Graduate Courses

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

3341.501 | Algebra 1 | 3 | 3 | 0 | This course studies algebraic structures (such as groups, rings, modules, and fields) and homological algebra. Important theorems and their applications are introduced. |

3341.502 | Algebra 2 | 3 | 3 | 0 | As a sequel to "Algebra 1", this course covers field theory and Galois theory, basics of commutative algebra, algebraic geometry and algebraic number theory. Various applications of the material are also discussed. |

3341.503 | Real Analysis | 3 | 3 | 0 | This course discusses such topics as Lebesgue measure and integration of Euclidean space, product measure and the Fubini theorem, complex measure and the Radon-Nykodim theorem, Lebesgue decomposition, measure of topological spaces and the Riesz representation theorem. |

3341.504 | Complex Analysis | 3 | 3 | 0 | This course first reviews basic theories of complex analysis including Cauchy-integral formula, convergence of power series, Taylor and Laurent series, residue theorem and its applications, and Schwarz lemma. The cours continues with more advanced topics such as Poisson integral formula and the boundary value problem for harmonic functions, partial fractions and Mittag-Leffler's theorem, infinite products and Weierstrass' theorem, normal families and Montel's theorem, and Riemann's mapping theorem. |

3341.505 | Differentiable Manifolds | 3 | 3 | 0 | Differential manifolds are discussed while providing concrete examples. Topics include differentiable structures, tangent vectors, tangent spaces, immersions, submersions, submanifolds, regular values, Sard's theorem, vector fields, distributions, Frobenius's theorem, Lie derivative, tensor fields, differential forms, Poincare lemma, orientation, integration of manifolds, Stokes's theorem, de Rham cohomology, and Lie groups. |

3341.601 | Commutative Algebra | 3 | 3 | 0 | This course covers how to study varieties from an algebraic perspective, dimensions and depths (of rings and modules) and related theorems. The course continues with a discussion of Cohen-Macaulay rings, Gorenstein rings, complete intersection rings and regular rings. |

3341.603 | Functional Analysis 1 | 3 | 3 | 0 | This course covers basic properties of topological vector spaces, semi-norms and locally convex spaces, weak topologies, Banach-Alaoglu's theorem, Krein-Milman's theorem, dual spaces, Stone-Weierstrass' theorem, the spectral theorem of compact operators, and Hilbert-Schmidt operators. |

3341.604 | Functional Analysis 2 | 3 | 3 | 0 | As a sequel to "Functional Analysis 1", this course focuses on test functions and distribution spaces, Fourier transform, Paley-Wiener's theorem, applications to PDE, Banach algebras, Gelfand transform of commutative Banach algebras, spectral theorem of normal operators, and unbounded operators. |

3341.605 | Differential Geometry 1 | 3 | 3 | 0 | This course coveres Riemannian manifolds, metrics, connections, geodesics, parallelism, structure equations, completeness, curvature, Jacobi fields, and first and second variations of length and volume. |

3341.606 | Differential Geometry 2 | 3 | 3 | 0 | As a sequel to "Differential Geometry 1", this course examines comparison theorems, submanifold theory, general relativity, holonomy groups, minimal submanifolds, constant mean curvature surfaces, harmonic maps, isoperimetric inequalities, Lagrangian geometry, and relationships between curvature and topology. |

3341.607 | Algebraic Topology 1 | 3 | 3 | 0 | This course covers basic topics from algebraic topology including the theory of fundamental group, covering space, and homotopy theories. |

3341.608 | Algebraic Topology 2 | 3 | 3 | 0 | As a sequel to "Algebraic Topology1", this course covers such topics as CW-complex, cohomology, orientation, Poincare duality, and cup product. |

3341.611 | Aalgebraic Number Theory | 3 | 3 | 0 | This course discusses various number fields, integer rings, ideals, ramifications, Dirichlet's unit theorem, valuations, localizations, ideal class groups and class numbers. |

3341.612 | Lie Algebra | 3 | 3 | 0 | This course covers semisimple Lie algebras, Cartan decomposition, Weyl's theorem, root systems and classification, Weyl groups, classical simple Lie algebras, universal enveloping algebras, the PBW theorem, representation theory and Verma module, and Chevalley groups. |

3341.613 | Algebraic Geometry | 3 | 3 | 0 | This course provides an introduction to algebraic geometry and is intended for graduate students entering this field of study. The main topics are as follows; affine and projective varieties, morphisms of projective varieties, rational functions on projective varieties, Hilbert polynomials, intrinsic and extrinsic properties of algebraic varieties. |

3341.621 | Operator Algebra | 3 | 3 | 0 | This course covers representations of C*-algebras, the basics of C*-algebras and von Neumann algebras, group C*-algebras and group von Neumann algebras, classification of von Neumann algebras, K-theory for operator algebras and classification of C*-algebras. |

3341.622 | Analytic Functions of Several Variables | 3 | 3 | 0 | This course covers Hartog's phenomenon, domain of holomorphy and the Levi problem, integral formula for polydisks, Bochner-Martinelli integral, Bergman kernel, plurisubharmonic functions, pseudo-convexity, and Hoermander's solution of the d-bar problem. |

3341.625 | Harmonic Analysis | 3 | 3 | 0 | Basic properties of topological groups, Haar measure on locally compact groupss, convolution for functions and measures, unitary representation of locally compact groups, Fourier transform and Pontyagin's duality theorems, representation of compact groups, Peter-Weyl's theorem, and Tanaka-Krein's duality theorem are discussed in the class. |

3341.626 | Numerical Analysis | 3 | 3 | 0 | Sobolev spaces, theory of elliptic partial differential equations, Lax-Mligram Lemma and Cea's lemma, polynomial approximation theory in Sobolev spaces, error estimates for elliptic problems, nonconforming finite element methods, and mixed finite elements are discussed in this course. |

3341.631 | Lie Groups | 3 | 3 | 0 | This course covers basic theory of Lie groups and other topics such as homogeneous space, covering groups, sub-Lie groups, Campbell-Hausdorff's theorem, the structure of compact Lie groups, and PBW theorem. |

3341.633 | Theory of Complex Manifolds | 3 | 3 | 0 | This course covers special properties of complex manifolds. The main topics include: complex structures, complexified tangent bundles, holomorphic tangent bundles, Dolbeault cohomology, Kaehler manifold, deformation of complex structures, and Kodaira's embedding theorem. |

3341.635 | Theory of Partial Defferential Equations 1 | 3 | 3 | 0 | In this course students explore classical theories of the Fourier series and Fourier integrals. Additional topics include the discrete cosine transform, fast Fourier transform, wavelet and multiresolution analysis, wavelet transform and the Fourier transform, signal and the image process, and applications to inverse problems. |

3341.636 | Theory of Partial Defferential Equations 2 | 3 | 3 | 0 | As a sequel to "Theory of Partial Differential Equations 1", this course examines nonlinear partial differential equations, fixed point methods, variational methods, methods of upper and lower solutions, regularity problems of nonlinear PDE, and concrete equations - such as Navier-Stokes equations, Euler equations, nonlinear wave equations and Einstein's field equations. |

3341.641 | Differential Topology | 3 | 3 | 0 | This course discusses how to analyze a topological space with a differentiable structure, and covers Sard's theorem, transversality, Euler characteristics. Depending on the lecturer, this course can also cover Hodge theory, de Rham theory and characteristic classes. |

3341.642 | Geometric Topology | 3 | 3 | 0 | This course discusses extra geometric structures of a topological space, and covers part of the following topics. Morse theory, vector bundles, homotopy theory, 3 or 4-manifold theory, hyperbolic geometry and symplectic topology. |

3341.651 | Methods of Homological Algebra | 3 | 3 | 0 | The course begins by introducing the language of category theory and homological algebra such as extension functors, torsion functors, and spectral sequences. It continues with a discussion of group cohomology, Lie algebra cohomology, and/or sheaf cohomology. It also gives some applications in various fields of mathematics and introduces recent topics such as derived categories. |

3341.714 | Topics in Algebraic Geometry | 3 | 3 | 0 | This course introduces relevant topics which vary each semester. |

3341.715 | Topics in Algebra | 3 | 3 | 0 | This course introduces relevant topics which vary each semester. |

3341.724 | Topics in Numerical Analysis | 3 | 3 | 0 | Topics relevant to the subtitle fixed in advance are studied. |

3341.725 | Advanced Numerical Linear Algebra | 3 | 3 | 0 | This course aims to teach advanced numerical linear algebra. We will cover the direct methods(i.e., the Frontal methods for matrix problems), LU, QR, SVD, decomposition methods for banded matrices, Jacobi iteration, Gauss-Seidel iteration, ADI method, Conjugate Gradient Methods, Lanczos Methods, Preconditioning, Eigenvalue Search Problems. The student will be expected to implement those algorithms with one of following program languages: Fortran, HPF, C/C++, JAVA, Matlab, Maple, Mathematica. |

3341.726 | Topics in Partial Differential Equations | 3 | 3 | 0 | Partial differential equations arise as basic models describing natural and social phenomenon in physics, engineering, biology, and economics. In addition recently some of partial differential equations have been designed and investigated as tools in the other area in mathematics, for example Geometry and Topology as well as Analysis. The topics include basic materials and recent development in P.D.E. and related areas. Each topic will be posted prior to the class. This class requires basic knowledge of the multivariable, real analysis and partial differential equations. |

3341.751 | Topics in Applied Mathematics | 3 | 3 | 0 | Topics relevant to the subtitle fixed in advance are studied. |

3341.752 | Computational Number Theory | 3 | 3 | 0 | This course deals with computational aspects of algebraic number theory. First, we learn basic computations including Euclid algorithm, Legendre Symbol, square-root computation, lattice reduction algorithm, and polynomial root finding algorithm. Second, we learn several algorithms for primality tests, integer factorizations, and discrete logarithms computations. Last, we also learn how to compute norm, trance, order, regulator, and class numbers in number fields. |

3341.753 | Topics in Mathematical Methods of Probability | 3 | 3 | 0 | Rigorous mathematical treatment of probability theory is the main objective
of this course. When time permits, a select topic from various areas of
application may also be covered. The content of the course may be selected
from the following topics: Measure theoretic foundation of probability theory, convergence theorems, Markov processes, Martingale theory, Brownian motion, stochastic integral, stochastic differential equations, various stochastic processes, stochastic analysis, Malliavin calculus, applications to physical sciences, biological sciences, social sciences and engineering |

3341.754 | Topics on Advanced Mathematical Physics | 3 | 3 | 0 | Mathematics and Physics have always been closely intertwined, with developments in one field frequently inspiring the other. This course is concerned with mathematical problems in fluid mechanics, statistical mechanics, quantum field theory, string theory and, in general, with the mathematical foundations of theoretical physics. This includes such subjects as Navier-stokes equation, Boltzmann equation, quantum mechanics, the theory of quantum field theory (both in general and in concrete models), mathematical foundation of string theory and mathematical developments in functional analysis and algebra to which such subjects lead. |

3341.755 | Mathematical models and methodology of neuroscience is presented. The content of the course may be selected from the following topics: ODE; dynamical systems, coupled oscillators and synchronization, PDE, Hodgkin-Huxley model and its variants, bio-fluid dynamics, numerical computations, Fourier analysis and signal processing, visual models, auditory models, speech models, hierarchical architecture of central nervous system, cognition models, models for learning and memory. | ||||

3341.761 | International videoconference lectures or intensive lectures by a short term visiting professors which exceeds 16 hours, can be offered as an official course with a credit, and the topic of the course is to be determined by the lecturer. | ||||

3341.762 | "International videoconference lectures" or intensive lectures by a short term visiting professors which exceeds 16 hours, can be offered as an official course with a credit, and the topic of the course is to be determined by the lecturer. | ||||

3341.781 | Calculus TA Seminar | 1 | 0 | 2 | This course offers training of? first year graduate teaching assistants for the undergraduate calculus courses to develop their teaching skills and to enhance their understanding of the course materials. |

3341.803 | Reading and Research | 3 | 3 | 0 | |

3341.721A | Topics in analysis | 3 | 3 | 0 | Analysis began with the invention of calculus and its rigorous study (Real and Complex Analysis) and has been developed to the following areas: the study on the space of functions(Functional Analysis), operators between spaces(Operator theory), Fourier series and harmonic functions (Harmonic Analysis), and its application to ordinary and partial differential equations. The topics include basic materials and recent development in Analysis and its related areas. Each topic will be posted prior to the class. This class requires basic knowledge of Mathematical Analysis. |

3341.731A | Topics in Geometry | 3 | 3 | 0 | Analysis began with the invention of calculus and its rigorous study (Real and Complex Analysis) and has been developed to the following areas: the study on the space of functions(Functional Analysis), operators between spaces(Operator theory), Fourier series and harmonic functions (Harmonic Analysis), and its application to ordinary and partial differential equations. The topics include basic materials and recent development in Analysis and its related areas. Each topic will be posted prior to the class. This class requires basic knowledge of Mathematical Analysis. |

3341.741A | Topics in Topology | 3 | 3 | 0 | It aims at understanding advanced knowledge and recent development in the research of manifolds and spaces. This course teaches one of the following topics in every semester: Low-dimensional manifolds theory, Homotopy and Homology theory, Characteristic classes, Differential topology, Geometric topology, Knot theory, Projective and affine manifolds, Hyperbolic 3-manifolds, Loop spaces, Seiberg-Witten theory, Gromov-Witten theory, Mirror symmetry, and so on. |

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