# Korea university College of Science

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### Courses & Syllabuses

Courses and Syllabuses

College of Science, Department of Mathematics

MATH 003 TOPIC COURSE 〔3〕
The topic of this course is determined each semester.

MATH 201 SET THEORY 〔3〕
Axiom System, Algebraic Structure, The Foundations of Geometry, The Foundations of Continuity.

MATH 203 DISCRETE MATHEMATICS 〔3〕
Mathematical Induction, Basic Counting Techniques, Graphs and Trees, Algorithm, Generating Functions, Algebraic Structure.

MATH 211 ANALYSIS I WITH LAB 〔3〕
Basic Topology on Euclidean Space. Continuity, Convergence, Uniform Convergence, Series, Convergence Tests. Differentiation, Inverse and Implicit Function Theorems.

MATH 212 ANALYSIS II WITH LAB 〔3〕
Riemann-Stieltjes Integrals, Sequences and series of functions, Uniform convergence, Fourier series, Functions of Several Variables, Integration of Differential Forms, Stokes’ Theorem.

MATH 221 LINEAR ALGEBRA I WITH LAB 〔3〕
Vector Spaces & Systems of Linear Equations, Linear Transformations & Matrices, Vector Spaces with an Inner Product, Determinants.

MATH 222 LINEAR ALGEBRA II WITH LAB 〔3〕
The Theory of Single Linear Transformation, Dual Vector Spaces & Multilinear Algebra, Orthogonal & Unitary Transformations.

MATH 223 NUMBER THEORY 〔3〕
Divisibility, Primes, Congruences, Quadratic Reciprocity, Farey Sequences, Euler Phi-Function, Fermat's Last Theorem.

MATH 232 INTRODUCTION TO GEOMETRY 〔3〕
We study modern methods of mathematics through the various theories of classical geometry. Non-euclidean geometry shows the concepts of duality, projectivity, hyperbolicity and also shows the role of group structures and metric structures in geometry. The study of these concepts also reveals how mathematics is applied to other sciences and solving practical problems.

MATH 240 DIFFERENTIAL EQUATIONS WITH LAB 〔3〕
First order differential equations, Second order linear differential equations, Series solution of second order linear Equations, Higher order linear differential equations, The Laplace transform.

MATH 282 ADVANCED DIFFERENTIAL EQUATIONS 〔3〕
Boundary value problem, Sturm-Liouville theory, Linear Systems theory, Oscillation theory, Stability theory.

MATH 315 COMPLEX ANALYSIS I 〔3〕
Complex Line Integral, Holomorphic Functions, Cauchy's Theorem, Power Series Representation, Zeros of Holomorphic functions, Maximum Modulus Theorem, Singularities, Residue Calculus.

MATH 321 ALGEBRA I 〔3〕
Groups, Subgroups, Cosets, Homomorphism Theorems, Sylow Theorems, Ring, Ideal, Field of Quotients, Ring of Polynomials, Principal Ideal Domain, Euclidean Rings.

MATH 331 DIFFERENTIAL GEOMETRY I 〔3〕
Tangent vectors and differential forms in euclidean spaces, Elementary vector calculus, Curves and Frenet formula, Visualization of curves, Examples of surfaces, Visualization of surfaces, Gauss curvature and mean curvature, Computation of curvatures, Geodesics.

MATH 333 TOPOLOGY I 〔3〕
This is an introduction to topology with an emphasis on the set-theoretical aspect of the subjects. The topics covered are topological and metric spaces, continuous functions, homeomorphisms, compactness, connectness and separation axioms.

MATH 342 NUMERICAL ANALYSIS AND COMPUTER EXPERIMENT 〔3〕
Root finding for nonlinear equations, interpolation theory, approximation theory, numerical integration, numerical solutions of system of linear equations.

MATH 343 PROBABILITY AND STATISTICS WITH LAB 〔3〕
We introduce basic concepts in probability such as random variable, distribution function, expectation, conditional expectation and limit theorems. Basic theory of mathematical statistics including estimation and testing hypotheses is developed.

MATH 344 STOCHASTIC PROCESSES 〔3〕
This course is intended to introduce stochastic processes by using elementary probability theory. We present basic important topics such as Poisson processes, Markov processes, and Brownian motion as well as their applications.

MATH 358 COMPLEX ANALYSIS Ⅱ 〔3〕
Mapping properties of holomorphic functions, Riemann mapping theorem, Harmonic functions, Maximum principle, Mean value property, Poisson integral formula, Analytic continuation, Infinite products, Special functions.

MATH 362 ALGEBRA II 〔3〕
We study the topics of fields, extension fields, algebraic elements, transcendental elements, finite fields, function fields, Galois theory and algebraic solutions of polynomial equations.

MATH 372 DIFFERENTIAL GEOMETRY II 〔3〕
Calculus of variations and geodesics, Completeness, Isometries, Parallel translation and geodesic curvatures, Riemann curvature tensor, Gauss-Bonnet theorem, Geodesic coordinates. Also some topics chosen from the followings: Minimal surfaces and surfaces of constant mean curvature(CMC), Visualization of minimal and CMC surfaces, Calculus of variations and applications to physics, Geometry of manifolds, Geometry of metric spaces.

MATH 374 TOPOLOGY II 〔3〕
Beginning with the classification of surfaces, this course introduces Euler characteristic, fundamental groups and its applications to 2 and 3 manifolds.

MATH 453 REAL ANALYSIS 〔3〕
Lebesgue measure, Lebesgue integrals, Lpspaces, Hilbertspaces, Duality.

MATH 458 TOPICS IN ANALYSIS 〔3〕
Topics will be selected from various advanced subjects of functions of complex or real variables. For functions of complex variables, topics include conformal mappings, analytic continuation, properties of analytic functions, harmonic functions, infinite products. For functions of real variables, Topics include complex measures, Hilbert spaces, Dual spaces and normed linear spaces.

MATH 462 APPLIED NUMBER THEORY 〔3〕
The theory of finite fields, the theory of polynomial rings, the theory of elliptic curves, RSA system, ECC system, XTR system, NTRU system, coding theory.

MATH 464 COMBINATORICS 〔3〕
Permutations and combinations, generating functions, recurrence relations, inclusion and exclusion, Polya's enumerations.

MATH 469 TOPICS IN ALGEBRA 〔3〕
We study the topics of representations of groups, mathematical cryptology, computational algebra, algebraic geometry.

MATH 476 TOPICS IN GEOMETRY 〔3〕
Topics will be chosen from the subjects of recent interests related to geometry. Topics vary accordingly and may include one of may the followings : Metric geometry, Minimal surfaces, Combinational geometry, Geometric analysis, Complex geometry, CAGD(Computer Aided Geometric Design), Mathematical mechanics, etc.

MATH 477 TOPICS IN TOPOLOGY 〔3〕
In this course, we are going to discuss the Knot theory which is applied to physics and biology. Moreover we will explore other related areas of mathematics such as group theory, differential geometry and algebraic topology.

MATH 481 PARTIAL DIFFERENTIAL EQUATIONS AND COMPUTER EXPERIMENT 〔3〕
Fourier series, Wave equations, Heat equations, Laplace equations, Special functions, Existence and Uniqueness theorem.

MATH 483 MATHEMATICAL FINANCE 〔3〕
We study the important problems in finance such as derivatives, interest rate models, and risk managements based on probability theory, PDE, and numerical computations.

MATH 484 ACTUARIAL MATHEMATICS 〔3〕
Probability theory, Statistical Inference, Risk model, Loss distribution, Risk premium, Experience rating, Life insurance, Annuities, Population theory.

MATH 485 TELECOMMUNICATION MATHEMATICS 〔3〕
Basic queueing theory and its applications to telecommunication systems are covered: Introduction to queueing theory and telecommunication systems, M/M/C queue, M/G/1 queue, modeling of various traffics such as voice and video, delay and loss probability of networks, performance analysis of various transmission schemes.

MATH 487 TOPICS IN APPLIED MATHEMATICS 〔3〕
We study mathematical theories and methods which are used to analyze the physical and industrial problems.

MATH 488 TOPICS IN PROBABILITY THEORY 〔3〕
Special topics are chosen from recent developments of research in probability theory. Topics covered vary from year to year.