Homepage of Department of Mathematical Sciences
Address: Department of Mathematical Sciences, Seoul National University San56-1 Shinrim-dong Kwanak-gu Seoul 151-747, Korea
Tel:+82 2 880 6530~1 Fax:+82 2 887 4694 E-Mail: web@math.snu.ac.kr

Welcome to the Department of Mathematical Sciences of Seoul National University. Here is some information that you might be looking for. If you can read Korean, please click the Korean homepage here which has a lot more information and is updated more frequently. If there is any other question regarding us, do not hesitate to send us an e-mail.

¢Ã Introduction

The Department of Mathematics was established in the College of Liberal Arts and Sciences when the Seoul National University was founded in 1946. In 1975 when the university moved to the present campus, the Kwanak Campus, it was integrated with the Department of Applied Mathematics in the College of Engineering to become the Department of Mathematics in the College of Natural Sciences. At that time, Statistics and Computer Sciences became separate from the Department of Mathematics. Its name was changed to the Department of Mathematical Sciences in 1999.

The Department of Mathematical Sciences aims at education and research in the major areas of mathematics: Algebra, Analysis, Topology, Geometry, and Applied Mathematics. It is in charge of education in basic mathematics for non-math majors as well as in more specialized mathematics for math majors. It has two organizations for mathematical research: the Research Institute of Mathematics and the Global Analysis Research Center. They are centers of active research on diverse areas of mathematics.

In the Department of Mathematical Sciences at present, there are 40 faculty members, including 8 temporary members, 16 post-doctoral researchers, about 136 graduate students, and about 158 undergraduate students.

The Department of Mathematical Sciences has sole occupancy of the Sangsan Mathematical Building for the purpose of mathematical education and research. The building is the home of the Research Institute of Mathematics of Seoul National University, which is Korea's leading Center for research in diverse areas of mathematics education. It was built in 1998 through the donation of an alumnus. It has a Library of Mathematics which contains thousands of mathematical books and more than 200 journals. The library is used not only by faculty and students of the Department of Mathematical Sciences but also by mathematicians from other universities. The computer facilities of the Department of Mathematical Sciences are outstanding. It has a mini-computer system as the main system and every office has one or more high-specification PCs. It also has three computer laboratories solely for the purpose of mathematical research and education.

Alumni of the Department of Mathematical Sciences play an important role in our society. After graduation, more than half of the students enter the graduate school for more specialized mathematical studies and proceed to become experts not only mathematics but also in other academic fields such as statistics, computer sciences, physics, economics, and sociology. Others go on to utilize their mathematical ability in more practical fields such as finance, industrial research laboratories, and educational institutes.

A new era of information technology is dawning in today's world and, in such an environment, the role of advanced mathematics is constantly assuming greater importance. In this brave new world, ever increasing contributions are expected from our graduates.

¢Ã Transportation from Airport

Please refer to SNU homepage first to find out how to get to the campus from Incheon international airport. Once you get inside of campus, you can find us at building # 27 and #129 (see the campus map).

¢Ã Staff Office

Address School of Mathematical Sciences, Seoul National University San56-1 Shinrim-dong Kwanak-gu Seoul 151-747, Korea Location Bld. 27, Room 403
Phone +82 2 880 6530~1 Fax +82 2 887 4694
E-Mail math@cns.snu.ac.kr Homepage www.math.snu.ac.kr
Office Hour 09:00 - 18:00 (Monday through Friday)
Office
Members		 Lee, Kook-hyun (Department Administrator) E-mail gelkh@math.snu.ac.kr
Choi, Suk-hee sukhee1@snu.ac.kr
Oh, Hai-ja ohhaija@math.snu.ac.kr

¢Ã TA Office

Address School of Mathematical Sciences, Seoul National University San56-1 Shinrim-dong Kwanak-gu Seoul 151-747, Korea Location Bld. 27, Room 416
Phone +82 2 880 6555 Fax  
E-Mail taoffice@math.snu.ac.kr Homepage www.math.snu.ac.kr/taoffice
Office Hour 09:00 - 18:00 (Monday through Friday)
Office
Members		 Kim, Woochan E-mail wckim@math.snu.ac.kr
Kim, Jaewoong kim2@math.snu.ac.kr
Kim, Joonhyung calvary@snu.ac.kr
Cho, Jinseok jindol@math.snu.ac.kr

¢Ã Mathematical Computation Lab

Address School of Mathematical Sciences, Seoul National University San56-1 Shinrim-dong Kwanak-gu Seoul 151-747, Korea Location Bld. 129, Room 104
Phone +82 2 880 5731 Fax  
E-Mail root@math.snu.ac.kr Homepage mcl.snu.ac.kr
Office Hour 09:00 - 18:00 (Monday through Friday)
Lab
Members		 Prof. Sheen, Dongwoo (Head of the lab) E-mail sheen@math.snu.ac.kr
Chung, Cheol-ho (Administrator) chiro@mcl.snu.ac.kr

¢Ã Mathematics Library

Address School of Mathematical Sciences, Seoul National University San56-1 Shinrim-dong Kwanak-gu Seoul 151-747, Korea Location Bld. 129, Room 501
Phone +82 2 880 5739 Fax +82 2 888 4439
E-Mail lib@math.snu.ac.kr Homepage www.math.snu.ac.kr/lib
Office Hour

09:00 - 18:00 (Monday through Friday)

Office
Members		 Prof. Lee, In-Sok (Head of the library) E-mail islee@math.snu.ac.kr
Na, Hyeran (Librarian) lib@math.snu.ac.kr
Description The library has a collection of 12,000 monographs, 6,000 bound journals, and subscribes to 152 mathematics-related journal titles. There are 63 seats for library users. Catalogue can be searched through SNU main library(http://library.snu.ac.kr/).

¢Ã Global Analysis Research Center

Address Global Analysis Research Center, College of Natural Sciences, Seoul National University Seoul 151-747, Korea Location Bd. 129, Room 305
Phone +82 2 880 6562 Fax +82 2 877 8435
E-Mail garc@math.snu.ac.kr Homepage garc.snu.ac.kr
Office Hour 09:00 - 18:00 (Monday through Friday)
Office
Members		 Prof. Kim Sang Moon(Head of the center) E-mail ckhan@math.snu.ac.kr
Cho, Mikyung (Administrator) mkcho@math.snu.ac.kr
Kim, Sunwoo swkim73@math.snu.ac.kr
Description

The Global Analysis Research Center (GARC) was initiated in 1991 as one of the Science Research Center funded by the Korea Science and Engineering Foundation (KOSEF), and it was given an Excellent mark by KOSEF at its first three year evaluation in 1993. Through continued effort, GARC's output has increased in quality and quantity, and it is now recognized world-wide as one of the centers of intense mathematical activities.
* The KOSEF funding has been terminated as of the year 1999. However, the Center continues research activities such as managing the website, publishig lecture notes & preprints for the time being.

¢Ã Research Institute of Mathematics

Address Research Institute of Mathematics, College of Natural Sciences, Seoul National University Seoul 151-747, Korea Location Bd. 129, Room 305
Phone +82 2 880 6562 Fax +82 2 877 8435
E-Mail rim@math.snu.ac.kr Homepage www.rim.snu.ac.kr
Office Hour 09:00 - 18:00 (Monday through Friday)
Office
Members		 Prof. Choi, Hyeong-In(Director) E-mail hichoi@math.snu.ac.kr
Cho, Mikyung (Administrator) mkcho@math.snu.ac.kr
Kim, Sunwoo swkim73@math.snu.ac.kr
Description

The Research Institue of Mathematics was launched on January 5, 1990 as an affiliate of College of Natural Sciences, Seoul National University. Its main activites consist of arranging seminars on pure and applied mathematics, supporing industrial-academic cooperation regarding mathematics, publicating lecture notes and preprint, and promoting scholarly exchange nationwide as well as abroad.

¢Ã Undergraduate Course Listing

code
course title
course description
006.001 The World of Mathematics Early mathematics, numbers and algebra, the number system, vectors and matrices, geometry and plane diagrams, the parallel axiom, functions and analysis, limits and continuity, set theory, the formalization of mathematics are introduced.
010.101 Calculus 1 As a basic mathematics course for students in science and engineering, properties of real numbers, series, Taylor expansions, vectors, matrices and determinants, curves, and their applications are discussed.
010.102 Calculus 2 As a sequel to of "Calculus 1", derivatives and integrals of several variable functions, vector fields, Green theorem and Stokes theorem and their applications are discussed.
010.103 Honor Calculus and Practice 1 As an honor course for "Calculus 1", properties of real numbers, series, Taylor expansions, vectors, matrices and determinants, curves are discussed in detail with depth.
010.104 Honor Calculus and Practice 2 As a sequel to of "Honor Calculus and Practice 1" and an honor course for "Calculus and Practice 2", derivatives and integrals of several variable functions, vector fields, Green theorem and Stokes theorem are discussed in detail with depth.
010.105 Calculus for life Science 1 As a basic mathematics course for students in life science, differential equations describing various natural phenomena related to life science and their solutions are introduced. For example, a system of differential equation model for spread of epidemics and and its solution using successive approximation are discussed. Mathematical computer programming is used.
010.106 Calculus for life Science 2 As a sequel to of "Calculus for Life Science 1", periodic behavior of pendulum and dynamical systems, functions of several variables, series and approximations, Poisson distribution and Fourier series are discussed.
010.107 Calculus for Humanities and Social Sciences As a basic mathematics course for students in liberal arts and social sciences, set theory and logic, algebraic systems, matrices and determinants, vector spaces, basic calculus are discussed.
026.001 Mathematics and Civilization Throughout history, mathematics has been one of the most important part in the development of mental world and civilization. Topics for example: axioms for Geometry by Euclid, Calculus by Newton and Leibniz, concept of computation by Turing and von Neumann, art and mathematics, society and mathematics, science/technology and mathematics, Oriental/European culture and mathematics.
026.002 Mathematics in information Age The aim of this course is to help the educated public understand what role mathematics has played in the inception of the computing machinery and in the birth of the modern information-based society. Main focus will be given to the mutually reinforcing interaction between mathematics and computer. Furthermore, some issues of the science and technology of the modern society will be examined from the angle of mathematics as an enabler. Finally, in return, some speculation on the future direction of computer will also be given.
300.123 Basic Calculus Basic mathematics necessary for "Calculus and Practice 1" are studied.
300.203 Linear Algebra As a basic tool for all scientific study, vector spaces, bases and dimensions, linear transformations, matrices and determinants, eigenvalues and eigenvectors, diagonalization of matrices, Hamilton-Cayley theorem, inner product spaces, orthonormal bases, and their applications are discussed.
300.204 Differential Equation Natural and social phenomena are often represented by differential equations. Thus study of properties of solutions of various differential equations is very important in almost of all sciences. In this course we study the basics of the method of solving fundamental differential equations.
300.206 Introduction to Modern Mathematics We study how modern mathematics is developed and how future mathematics will look like. Topics for example: infinite sets, Russell's paradox, Turing machine, symmetries in various figures and phenomena, packing problem, calculus of variations, transcendental numbers, distribution of prime numbers, Riemann hypothesis.
300.246 Vector Calculus Differentiation and Integrations of vector-valued functions are treated. Topics include Differentiation of multi-variable functions, implicit function theorem, maxima and minima of multi variable functions, multiple integrations, Fubini theorem, change of variables in integrations, Green's theorem, Stokes theorem, and Gauss divergence theorems.
300.247 Function of Complex Variables We study complex differentiation and integration and properties of complex analytic functions: Moebius transformations, Exponential and logarithmic functions, Cauchy-Riemann equations, Harmonic functions, Taylor series, Line integrals, Cauchy integral formula, maximum principle Laurent series, residue theorem, real integrals by means of residue calculus.
881.001* Applied Mathematics 1 First order ODE, Linear ODE, power series solution of ODE, Sturm-Liouville theorem, Laplace transform, vector calculus are studied.
881.002* Applied Mathematics 2 As a sequel to of "Applied Mathematics 1", Fourier series and integral, complex analytic function, conformal mapping, Taylor series and Laurent series, residue theorem are studied.
881.003* Differential Equation Methods of solving ordinary differential equations, series methods, Laplace transform methods, Theorems on existence and uniqueness theorems are discussed.
881.004* Complex Variables The following topics will be covered: Cauchy-Riemann equations, Harmonic functions, Taylor series, Moebius transformations, Line integrals, Cauchy integral formula, maximum principle, Laurent series, real integrals by means of residue calculus, conformal mapping, Poisson integral formula, Dirichlet problem, Riemann's zeta function, etc.
881.006* Applied Mathematics Linear ODE, Power series solution of ODE, Fourier series, complex analytic functions, residue theorem are studied.
881.007* Introduction to Linear Algebra Vector spaces, linear transformations, bases and dimensions, matrices and determinants, eigenvalues and Hamilton-Cayley theorem, diagonalization of matrices, inner product spaces, Gram-Schmidt method, least square method are discussed.
881.008* Mathematical Analysis Sequence of continuous and differentiable functions, uniform convergence, Arzela-Ascoli theorem, Weierstrass theorem, power series, analytic functions, trigonometric series, Fourier series are studied.
881.009* Algebra and Applications Definitions and examples of groups, rings, modules and fields, their sub-structures, quotient-structures, homomorphisms and isomorphism theorems, Sylow theory, ideal theory, field extensions and Galois theory are discussed.
881.301 Modern Algebra 1 Definitions and examples of groups, rings, modules and fields, their sub-structures, quotient-structures, and homomorphisms are studied. Some important theorems and applications are introduced.
881.302 Modern Algebra 2 As a sequel to of "Modern Algebra 1", important theorems on groups, rings, modules and fields (Jordan-Hoelder theorem, Sylow theorems, Galois theorems, etc.) are proved and various applications are discussed.
881.303 Introduction to Differential Geometry 1 We study 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, Frenet-Serret formula are discussed.
881.304 Introduction to Differential Geometry 2 As a sequel to of "Introduction to Differential Geometry 1", we study surfaces in the 3-dimensional Euclidean space. Tangent plane, normal vector field, helicoid, surfaces of revolution, area of surfaces, surface integral, 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 are discussed.
881.308 Metric Spaces We cover general topological properties like limit, continuity, connectedness, disconnectedness, compactness in metric space. We also cover completeness as well as some elementary concepts of general topological space.
881.313 Sets and Mathematical Logic Several topics among elementary set theory, construction of natural numbers, integers, rational numbers and real numbers, axiom of choice, cardinals and ordinals, methods of proofs are studied.
881.319 Numerical Linear Algebra 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, conjugate gradient methods are covered.
881.320 Introduction to Numerical Analysis 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, numerical methods for finding zeros of polynomial are covered.
881.321 Mathematical Analysis 1 Basic properties of real numbers and limits of sequences, compact and connected sets, precise definitions of limit and continuity, uniformly continuous functions, Riemann integral, Riemann-Stieltjes integral, fundamental theorem of calculus are studied.
881.322 Mathematical Analysis 2 As a sequel to 'Mathematical Analysis1" uniform convergence of sequence of functions, Weierstrass approximation theorem, Arzela-Ascoli theorem, power series and analytic functions, trigonometric series, gamma function, Fourier series are studied.
881.401 Introduction to Topology 1 We cover basic properties of topological spaces, Tietze extension theorem, metrizability, Hausdorff space and separability, compact spaces, etc.
881.402 Introduction to Topology 2 As a sequel to of "Introduction to Topology 1", we cover topology on manifolds, first fundamental groups, covering spaces, etc.
881.408 Geometric Algebra 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, Lie groups and examples of Lie groups are discussed.
881.409 History of Mathematics Ancient mathematics (Babylon, Egypt, Greece), Oriental mathematics (China, India, Arab), mathematics of European middle ages, 17c mathematics, 18c mathematics, 19c geometry and algebra, 19c analysis, 20c abstract mathematics are discussed.
881.410 Introduction to Algebraic Geometry This course is intended for students who have mastered the basics of undergraduate abstract algebra. As an easy introductory course in algebraic geometry, the following topics will be covered: 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, resolution of singularities of projective algebraic curves.
881.423 Partial Differential Equations We introduce basic theories of partial differential equations. First order quasilinear PDE, local existence, uniqueness, Cauchy-Kovalevsky theorem, Laplace equation, maximum principle, Harnack's inequality, Hilbert space methods, variational principle are discussed.
881.424 Applications of Partial Differential Equations In this course we study how the theories of partial differential equations are applied to the problems in physics and mechanics. In particular we study the following topics: Dirac equations, Maxwell equations, self-dual equations in the nonlinear field theories, their soliton solutions, tensor analysis and the Einstein field equations. We also study the Navier-Stokes and the Euler equations arising in the mathematical fluid mechanics.
881.425 Real Analysis 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, Applications to Fourier series and integral are discussed.
881.427 Algebraic Coding Theory Notion of entropy and Shannon theory are introduced, and basic properties and error-correcting function of various codes (linear codes, cyclic codes, Hamming codes, Reed-Muller codes, etc.) are studied.
881.431 Fourier Analysis and Applications We study the classical theories of the Fourier series and the Fourier integrals. We also study the discrete cosine transform, fast Fourier transform, wavelet and the multiresolution analysis, wavelet transform and the Fourier transform, signal and the image process, applications to the inverse problems.
881.433 Number Theory and Cryptography Elementary number theory necessary is introduced first. Then encryption and decryption algorithms, complexity, security, advantages and disadvantages of various cryptosystems are discussed.
881.434 Chaos and Dynamical Systems Kepler motion, ecological problem, Hamiltonian system, stability and chaos, limit cycles, Poincare map, strange attractors are covered.
881.435 Financial Mathematics In this lecture, we present the core ideas and techniques that are currently used in mathematical finance. Among them are: discrete multi-period model, principles of arbitrage pricing theory, Black-Scholes model and formula, various financial derivative instruments, introduction to interest rate model, martingale measure and fundamental theorem of finance. Necessary basics of probability theory - Brownian motion, stochastic integral, Ito formula and Girsanov theorem, etc. - are covered.
* indicates courses for the students of other departments

¢Ã Graduate Course Listing

code
course title
course description
3341.501 Algebra 1 Algebraic structures (groups, rings, modules, and fields) and homological algebra are studied. Important theorems and their applications are introduced.
3341.502 Algebra 2 As a sequel to of "Algebra 1", field theory and Galois theory are studied and basics of commutative algebra, algebraic geometry and algebraic number theory are introduced together with various applications.
3341.503 Real Analysis Lebesgue measure and integration on the Euclidean space, product measure and Fubini theorem, complex measure and Radon-Nykodim theorem, Lebesgue decomposition, measure on topological spaces and Riesz representation theorem are discussed.
3341.504 Complex Analysis We first review basic theory of complex analysis: Cauchy-integral formula, convergence of power series, Taylor and Laurent series, residue theorem and applications, Schwarz lemma. And then study advanced topics: Poisson integral formula and boundary value problem for harmonic functions, partial fraction and Mittag-Leffler's theorem, infinite product and Weierstrass' theorem, normal family and Montel's theorem, Riemann mapping theorem.
3341.505 Diffrentiable Mainfolds Basic concepts and knowledge of differential manifolds and concrete examples are discussed. 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 on manifolds, Stokes's theorem, de Rham cohomology, Lie groups.
3341.601 Commutative Algebra To study varieties in algebraic view point, dimensions and depths (of rings and modules) and theorems related to them are discussed. Based on these, Cohen-Macaulay rings, Gorenstein rings, complete intersection rings and regular rings are discussed.
3341.602 Homological Algebra First, categories, functors, abelian categories are studied. And then extension functors and torsion functors are introduced. Kunneth formula, group cohomology, Lie algebra cohomology. and some theorems on spectral sequences are discussed.
3341.603 Functional Analysis 1 Basic properties of topological vector spaces, semi-norms and locally convex spaces, weak topologies, Banach-Alaoglu theorem, Krein-Milman theorem, dual spaces, Stone-Weierstrass theorem, spectral theorem of compact operators, Hilbert-Schmidt operators are discussed.
3341.604 Functional Analysis 2 As a sequel to "Functional Analysis 1", test functions and distribution spaces, Fourier transform, Paley-Wiener theorem, application to PDE, Banach algebras, Gelfand transform of commutative Banach algebras, spectral theorem of normal operators, Unbounded operators are discussed.
3341.605 Differential Geometry 1 Riemannian manifolds, metrics, connections, geodesics, parallelism, structure equation, completeness, curvature, Jacobi fields, first and second variations of length and volume are discussed.
3341.606 Differential Geometry 2 As a sequel to "Differential Geometry 1", relation between curvature and topology, comparison theorems, submanifold theory, general relativity, holonomy groups, minimal submanifolds, constant mean curvature surfaces, harmonic maps, isoperimetric inequalities, Lagrangian geometry are discussed.
3341.607 Algebraic Topology 1 This course covers the basic topics in algebraic topology including the theory of fundamental group and covering space, homotopy theories
3341.608 Algebraic Topology 2 This is a sequel to "Algebraic Topology1" and covers the topics such as CW-complex, cohomology, orientation, Poincare duality, and cup product.
3341.611 Algebraic Number Theory For various number fields, their integer rings, ideals, ramifications, Dirichlet's unit theorem, valuations, localizations, ideal class groups and class numbers are discussed.
3341.612 Lie Algebra Semisimple Lie algebras, Cartan decomposition, Weyl's theorem, root systems and classification, Weyl groups, classical simple Lie algebras, universal enveloping algebras, PBW theorem, representation theory and Verma module, Chevalley groups are discussed.
3341.613 Algebraic Geometry This course covers basics in algebraic geometry intended for graduate students beginning to study algebraic geometry. 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 Representation of C*-algebras, basics for 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 are discussed.
3341.622 Analytic Functions of Several Variables Hartog's phenomenon, domain of holomorphy and the Levi problem, integral formula for polydisks, Bochner-Martinelli integral, Bergman kernel, plurisubharmonic functions, pseudo-convexity, Hoermander's solution of d-bar problem are discussed.
3341.623 Nonlinear Functional Analysis Set valued function, fixed point theory, monotone operator, nonlinear semigroup, semicontinuous function, various differentiability variation principle, convex analysis, degree theory, nonexpansive mapping, analysis on locally convex space, convex optimization, application to partial differential equations are discussed.
3341.625 Harmonic Analysis Basic properties of topological groups, Haar measure on the locally compact group, convolutions for functions and measures, unitary representation of the locally compact group, Fourier transform and Pontyagin's duality theorems, representation of compact group and Peter-Weyl theorem, Tanaka-Krein duality theorem are discussed.
3341.626 Numerical Analysis 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, mixed finite elements are discussed.
3341.631 Lie Groups We cover the basic theory of Lie groups and topics such as homogeneous space, covering group, sub-Lie group, Campbell-Hausdorff theorem, the structure of compact Lie groups, and PBW theorem.
3341.633 Theory of Complex Mainfords Special properties of complex manifolds are studied. Topics include: complex structures, complexified tangent bundle, holomorphic tangent bundle, Dolbeault cohomology, Kaehler manifold, deformation of complex structures, Kodaira embedding theorem, etc.
3341.635 Theory of Partial Differential Equations 1 We study the classical theories of the Fourier series and the Fourier integrals. We also study the discrete cosine transform, fast Fourier transform, wavelet and the multiresolution analysis, wavelet transform and the Fourier transform, signal and the image process, applications to the inverse problems.
3341.636 Theory of Partial Differential Equations 2 As a sequel to "Theory of Partial Differential Equations 1", nonlinear partial differential equations, fixed point methods, variational methods, method of upper and low solutions, regularity problems of the nonlinear PDE, and concrete equations - Navier-Stokes equations, Euler equations, nonlinear wave equations, and the Einstein's field equations - are discussed.
3341.641 Differential Topology We cover definition of differentiable manifolds, Sard theorem, transversality, Euler number, integration on manifolds and differential forms on manifolds
3341.642 Geometric Topology We cover 3 dimensional topology, application of minimal surface theory to 3 manifold topology, Alexander invariant and rigidity of symmetric spaces
3341.643 Mathematical Logic and Theory of Computation The course will cover fundamentals of mathematical logic and theory of computation.
3341.644 Advanced Discrete Mathematics Basic notions of discrete structures are introduced, and related mathematical logic, set theory, algebra, combinatorics, graph theory, algorithms, and various applications are discussed.
3341.714 Topics in Algebraic Geometry Topics relevant to the subtitle fixed in advance are studied.
3341.715 Topics in Algebra Topics relevant to the subtitle fixed in advance are studied.
3341.716 Topics in Applied Algebra Topics relevant to the subtitle fixed in advance are studied.
3341.721 Topics in Analysis 1 Topics relevant to the subtitle fixed in advance are studied.
3341.722 Topics in Analysis 2 Topics relevant to the subtitle fixed in advance are studied.
3341.724 Topics in Numerical Analysis Topics relevant to the subtitle fixed in advance are studied.
3341.731 Topics in Geometry 1 Topics relevant to the subtitle fixed in advance are studied.
3341.732 Topics in Geometry 2 Topics relevant to the subtitle fixed in advance are studied.
3341.741 Topics in Topology 1 Topics relevant to the subtitle fixed in advance are studied.
3341.742 Topics in Topology 2 Topics relevant to the subtitle fixed in advance are studied.
3341.751 Topics in Applied Mathematics Topics relevant to the subtitle fixed in advance are studied.
3341.803 Reading and Research  
3341.902* Advanced Applied Mathematics 1 Finite dimensional vector spaces, complete metric spaces, approximation in Hilbert spaces, analysis and numerical methods for integro-differntial equations, Green's function, least squares method, calculus of variations, complex variable theory are covered.
3341.903* Advanced Applied Mathematics 2 As a sequel to "Advanced Applied Mathematics 1", transformation and spectral theory, scattering theory partial differential equations, numerical solutions of partial differential equations, inverse scattering theory, numerical methods for inverse scattering problems, asymptotic expansion, perturbation theory are covered.
3341.904* Dynamics Systems Nonlinear dynamics, chaos, one dimensional flow, two dimensional flow, phase portraits, rabbits versus sheep, Poincare map, logistic maps, fractals, strange attractors are discussed.
3341.905* Theory of Approximations Polynomial interpolation, splines, LU-decomposition, least squares approximation, QR-factorization, orthogonal polynomials, minimax approximation, PAD¡ç, Ai¡ç (B approximation, Toeplitz systems) are discussed.
3341.907* Advanced Numerical Analysis This course covers: introduction to parallel computing, parallel algorithms for linear systems, introduction to domain decomposition methods, Schwarz methods, multilevel methods, full multigrid method, substructuring methods, preconditioners, etc.
3341.908* Non-Deterministic Mathematics Probability, Markov chain, ergodic theory, random walks, hitting time, martingale, Brownian motion, potential theory, electrical network are discussed.
3341.909* Optimization Theory and Practice Optimality conditions, gradient methods, Newton methods, method of conjugate directions, minimax problems, constrained optimization, quadratic programming, optimal control are discussed.
* indicates courses for the students of other departments

go to [top]