 Abstract Algebra I

MATH 321 Probability

MATH 322 Mathematical Statistics

MATH 351 Numerical Methods

MATH 361 Introductory Real Analysis I

Any Math 300- or 400-level courses. At least 24 ECTS must be from 400-level courses. Furthermore, at least 12 ECTS must be research oriented courses, which could be selected from list of courses including MATH 407, MATH 411, MATH 440, MATH 441, MATH 446, MATH 449, MATH 477, and MATH 499.

Kazakh Language (KAZ)

PHIL 210 Ethics

HST 100 History of Kazakhstan

WCS 150, one 200-level writing course

Research Method (MATH 350)

Social Science (SOC, PLS, ANT, or ECON)

Natural Science (PHYS, CHEM, BIO, or GEOLOGY)

Math (MATH 161)

Computer Programming (Any CSCI course or Programming course)

Business (BUS)

A technical elective is a course taken in the following: BIOL, CHEM, ECON, PHYS, SEDS, SMG (One technical elective must be 300-level or above

In addition to the requirements above, students must take any courses offered at NU to reach the total 240 ECTS requirement.

**MATH 109: Mathematical Discovery (6)**. This course is to explore applications of mathematics in real life, including social choice and management science, showing that math is more than a set of formulas.

Pre-requisite: none

**MATH 161: Calculus I (8)**. This course covers limits and continuity as well as differentiation and integration of polynomial, rational, trigonometric, logarithmic, exponential and algebraic function. The application areas include slope, velocity, extrema, area, and volume.

**MATH 162: Calculus II (8)**. This course covers transcendental functions, advanced integration techniques, indeterminate forms, improper integrals, area and arc length in polar coordinates, infinite series, power series and Taylor’s theorem.

Pre-requisite: MATH 161 Calculus I

**MATH 251: Discrete Mathematics (6)**. This course focuses on logical reasoning and mathematical thinking, as well as to familiarize the students with the precision of mathematical language. You will learn how to write mathematical proofs and several common proof techniques. We will also cover some topics such as set theory, induction and recurrence relations.

Pre-requisite: MATH 162 Calculus II

**MATH 263: Calculus III (8)**. This course covers analytic geometry in 3-space, partial and directional derivatives, extrema, double and triple integrals, line integrals, surface integrals, gradient, divergence, curl, multi-integral applications, as well as cylindrical and spherical coordinates.

Pre-requisite: MATH 162 Calculus II

**MATH 273: Linear Algebra with Applications (8)**. This course covers systems of linear equations, matrices and inverses, determinants, eigenvalues and eigenvectors, inner product spaces and orthogonal matrices, and an introduction to vector spaces as the theoretical portion of the course. Basic Matlab commands are used in application problems. There are 3 hours of lecture and 1 hour of recitation per week.

Pre-requisite: MATH 161 Calculus I. Co-requisite: MATH 162 Calculus II

**MATH 274: Introduction to Differential Equations (6)**. This course covers first order differential equations; mathematical models and numerical methods; linear systems and matrices; higher-order linear differential equations; linear systems of differential equations; and Laplace transform methods.

Pre-requisite: MATH 162 Calculus II

**MATH 301: Introductory Number Theory (6)**. An introductory course in the theory of numbers covering such topics as: Euclidean algorithm, Fundamental Theorem of Arithmetic, congruences, diophantine equations, Fermat and Wilson Theorems, quadratic residues, continued fractions, Prime number theorem, and applications such as cryptography.

Pre-requisite: MATH 273 Linear Algebra with Applications and MATH 162 Calculus II

**MATH 310: Applied Statistical Methods (6)**. This is a course serving students from a variety of majors, including biology, mathematics, and computer science. Biology students make up the majority of the audience and they come from the only major for which this course is required. Mathematics students take the class as an elective in preparation to future courses in statistics. Computer science students take this class only if they were not able to satisfy the calculus prerequisite for Math 321.

Prerequisites: Math 161 Calculus I and sophomore standing or higher

**MATH 321: Probability (6)**. This course covers foundations of probability theory, combinatorial and counting methods, conditional probability, random variables, discrete and continuous distributions, expectation, moment generating functions, multivariate distributions, variable transformations, the Law of Large Numbers and the Central Limit Theorem.

Pre-requisite: MATH 162 Calculus II

**MATH 322: Mathematical Statistics (6)**. The course starts where Probability ends and considers point estimation, sampling distributions, confidence intervals, unbiased estimators, hypothesis testing, t-test, most powerful tests, categorical data inference, regression and analysis of variance.

Pre-requisite: MATH 321 Probability

**MATH 323: Actuarial Mathematics I (6)**.This course covers the topics for the Financial Mathematics (FM) exam of the Society of Actuaries. They include the fundamental concepts and methods for calculating present and accumulated values for reserving, valuation, pricing, asset/liability, investment income, capital budgeting, and contingent cash flows.

Pre-requisite: MATH 321 Probability or both MATH 162 and Math 310

**MATH 351: Introduction to Numerical Methods with Applications (6)**. This course covers the fundamentals of numerical methods for students in science and engineering; floating-point computation, roots of equations, least-squares approximations, systems of linear equations, approximation of functions and integrals, the single nonlinear equation, and the numerical solution of ordinary differential equations; various applications in science and engineering; programming exercises in multiple computing environments including C, Fortran, Matlab and Mathematica.

Pre-requisite: MATH 274 Introduction to Differential Equations

**MATH 361: Introductory Real Analysis I (6)**. This course covers a careful treatment of the theoretical aspects of the calculus of functions of a real variable, including the real number system, the topology of the real line, continuity, derivatives and the Riemann and Riemann-Stieltjes integral. This course covers the first half of a book such as Rudin’s Principles of Mathematical Analysis.

Pre-requisite: MATH 263 Calculus III

**MATH 362: Introductory Real Analysis II (6)**. This course includes topics such as Fourier series, limits of sequences of functions, multivariable analysis and alternative integration techniques. This course completes the treatment of the topics in Rudin’s Principles of Mathematical Analysis and is designed to prepare the student for graduate school in mathematics.

Pre-requisite: MATH 361 Introductory Real Analysis I

**MATH 371: Introduction to Mathematical Biology (6)**. This course provides an introduction to the use of discrete and differential equations 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.

Pre-requisite: MATH 273 Linear Algebra with Applications and MATH 274 Introduction to Differential Equations

**MATH 302: Abstract Algebra I (6)**. This course covers modular arithmetic, permutations, group theory through the isomorphism theorems and Sylow theorems, ring theory through the notions of prime and maximal ideals, factorization domains and classification of groups of small orders.

Pre-requisite: MATH 273 Linear Algebra with Applications

**MATH 403: Abstract Algebra II (6)**. This course covers topics from coding theory, Galois Theory, multilinear algebra, advanced group theory, and advanced ring theory.

Pre-requisite: MATH 302 Abstract Algebra I

**MATH 407: Introduction to Graph Theory (6)**. Examines basic concepts and applications of graph theory, where graph refers to a set of vertices and edges that join some pairs of vertices; topics include subgraphs, connectivity, trees, cycles, vertex and edge coloring, planar graphs and their colorings. Draws applications from computer science, operations research, chemistry, the social sciences, and other branches of mathematics, but emphasis is placed on theoretical aspects of graphs.

Pre-requisite: (MATH 301 Introduction to Number Theory OR MATH 251 Discrete Mathematics) AND MATH 273 Linear Algebra with Applications

**MATH 411: Linear Programming (6)**. This course introduces the concepts and theoretical aspects of linear optimization. The course starts with formulation of real problem into a linear program (LP), geometric interpretation of linear optimization, integer programming and LP relaxation, duality theory, simplex method, sensitivity analysis, robust optimization, ellipsoid and interior point methods, and semidefinite optimization, applications (e.g., game theory and network optimization).

Pre-requisite: MATH 273 Linear Algebra with Applications and MATH 263 Calculus III

**MATH 412: Nonlinear Optimization (6)**. The course covers an introduction into convex sets and convex functions, unconstrained and constrained optimization, existence of solutions (Weierstrass’ theorem), unconstrained optimization: first and second order conditions, equality constraints: Lagrange’s optimality theorem, second order conditions, inequality constraints: Karush-Kuhn-Tucker theorem, convex and non-convex optimization, some computational methods: line search, steepest descent, Newton, quasi Newton, trust region, feasible points, penalty methods, and augmented Lagrangian methods.

Pre-requisite: MATH 273 Linear Algebra with Applications and MATH 263 Calculus III

**MATH 417: Cryptography (6)**. The course is an introduction to modern cryptographic algorithms and to their cryptanalysis, with an emphasis on the fundamental principles of information security. Topics include: classical cryptosystems, modern block and stream ciphers, public key ciphers, digital signatures, hash functions, key distribution and agreement.

Pre-requisites: MATH 273 Linear Algebra with Applications and either MATH 251 Discrete Mathematics or MATH 301 Introduction to Number Theory

**MATH 423: Actuarial Mathematics II (6)**.

Claim frequency models, claim severity models and empirical loss distributions, time-to-failure and survival models, life tables, life insurance, contingent payment models, contingent annuity models, contingent contract reserves.

Pre-requisite: MATH 321 Probability

**MATH 424: Mathematical Finance (6).**

The course aims to cover knowledge of basic financial actuarial models with applications and to prepare students pass the actuarial financial economics (MFE) exam.

Pre-requisite: MATH 321 Probability

**MATH 425: Stochastic Processes (6)**. There first part of this course deals with discrete time Markov chains, including Markov property, transition probabilities, classification of states, visits to a fixed state, notion of limiting behavior, reducibility, recurrence. The course continues to Poisson processes, continuous-time chains, birth-and-death processes, renewal processes, branching processes, and queuing theory with applications. There is also a brief introduction to martingales and Brownian motion.

Pre-requisite: MATH 321 Probability

**MATH 440: Regression Analysis (6)**. The course starts with simple linear regression, diagnostic tests and plots, quality measures, matrix description of regression model. It continues with the multiple regression, predictor subset selection, interactions, variable transformations, use of categorical predictors, model validation, remedial measures. Some other topics that are considered are autocorrelation and logistic regression.

Pre-requisites: MATH 273 Linear Algebra with Applications and either MATH 240 Applied Statistical Models or MATH 322 Statistics

**MATH 441: Design of Experiments (6)**. The course begins with one-way and two-way ANOVAfor fixed effects, diagnostics and remedial measures, multiple comparisons. We proceed to covering studies with unequal sample sizes, multifactor ANOVA, random effects, randomized block designs, nested designs, repeated measure designs, Latin Square and related designs.

Pre-requisites: MATH 273 Linear Algebra with Applications and either MATH 240 Applied Statistical Models or MATH 322 Statistics

**MATH 444: Nonparametric methods (6)**. A large part of this course is dedicated to considering nonparametric analogs of parametric tests. These analogs include the sign test, Wilcoxon signed-rank test, permutation test, Wilcoxon rank-sum test, Kruskal-Wallis test, Friedman test and various chi-square tests. The course also considers goodness-of-fit tests, various permutation procedures, bootstrapping methods, procedures for censored data.

Pre-requisite: MATH 322 Statistics

**MATH 446: Time Series Analysis (6)**. Time series data trend estimation, seasonality analysis, stationary models, moving average, autoregressive and ARMA modeling, model identification, forecasting, intervention analysis.

Pre-requisite: MATH 322 Statistics

**MATH 449: Introduction to Statistical Programming (6)**. 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.

Pre-requisite: MATH 322 Statistics

**MATH 455: Stochastic Calculus (6).** This class introduces stochastic calculus and its applications. In the first half of the class, the course introduces the measure theoretic probability concepts. Next, we move to Ito lemma and stochastic differential equations. Examples of stochastic differential equations with explicit solutions are provided.

Pre-requisite: MATH 321 Probability

**MATH 460: Introduction to Topology (6)**. A first course in topology, the first part of the course will deal with metric spaces, including limits of sequences, cluster points of sets, the two definitions of compactness, continuity and convergence. The later part of the course will cover topological spaces and their properties. Elementary properties of homotopy theory will be covered at the end.

Pre-requisite: MATH 361 Introductory Real Analysis I

**MATH 462 Advanced Linear Algebra**

This course is designed to introduce some advanced topics not covered in MATH 273 and study further some concepts covered in MATH 273. This course is a theorem-proof style course, emphasizing theories. We will cover vector spaces; quotients; dual space; determinants; traces; eigenvalues, eigenvectors and diagonalization; inner product spaces and the spectral theorem for normal operators; singular value decomposition; and Jordan canonical form.

**MATH 471: Nonlinear Differential Equations (6)**. The course studies nonlinear differential equations and the time-behavior of their solutions. The course begins with a quick review on first order (logistic) differential equations, and continues with classification into (non) autonomous and (non)linear equations, interpretation of the solution and its general and near equilibrium behavior, the phase-plane, Poincare map, critical points, periodic solutions, Poincare-Bendixson theorem, the concept of stability (Lyapunov), stability analysis by linearization, perturbation theory, Poincare-Lindstedt method, averaging, bifurcation, one dimensional chaos, fractal sets, and Hamiltonian systems.

Pre-requisite: MATH 273 Linear Algebra with Applications and MATH 274 Introduction to Differential Equations

**MATH 476: Numerical methods for partial differential equations (6)**. The course is a blend of conceptual and practical aspects of numerical solutions of partial differential equations, and aims at students of senior level. Various numerical methods are discussed: finite difference, finite volume, finite element, and spectral methods, and explicit/implicit methods for time integration. Conceptual aspects include convergence, consistency, stability, Courant-Friedrichs-Lewy criteria, Lax equivalence theorem, error analysis, Fourier-von Neumann stability analysis. Discussions are tailored towards numerical solutions of model problems: Poisson equation, heat equation, advection equation, wave equation, advection-diffusion equation, and Stokes equation. Students are required to implement some methods in Matlab for 1D/2D Poisson, advection, and advection-diffusion equations.

Pre-requisite: MATH 351 Introduction to Numerical Methods with Applications and MATH 481 Partial Differential Equation

**MATH 477: Applied Finite Element Methods (6)**. Piecewise polynomial interpolations, basis shape functions in natural coordinates, local and global shape functions in spatial one and two dimensions, initial/boundary value problems in science and engineering with industrial applications, Galerkin method, Rayleigh-Ritz method, local and global finite element matrices, connectivity and nodal degrees of freedom, numerical integration, basic numerical linear algebra, Newton’s method, conjugate gradient method, industrial applications of finite elements for heat transfer, structural, and fluid flows, industrial models and software application, validation and presentation of simulation results.

Pre-requisite: MATH 351 Numerical Methods

**MATH 480: Complex analysis (6)**. This will be first look at complex analysis. In the introduction, properties of complex numbers, limits and topology in the complex plane will be covered. Then the properties of analytic functions will be studied, ending in a proof of Cauchy’s theorem. Power series, Laurent series and residues will finish the core part of the course. Applications of contour integration will be covered. As time permits, the instructor may cover basic conformal mappings.

Pre-requisite: MATH 263 Calculus III

**MATH 481: Partial Differential Equations (6)**. An introductory course on partial differential equations, it focuses on classical linear and quasi-linear partial differential equations. The course begins with the classification of partial differential equations (hyperbolic, parabolic, and elliptic) and derivation of some model problems (heat, wave, and Laplace equation) that fall under this classification. Solution methods for model problems are discussed, which include separation of variables, Fourier series and transforms, method of characteristics, Sturm-Liouville eigenvalue problem, and Green’s function. Existence and uniqueness of the solutions for the model problem will also be discussed.

Pre-requisite: MATH 274 Introduction to Differential Equations and MATH 480 Complex Analysis

**MATH 482: Fourier Analysis (6)**. This course introduces the topics of Fourier series and Fourier transform, with applications to differential equations and orthogonal polynomials.

Pre-requisite: MATH 263 Calculus III and MATH 274 Introduction to Differential Equations

**MATH 490/491 Special Topics in Mathematics I,II (6)**: Selected topics in mathematics not included in existing courses.

Pre-requisite: Instructor’s approval through the add form

**MATH 497/498 Directed Study in Mathematics I,II (6)**. Under the supervision of a faculty advisor, a student pursues independent research on a topic in, or related to, Mathematics.

Pre-requisite: Instructor’s approval through the add form

**MATH 350: Research Methods (6)**. This course is designed to teach students basic skills that are essential in research work in mathematical sciences.

Pre-requisite: MATH 321 Probability

**MATH 499: Graduation Project (6)**

A short description of the project can be found here.

Pre-requisite: Senior standing

**MATH 399 Internship (6)** is an elective, which provides an opportunity for a student to gain practical hands-on work experience in a related field of interest. A student can do an internship after the completion of at least 120 ECTS, with at least 50 ECTS of math courses. The student either works directly with a faculty or through collaboration with scientifically recognized industry or another academic institution, on a research project with a significant mathematics component. The expected workload should be equivalent to between 180-240 hours of total workload in 6-8 weeks. The credits are not counted towards the math core or math elective requirement. Upon the completion of the internship, students should be able to integrate some aspects of mathematics into practice, demonstrate communication skills through presentation and written report, recognize the importance of professional development and attitude.

**Required**MATH 161 Calculus I- MATH 162 Calculus II
- MATH 263 Calculus III
- MATH 273 Linear Algebra with Applications
**Elective**At least four electives, with two from among Math 251, 274, 321, 322, 351, and 361 and two at the 4xx level.

– Any mathematics course that is being used as a

– For courses such as Real Analysis and Abstract Algebra, Numerical Methods, Mathematical Statistics for graduation it is required to get C- or above for graduation

– If a student gets D+ in a math elective, it cannot be counted as a math elective, but it can be counted as a general elective.

-The passing grade for a general elective is D.