| G10464 |
AlgebraⅠ |
Groups, classification of finite abelian groups, theory of commutative rings, theory of categories. |
| G10744 |
Algebraic Topology |
We study Fundamental groups, Covering space, Simplicial and Singular homology, Cohomology theory. |
| G10469 |
Real AnalysisⅠ |
Measures, Integration, Decomposition of measures Lp-spaces, Complex measure, integration on product space. |
| G10749 |
Complex AnalysisⅠ |
Complex numbers, Complex functions, Analytic functions, Cauchy's theorem and formula, Harmonic functions, Poisson integral, Conformal Mapping, Normal family. |
| G10472 |
Differential Topology |
Differential Topology, tangent spaces, Immersion, Submersion, Topology on function spaces, Approximations, Morse-Sard theorem, Transversality. |
| G10752 |
Numerical Differential Equations |
Numerical Solutions of Ordinary Differential Equations: Initial Value Program, Stiff Systems, Boundary Value Problems, Well-posed Initial Value Problems for Partial Differential Equations. The Choice of norms and Stability definitions. The Fourier and energy method for finite difference Schemes. Discussion of finite element methods. |
| G10474 |
Algebraic Number TheoryⅠ |
Ring of integers of field extentions, closure in a field extension, prime factorization ideals, inertia group, ideal class group, cyclotomic fields, Minkovski's volume formula, generalized ideal class. |
| G10756 |
AlgebraⅡ |
Theory of fields, Galois theory, simple and semisimple rings, theory of group representations. |
| G10476 |
Commutative AlgebraⅠ |
Introduction to commutative algebra: rings, modules, ideals, Noetherian rings, primary decomposition, Artinrings, valuation rings, polynomial and power series rings. |
| G10763 |
Commutative AlgebraⅡ |
Commutative ring theory: chain conditions, local rings, dimension theory, regular rings, regular sequences, complete local rings, Cohen-Macaulay rings. |
| G10481 |
Algebraic Curves |
Affine algebraic curves, coordinate rings, projective curves, Bezout's theorem, Riemann Roch theorem. |
| G10766 |
Applied Algebra |
We study coding theory and cryptography as applications of algebra and number theory. |
| G10535 |
Selected Topics in AlgebraⅠ |
Discuss interesting recent papers in algebra. |
| G10771 |
Lie Group |
Lie groups, Lie algebras, Lie subgroups, exponential maps, homomorphisms, adjoint representations, bilinear form, derivation homogeneous manifolds. |
| G10488 |
Linear and Multilinear Algebra |
Bilinear form, duality, sesquilinear form, orthogonal sum, quadratic map, symmetric form, orthogonal basis, hyperbolic space, Witt's theorem, Witt group, Clifford algebra, alternating form, Paffian, Hermitian form, spectral theorem, tensor algebra, alternating product, symmetric product, Euler Grothendieck ring. |
| G10774 |
Riemann Surface |
Riemann surfaces, holomorphic and meromorphic functions and differentials, singularities, Riemann-Hurwitz formula, Riemann-Roch theorem. Jacobian varieties and Abel's theorem, The Riemann bilinear form, Jacobi inversion theorem, moduli space. |
| G10491 |
Differential GeometryⅠ |
Exterior algebra, Submanifolds of PRn, Moving frames, Differential Manifolds, Vector Bundles, Forms and Metrics, Integration on manifolds, Connections. |
| G10777 |
Differential GeometryⅡ |
Applications of Differential GeometryⅠ. |
| G10610 |
Selected Topics in TopologyⅠ |
Chern classes, Complex vector bundle, Splitting principle, Frag Manifold, Pontrjagin classes, Grassmann manifold, Universal bundle. |
| G10635 |
Riemannian GeometryⅠ |
Toponogove's Theorem,Homogeneous space, closed geodesics cut locus |
| G10782 |
Real AnalysisⅡ |
Interpolation of Lp-spaces, convolutions, Fourier transforms. |
| G10497 |
Fourier Analysis |
Maximal functions, Singular integrals, Poisson integral, Littlewood-Paley theory and multiplies. |
| G10788 |
Introduction to Functional Analysis |
Banachspaces, Hilbertspaces, Linearfunctionalsandoperatorsinsuchspaces, Duality and spectral theory. |
| G10792 |
Topics in Analysis |
Selected topics in real analysis and functional analysis. |
| G10507 |
Topics in Functional Analysis |
Banachspaces, Hilbertspaces, Linearfunctionalsandoperatorsinsuchspaces, Duality and spectral theory. |
| G10511 |
Complex AnalysisⅡ |
Infinite product, Entire and meromorphic functions, Analytic continuation, Weierstrass factorization theorem, Mittag Lefflers' Theorem Univalent functions. |
| G10799 |
Topics in Complex Variables |
Selected topics in complex variables. |
| G10514 |
Approximation Theory |
Polynomial interpolation, best approximation, quasi-interpolation, least square method, spline and wavelet theories will be studied as well as some topics related to image processing will be covered |
| G10517 |
Numerical Linear Algebra |
Matrix computation, QR factorization, Given's rotation, Householder method, Cholesky decomposition, Numerical methods for special type matrices: Band matrix, Sparse matrix, Triangular matrix, Eigenvalue problem, Nonlinear equations, Iterative methods. |
| G10521 |
Applied Differential Equations |
Existence and stability of the nonlinear system, second-order linear equations, the introduction of spectral theory and spectral functions, geometrical interpretation of nonlinear system in 2-dimensional space, stability, bifurcation, equations with periodic coefficient. |
| G10801 |
Applied Partial Differential Equations |
Properties of first order PDEs, classification of PDEs, Laplace, equation, Fourier series and integrals, hyperbolic system, semigroups for parabolic equations, a priori estimate for elliptic equation, fundamental solutions, spectral analysis, nonlinear problems. |
| G10523 |
Scientific Computing |
Design and implementation of scientific programs, Efficient algorithms and program design, efficiency, reliability, portability, Simulation of physical, biological, chemical systems. |
| G10525 |
Number theory and Cryptography |
We study the basic number theory such as Fermat theorem, finite field and modular properties. We study the complexity theory, public-key systems like RSA and Elliptic curve cryptosystem. |
| G10802 |
Advanced Numerical Analysis |
Approximation of data, polynomial interpolation, iterative methods, nonlinear equations, line search, univariate minimization, convergence issues. |
| G10526 |
Cryptography and Communication |
Motivation for studying cryptography, definitions and terminology are illustrated with simple examples. In the public-key cryptography, we learn the modern cryptography schemes based on the factoring problem, discrete log problem and pairings. |
| G10815 |
Elliptic Curve Cryptosystems |
We study the structure of the elliptic curve, discrete log problem of the elliptic curve, elliptic curve cryptosystem and implementation. |
| G10819 |
Image Processing |
General mathematical the orison Image reconstruction, compression, edge detection will be studied. In particular, the following topics will be covered: Fourier transform, spline, subdivision, refinable function, Multi-resolution, wavelet and general theories of approximation. |
| G10824 |
Topics in Modern Cryptography |
We study the recently suggested cryptosystem and provable security. |
| G10528 |
Mathematical Finance |
In recent years, the mathematical complexity has been so involved in financial markets that understanding financial markets and financial derivatives require deep analytics of mathematics, in particular, stochastic mathematics. The aim of this course is to make students familiar with various financial derivatives and to equip them with mathematical tools for pricing financial derivatives. With a short introduction of discrete-time finance, the course covers stochastic integration, Ito's Lemma, Black-Scholes formula of option pricing, the pricing of various options such as American options and knok-out options, and the applications to risk management and capital structure. |
| G10529 |
Ring Theory |
Radical theory, primitive rings and density theorem, simple algebra and Wedderburn theorem, division rings. |
| G10530 |
Theory of Commutative Algebra |
Various recent topics in commutate ring theory: Rees algebras, Cohen-Macaulay rings, Grothendieck rings, Valuation theory. |
| G10532 |
Homology Theory |
This course concerns modules, categories and functors, homology functors, torsion and extension functors, homology and cohomology theories of groups and monoids. |
| G10538 |
Algebraic GeometryⅠ |
Affineandprojectivevarieties, nonsingular varieties, schemes, separated and proper schemes, projective schemes |
| G10542 |
Arithmetic Geometry |
Abelian varieties, complex Tori, dual abelian varieties, Cohomology of abelian varieties, theory of heights, Modell-Weil's Theorem for abelian varieties, heights and metrized line bundles, distance functions and logarithmic singularities. |
| G10544 |
Theory of Modular functions |
Weierstrass function, discriminant, Klein'smodularfunction, Mobiustransformation, modulargroup, modularfunction, J-function, Eisensteinseries, Dedekindfunction, congruencesubgroup, Radmacherseries, modularformsofweightk, Heckeoperator, Derichletseries, Kroneckertheorem, Riemannzeta function, Bohr's function. |
| G10553 |
Elliptic curvesⅠ |
Weierstrass equation, isogenies, Tatemodule, endomorphism ring, formal group, elliptic curve over the complex numbers and finite fields, Hasseinvariant, elliptic curves over a local field, ModellWeil theorem, height, integral points on elliptic curves, Selmer group, Shafarevich group, Fermat's last theorem. |
| G10556 |
Functional AnalysisⅠ |
Topological vector spaces, Locally convex spaces, Linear functionals and operators, Weak topologies, Spectral theory, Banach algebras, and C*-algebras. |
| G10559 |
Harmonic AnalysisⅠ |
Convergence of Fourier series, Summability, Differentiation of integrals, Oscillation of functions, Ap-weights. |
| G10563 |
Theory of Partial Differential EquationsⅠ |
Local existence theorem, Laplace operator, Dirichlet and Newmann problems via integral equations, Heat operator, Sobolev space, etc. |
| G10567 |
Selected Topics in Harmonic AnalysisⅠ |
Selected current topics in Harmonic analysis. |
| G10571 |
Theory of Singular IntegralsⅠ |
The singular integral of Calderon-Zygmund and Riesz transforms Poisson integrals, Ap-weight and singular integral. |
| G10574 |
Operator TheoryⅠ |
General spectral theory, Classes of compact operators and Fredholm operators, Classes of Normal, Subnormal, Hyponormal operators, Classes of unbounded operators. |
| G10579 |
Selected Topics in Functional AnalysisⅠ |
Selected current topics in Functional analysis. |
| G10594 |
Theory of Hp SpaceⅠ |
Basic structures of Hp functions, Conjugate functions, Mean Growth and Smoothness, Hp as a Linear space, etc. |
| G10601 |
Analytic Function of Several Complex VariablesⅠ |
Weierstrass preparation theorem, Analytic spaces, Stein spaces and sheaf theory, etc. |
| G10603 |
Selected Topics in Complex Analysis |
Selected current topics in Complex analysis. |
| G10607 |
Algebraic TopologyⅠ |
Westudysingularhomologytheory, Mayer-Vietoriessequence, CW-complex, Application of homology theory and Cohomology theory. |
| G10616 |
Index TheoryⅠ |
Characteristic classes, Index Theorem, de Rham Operator, Hodge Operator, Dolbeault operator, Dirac operator, K-theory. |
| G10628 |
Homotopy TheoryⅠ |
Higher homotopy group, Hurewicz isomorphism theorem, CW complex, Spectrum, The relation between homotopy and ordinary homology, obstruction theory. |
| G10631 |
Differential TopologyⅠ |
Differentiable structure, Tangent bundle, Embedding, Strong and weak topology, Analytic approximation, Morse-Sard theorem, Transversality. |
| G10633 |
Bundle TheoryⅠ |
Coordinate bundle, fiber bundle, Construction of bundle, product bundle, differentiable manifold and tensor bundle, principal bundle, Associated bundle, Construction of cross-section, Bundle homotopy. |
| G10637 |
Global AnalysisⅠ |
Jet Bundle, Differential operator, Banach space valued section Functors, Derivative Functor, Dual Functor, Examples, moduli space. |
| G10640 |
Harmonic Theory |
Differential operators, Laplacian, Hodge theorem, Green theorem, Weitzenbock formula, Chern's formula for the Laplacian. |
| G10645 |
Numberical Method and Scientific ComputationⅠ |
Error analysis, Interpolation, Extrapolation, Systems of linear equations, Iterative methods, Convergence analysis, Mathematical stability and ill-conditioning, Systems of nonlinear equations. |
| G10648 |
OptimizationⅠ |
Fundamentals of unconstrained optimization, line search methods, trust-region methods, conjugate gradient methods, practical Newton method, calculating derivatives, quasi-Newton methods, large-scale quasi-Newton method and partially separable optimization. nonlinear least squares problems, nonlinear equations. |
| G10652 |
Principles of Applied MathematicsⅠ |
Finite-dimensional vector spaces, function spaces, Integral equations, Differential operators, Calculus of variations, Complex variable theory. |
| G10654 |
Selected Topics in Numerical Analysis |
Linear algebra on vector and parallel processing, High-performance computers, Implementation, Performance, Direct solution of sparse linear systems on parallel computers, Iterative solution of sparse linear systems. |
| G10657 |
Graph Theory |
Trees, Connectivity, Euler tours and Hamilton cycles, Matchings, Edge colorings, Independent sets, Vertex coloring, Planer graphs. |
| G10661 |
Initial value problem in Differential Equations |
Design and analyze time-dependent numerical methods, analysis of stability using CFL, Fourier, energy estimate, properties of existing methods, handling of physical and artificial boundaries. |
| G10663 |
Mathematical Modeling |
Formulation of mathematical modeling and analysis based on optimization, covers optional control, probability, queues, difference equations, differential equations, dimensional analysis, traffic flow, economic model. |
| G10665 |
Modern Cryptography I |
The secret key system, public-key system, quantum cryptography, zero-knowledge problem, multi-party key agreement, standard cryptography and current schemes. |
| G10669 |
Public Cryptography I |
Applied cryptographic schemes based on RSA and ECC such as ID-based scheme for pairings. |
| G10675 |
Cryptography Algorithm I |
We analyze and implement the schemes in the symmetric key systems and the stream key systems. |
| G10676 |
Computer Algorithm |
This subject covers the algorithms related to the design and analysis. We also cover P, NP problems etc. |
| G10678 |
Selected Topics in Cryptography I |
We discuss the recent hot results in the topics of public-key cryptography or cryptanalysis. |
| G10682 |
Coding theory and cryptography I |
We study coding theory and its application to cryptography. |
| G10684 |
Selected topics in Number Theory I |
Basic foundations on some selected topics of number theory are covered. For the possible topics, we can select any specialized area of number theory such as Function Field Arithmetic, the general theory of Drinfeld modules, Class Field Theory, Class group structures of number fields, Class group structures of global function fields and so forth. |
| G10760 |
Algebraic Number TheoryⅡ |
Based on the study of Algebraic number theory I, we learn about local fields, totally ramified extension, unramified extension, tamely ramified extension, valuation, global fields. |
| G10792 |
Topics in Analysis |
Selected topics in real analysis and functional analysis. |
| G10799 |
Topics in Complex Variables |
Selected topics in complex variables. |
| G10831 |
Group Representation |
Schur index, Projective representation, Cartan invariants, Green's theorem, Brauer's main theorems. |
| G10839 |
Selected Topics in AlgebraⅡ |
Discuss interesting recent papers in algebra. |
| G10845 |
Algebraic GeometryⅡ |
Cohomology on schemes, Serre duality, dualizing sheaf, ruled surface, monoidal transformation, cubic surfaces, classification of algebraic surfaces. |
| G10852 |
Algebraic K-Theory |
Projective modules, K0-groups, K1-groups, Steinberg group, Matsumoto's theorem, K2-groups of Milnor. |
| G10858 |
Complex Algebraic Geometry |
Analytic varieties, the De Rham and Dolbeault theorems, Complex vector bundles, the Hodge theorem, Kahler manifolds, Kodaira vanishing theorem, Kodaira embedding theorem, Riemann surfaces and their Jacobians. |
| G10867 |
Elliptic curvesⅡ |
Weierstrass equation, isogenies, Tatemodule, endomorphism ring, formal group, elliptic curve over the complex numbers and finite fields, Hasseinvariant, elliptic curves over a local field, good and bad reduction, ModellWeil theorem, height, integral points on elliptic curves, Selmer group, Shafarevich group, Fermat's last theorem. |
| G10877 |
Functional AnalysisⅡ |
Topological vector spaces, Locally convex spaces, Linear functionals and operators, Weak topologies, Spectral theory, Banach algebras, and C*-algebras. |
| G10885 |
Harmonic AnalysisⅡ |
Convergence of Fourier series, Summability, Differentiation of integrals, Oscillation of functions, Ap-weights. |
| G10892 |
Theory of Partial Differential EquationsⅡ |
Local existence theorem, Laplace operator, Dirichlet and Newmann problems via integral equations, Heat operator, Sobolev space, etc. |
| G10897 |
Selected Topics in Harmonic AnalysisⅡ |
Selected current topics in Harmonic analysis. |
| G10916 |
Theory of Singular IntegralsⅡ |
The singular integral of Calderon-Zygmund and Riesz transforms Poisson integrals, Ap-weight and singular integral. |
| G10920 |
Operator TheoryⅡ |
General spectral theory, Classes of compact operators and Fredholm operators, Classes of Normal, Subnormal, Hyponormal operators, Classes of unbounded operators. |
| G10925 |
Selected Topics in Functional AnalysisⅡ |
Selected current topics in Functional analysis. |
| G10932 |
Theory of Hp SpaceⅡ |
Basic structures of Hp functions, Conjugate functions, Mean Growth and Smoothness, Hp as a Linear space, etc. |
| G10939 |
Analytic Function of Several Complex VariablesⅡ |
Weierstrass preparation theorem, Analytic spaces, Stein spaces and sheaf theory, etc. |
| G10944 |
Algebraic TopologyⅡ |
We study Orientation of manifold, Singular cohomology, Cup and Cap product, Poincare duality, Alexander duality, Lefschtz, Product and Lefschtz fixed point theorem, Applications Cohomology theory and Homotopy theory. |
| G10950 |
Selected Topics in TopologyⅡ |
Spectral Sequence Of A Filtered Complex(fiber bundle), Gysinsequence, Leray'sconstruction, Homology Spectral Sequence, path fibration, EilenbergMaclanespaces, transgression, Postnikov Approximation, some computation of fundamental group of spheres. |
| G10956 |
Topological K-Theory |
Operation on vector bundles, G-bundles over G-manifolds, Bott periodicity Theorem, Cohomology Properties of K, Computation of K*(X) isomorphism, Adams operation. |
| G10960 |
Index TheoryⅡ |
Topological G-index, G-index theorem, Atiyah-Singer Fixed point Theorem, Lefschetz Fixed point Theorem, Holomorphic Fixed point Theorem, G-signature Theorem, G-Spin Theorem. |
| G10965 |
Homotopy TheoryⅡ |
Cohomology operations, Construction of the Steenrod squares, Steenrod algebra, Fibre spaces, Cohomology of K(Π,n), n-type and Postnikov system, Spectral seqences and Postnikov systems, Spectral sequences. |
| G10971 |
Differential TopologyⅡ |
Vector bundle, Classification of vector bundle, Coller and Tublar neighborhood, Degree of Map, Intersection number, Euler Characteristic, Symplectic topology. |
| G10973 |
Bundle TheoryⅡ |
Universal bundle and classification theorem, Homotopy group of spheres, Cohomology groups based on a bundle of a coefficient obstruction cocycle, characteristic classes as obstruction classes. |
| G10977 |
Riemannian GeometryⅡ |
Symmetric spaces, Hilbert Manifolds, Loopspaces, Index and Curvature, Injective radius, Comparison Theorems of Riemannian geometry, Sphere Theorem. |
| G10980 |
Global AnalysisⅡ |
Basis for non-linear Analysis,Vector bundle Neighborhoods, Nonlinear Differential operators,polynomial Differential operator,Index of a non-linear Elliptic operator,Symplectic geometry. |
| G10983 |
Gauge Theory |
4-dimensional manifolds, Geometry of connections, Self-dual Yang-Mills equation, Sobolev Embedding theorem, Weitzenbock Formula, Removable-Singularity theorem, Existence Theorem of Self-dual connection Muli space. |
| G10998 |
Numberical Method and Scientific ComputationⅡ |
Hyperbolic Partial Differential Equations, Analysis Of Finite Difference Scheme, Order Of Accuracy Of Finite Difference Schemes, Stability Of Multistep Schemes, Dissipation And Diversion, Parabolic Partial Differential Equations, Systems of partial differential equations in a higher dimension, Second-order equations. |
| G11005 |
OptimizationⅡ |
Theory Of Constrained Optimization, linear programming, The Simplex Method, Interior-point methods, fundamentals of algorithms for nonlinear constrained optimization, quadratic programming, penalty, barrier, augmented Lagrangian methods, sequential quadratic programming. |
| G11017 |
Principles of Applied MathematicsⅡ |
Transform and spectral theory, Partial differential equations, Inverse scattering transform, Asymptotic expansion, Regular perturbation theory, Singular perturbation theory. |
| G11026 |
Computational fluid dynamics |
Physical phenomena of fluid, incompressible flow, water waves compressible flow, acoustics, shock waves, stability theory, turbulence and chaos, their numerical computation. |
| G11030 |
Combinatorics |
Combinational problem solving, Basic counting principles, The principles of inclusion-exclusion, Combinational algorithms, Generating functions, Recurrence relations. |
| G11034 |
Finite Element Methods |
Finite Element Methods For Parabolic partial differential equations, calculus of variations, conformal elements, interpolation using polynomials and approximation, error bounds, numerical integration, nonconforming isoparametric element. |
| G11039 |
Biomathematics |
Differential equations in call motion, differentiation, the mechanical and biochemical relationship between call and organs, cell proliferation, cellular automata, chemotaxis, differential adhesion, tissue rheology, reaction-diffusion system. |
| G11043 |
Modern Cryptography II |
Provable security, cryptanalysis, probabilistic algorithms etc. |
| G11048 |
Public Cryptography II |
In this subject, we learn the current public key crypto schemes and provable securities. |
| G11054 |
Cryptography Algorithm II |
We analyze and implement the block cryptosystems. |
| G11059 |
Selected Topics in Cryptography II |
We research the recent hot results of the public key system, in design and analysis. |
| G11067 |
Coding theory and cryptography II |
We study the recent hot results of the coding theory and the cryptography. |
| G11076 |
Selected topics in Number Theory II |
"Selected Topics in Number Theory II" is a continuation of the course "Selected topics in Number Theory I" with more depth. We cover Function Field Arithmetic, Class Field Theory, Primes on function fields, Algebraic function fields, Cyclotomic function fields, Drinfeld modules, Geometric Goppa codes, Pairings related to Cryptography. |
