- Core Courses

No | Subject | Creit | Time of Lecture | Lab | Instroduction of Course |

300.204 | Differential Equations | 4 | 3 | 2 | Natural and social phenomena are often represented by differential equations. Therefore, studying solutions of various differential equations is very important to almost all sciences. In this course, we study the basic methods of solving fundamental differential equations. |

881.301 | Modern Algebra 1 | 3 | 3 | 0 | This course deals with definitions and examples of groups, rings, modules and fields, their sub-structures, quotient-structures, and homomorphisms. Students are introduced to important theorems and applications. |

881.302 | Modern Algebra 2 | 3 | 3 | 0 | This course follows "Modern Algebra 1" and includes important theorems on groups, rings, proofs on modules and fields (Jordan-Hoelder theorem, Sylow theorems, Galois theorems, etc.) and various applications. |

881.303 | Introduction to Differential Geometry 1 | 3 | 3 | 0 | Course covers study of curves in Euclidean spaces, Euclidean space, rigid motions, rotations and reflections, orientations, cross product, tangent spaces and tangent maps, length of curves, tangent line, curvature, osculating circle, radius of curvature, curvature vector, rotation index, isoperimetric inequality, torsion, and the Frenet-Serret formula. |

881.304 | Introduction to Differential Geometry 2 | 3 | 3 | 0 | This course follows "Introduction to Differential Geometry 1" and deals with surfaces in 3-dimensional Euclidean space. Topics covered are: Tangent planes, normal vector fields, helicoid, surfaces of revolution, area of surfaces, surface integrals, the first fundamental form, geodesic, the second fundamental form, principal curvatures, Gaussian curvature, mean curvature, structure equations, Hilbert theorem, Gauss-Bonnet theorem, vector fields and Hopf's theorem. |

881.313 | Sets and Mathematical Logic | 3 | 3 | 0 | This course exposes students to several topics such as elementary set theory, construction of natural numbers, integers, rational numbers and real numbers, axiom of choice, cardinals and ordinals, and methods of proofs. |

881.319 | Numerical Linear Algebra | 3 | 3 | 0 | This course covers Gauss elimination, Cholesky decomposition, Householder and Gram-Schmidt methods, data fitting, nonlinear least squares problems, simplex methods, decomposition of matrices, Jacobi and Seidel iteration, relaxation methods, finite differences, ADI method, and conjugate gradient methods. |

881.32 | Introduction to Numerical Analysis | 3 | 3 | 0 | Students study topics such as error analysis, polynomial interpolation, Newton divided difference, rational approximation, trigonometric interpolation, fast Fourier transform, spline, numerical integration, Peano error representation, Euler-Maclaurin formula, Gauss quadrature, Newton and quasi-Newton methods, and numerical methods for finding zeros of polynomials. |

881.401 | Introduction to Topology 1 | 3 | 3 | 0 | In this course, students are trained in the basic properties of topological spaces, Tietze extension theorem, metrizability, Hausdorff space and separability, and compact spaces. |

881.402 | Introduction to Topology 2 | 3 | 3 | 0 | As the continuation of Introduction to Topology 1, this course trains students in topology on manifolds, first fundamental groups, and covering spaces. |

881.408 | Geometric Algebra | 3 | 3 | 0 | This course trains students in the interpretation of linear algebra in terms of abstract algebra, orthogonal geometry and symplectic geometry over arbitrary fields, classical groups, topological groups, Zariski topology and algebraic groups, and the definition and examples of Lie groups. |

881.41 | Introduction to Algebraic Geometry | 3 | 3 | 0 | This course is for students who have mastered the basics of undergraduate abstract algebra. As an easy introductory course in algebraic geometry, it covers the following topics: affine and projective space; projective geometry on the plane; projective Nullstellensatz and dimension theorem; extrinsic properties of projective varieties; Riemann-Roch theorem for algebraic curves; and resolution of singularities of projective algebraic curves. |

881.423 | Partial Differential Equations | 3 | 3 | 0 | In this course, students are introduced to the basic theories of partial differential equations. In addition, first order quasilinear PDE, local existence, uniqueness, Cauchy-Kovalevsky theorem, Laplace equation, maximum principle, Harnack's inequality, Hilbert space methods, and variational principle are discussed. |

881.424 | Applications of Partial Differential Equations | 3 | 3 | 0 | In this course, students are introduced to ways in which the theories of partial differential equations are applied to problems in physics and mechanics. In particular, they will study the following topics: Dirac equations; Maxwell equations; self-dual equations in the nonlinear field theories and their soliton solutions; and tensor analysis and the Einstein field equations. In addition, the course covers the Navier-Stokes and the Euler equations derived from mathematical fluid mechanics. |

881.425 | Real Analysis | 3 | 3 | 0 | In this course, students are introduced to the Lebesgue integral and measure on the real line, absolutely continuous functions, functions of bounded variations, space of integrable functions, product of measures and Fubini theorem, and applications to Fourier series and integral. |

881.427 | Algebraic Coding Theory | 3 | 3 | 0 | In this course, students are introduced to the notion of entropy and Shannon theory and the basic properties and error-correcting functions of various codes (linear codes, cyclic codes, Hamming codes, and Reed-Muller codes). |

881.431 | Fourier Analysis and Applications | 3 | 3 | 0 | This class will study the classical theories of the Fourier series and its integrals. Included in the studied topics are the discrete cosine transform, the fast Fourier transform, wavelet and the multiresolution analysis, as well as the wavelet and the Fourier transform, the process of signals as well as the images and applications to the inverse problems. |

881.434 | Chaos and Dynamical Systems | 3 | 3 | 0 | The course will cover the Kepler motion, ecological problem, Hamiltonian system, stability and chaos, limit cycles, Poincare map, and strange attractors. |

881.436 | Discrete Mathematics | 3 | 3 | 0 | In this course, we will study discrete phenomena in computer sciences, operation research, and statistics, and practice solving problems on discrete structures. Starting from the basic mathematical tools such as sets, logic, functions, and probability, we will go on to mathematical reasoning and counting method using permutations, combinations, graph and tree. This course also deals with Boolean functions, turing machines, algorithms and complexity that form the basis of computer science. |

3341.201 | Introduction to Mathematical Analysis 1 | 3 | 3 | 0 | Basic properties of real number field including completeness axiom, limits of sequences, elementary topological properties of coordinate spaces, Cauchy sequences, compact and connected sets, precise definitions of limit and continuity, uniformly continuous functions, properties of monotone functions, Riemann integral, Riemann-Stieltjes integral, properties of functions of bounded variations, fundamental theorem of calculus are studied. |

3341.202 | Introduction to Mathematical Analysis 2 | 3 | 3 | 0 | As a sequel to Mathematical Analysis 1, uniform convergence of sequence of functions, differentiation and |

3341.211 | Number Theory | 3 | 3 | 0 | This is an introductory course for Number Theory. The course covers various subjects of number theory including prime numbers, congruence equations, sums of squares, multiplicative functions and Diophantine equations, to name a few, and some applications. The course will introduce not only arithmetic methods but also analytic methods of number theory. |

3341.347 | Complex Function Theory 1 | 3 | 3 | 0 | Some basic properties of complex analytic functions are studied, and some special functions such as the gamma function and the Riemann zeta function are introduced. The following topics will be covered: Moebius transformations, elementary functions, Cauchy-Riemann equations, analytic functions, harmonic functions, Taylor series, line integrals, Cauchy's theorem, Cauchy's integral formula, maximum modulus theorem, Laurent series, argument principle, real integrals by means of residue calculus, gamma function and the Riemann zeta function. |

3341.348 | Functions of Several Variables | 3 | 3 | 0 | Differentiation and integration of vector-valued functions are treated in this course. Topics include differentiation of multi-variable functions, the implicit function theorem, maxima and minima of multi-variable functions, multiple integrations, the Fubini theorem, change of variables in integrations, Green's theorem, Stokes's theorem, and Gauss's divergence theorems. |

3341.352 | Introduction to Stochastic Differential Equations | 3 | 3 | 0 | As a basic material, we first cover the following topics: - Probability theory based on Measure theory - Kolmogorov construction of Brownian motion - Martingale theory - Ito's stochastic integral and Ito formula Utilizing these tools, we cover the existence and the uniqueness of the solution of the stochastic differential equations driven by the Brownian motion. If time permits, we may also cover some topics related to Markov process, infinitesimal generator and the Feynman-Kac formula. |

3341.353 | Introduction to Scientific Computing | 3 | 3 | 0 | The methods of applied mathematics are necessary to understand the Scientific Computing. So, in this course, we introduce the Hilbert space and Sobolev space to understand the applied mathematics and analysis the integral-differential equations on the those spaces using a mathematical theory. Courses include Functional space, integral-differential equation, Fredholm Alternative, Variational principle, Fourier and Laplace Transforms and asymptotic analysis. |

3341.362 | Efficient Programming and Practice | 3 | 3 | 0 | This is a course intended for students without any previous programming experience, and will emphasize the efficiency of the written program. The course will start as a basic programming language course and will lead into skills for writing programs that are memory efficient and of high speed. |

3341.445 | Topics in Mathematics 1 | 3 | 3 | 0 | In recent years, mathematics is undergoing exciting new developments. The barriers between fields are being broken; many new unexpected applications are continually found; and out of this cross-fertilization, new kinds of mathematics are born. The objective of this course to introduce this exciting new developments to advanced mathematics undergraduate students in a timely manner. The current possibilities include but not confined to the following topics various new advances of pure mathematics and logic; computational science and numerical analysis; fluid mechanics and geophysics; wavelets and signal processing; cryptology; quantum computation; mathematical biology including bio-informatics, proteomics and neuroscience; intelligence science; financial mathematics and mathematical economics; probability theory with various applications. But ultimately, the topic to be covered will vary depending on the instructor and the circumstances. |

3341.446 | Topics in Mathematics 2 | 3 | 3 | 0 | In recent years, mathematics is undergoing exciting new developments. The barriers between fields are being broken; many new unexpected applications are continually found; and out of this cross-fertilization, new kinds of mathematics are born. The objective of this course to introduce this exciting new developments to advanced mathematics undergraduate students in a timely manner. The current possibilities include but not confined to the following topics: various new advances of pure mathematics and logic; computational science and numerical analysis; fluid mechanics and geophysics; wavelets and signal processing; cryptology; quantum computation; mathematical biology including bio-informatics, proteomics and neuroscience; intelligence science; financial mathematics and mathematical economics; probability theory with various applications. But ultimately, the topic to be covered will vary depending on the instructor and the circumstances. |

3341.451 | Financial Mathematics 1 | 3 | 3 | 0 | This course is designed to introduce the basic theoretical frameworks and methodologies of financial mathematics and then the Black-Scholes model. In particular, the following topics are covered: replicating portfolio; arbitrage pricing theory; introduction to the probability theory based on the measure theory; martingale measure and its application to the derivative pricing; Brownian motion; Ito integral; Ito formula; Black-Scholes market; Black-Scholes formula; numerical solution of partial differential equations. |

3341.452 | Financial Mathematics 2 | 3 | 3 | 0 | This course presupposes the prior knowledge of Financial Mathematics I or its equivalents. The topics covered in this course are selected from: American option; exotic option; interest rate models; risk management; other topics of interest chosen by the instructor. |

3341.453 | Mathematical Modeling and Simulation | 3 | 2 | 2 | Introduce the modeling equation arising from physics, biology, medical applications and economics. Each governing equations are mathematically analyzed by investigating equilibria solutions, stability, existence and uniqueness. Also we emphasis on practical issues of computational methods. |

3341.454 | Mathematical and Numerical Optimization | 3 | 3 | 0 | Optimization and its computational methods are very important on science, engineering and industry. In many cases, we may get the wrong solutions due to the instabilities of parameter optimizations or inverse problems. To understand and solve those problems, we will give a lecture on mathematical theories and numerical methods on those subjects. |

300.203A | Linear Algebra 1 | 3 | 3 | 0 | We learn basic concepts of linear algebra. Beginning with Gauss elimination and row-reduced echelon form, we study matrices and linear maps and define determinants. We also learn basic notions of vector spaces such as basis and dimension. We understand the matrix of a linear map corresponding to a basis change, and learn characteristic polynomial, diagonalization and triangularization. Moreover, we deal with inner product spaces and, more generally, spaces with bilinear forms, and then we begin studying elementary group theory in order to define orthogonal groups. We understand 2-dimensional and 3-dimensional orthogonal groups and their structures. Meanwhile, we introduce quotient spaces to utilize the induction on dimension. |

300.206A | Linear Algebra 2 | 3 | 3 | 0 | Based on the knowledge of Linear Algebra 1, we begin deeper and more abstract approach. We understand orthogonal and unitary operators, and study spectral theorems. We learn isomorphisms and homomorphisms of groups, and also normal subgroups and quotient groups. We learn various orthogonal groups corresponding to various bilinear forms, and then we try to understand linear algebra in terms of orthogonal groups. We learn the primary decomposition theorem and introduce the second decomposition theorem(Jordan normal form) briefly. Moreover, we select and study some interesting applications of linear algebra in various branches of mathematics. |

3341.301A | Complex Function Theory 2 | 3 | 3 | 0 | As a sequel to 'Complex Function Theory 1', some deeper results as well as various applications of the theory are introduced. The following topics are studied: proof of the prime number theorem by using the Riemann zeta function, conformal mappings, Riemann mapping theorem, Schwarz-Christoffel integrals, elliptic functions, Weierstrass functions, the Jacobi theta functions and their applications. |

881.433A | Introduction to Cryptography | 3 | 3 | 0 | This course, which is aptly titled Number Theory and Cryptography, will begin with an introduction to the essential elementary number theories. Afterwards, we will go on to learn about the various encryptive and decryptive algorithms. In addition, various cryptosystems, their complexity, security, and overall advantages as well as disadvantages will be discussed. |

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