교과목명

구분

학점

지도

교수

교과목 개요

현대대수학 입문 1

이론

3

Algebraic systems : groups, rings, fields, their actions, algebras, Subsystems, homomorphisms, congruence relations, Direct products, direct sums, Classical isomorphism theorems, Cyclic groups, Cosets and counting, Symmetric, alternating, and dihedral groups, Categories: products, coproducts, and free objects, Free groups, free products, generators, and relations, Free abelian groups, Finitely generated abelian groups, The krull- Schmit theorem, The Sylow theorems, Classification of finite groups, Nilpotent and solvable groups, Normal and subnormal series

현대대수학 입문 2

이론

3

Factorization in commutative rings, Rings of quotients and localization, Rings of polynomials and formal power series, Factorization in polynomial rings, The Jacobson radical, Semisimple rings, Division algebras, Free modules and vector spaces, Projective modules and injective modules, Hom and duality, Tensor products, Modules over a principal ideal domain, Field extensions, Splitting fields, The Galois group of a polynomial, Cyclic extensions, Radical extensions, Linear disjointness and separability

실복소해석학 1

이론

3

Lebesgue measure, abstract measure, measurable sets, nonmeasurable sets, mesaurable functions, lebesgue integration, Differentiations, functions of absolute continuity, general convegence theorems, signed measure, The Radon-Nikodym theorem, The Lp-spaces, Minkowski and Hölder inequalities, outer measures. The Lebesgue-Stieltjes integrals, product measure, Fubini theorem

실복소해석학 2

이론

3

Complex number system, power series, anytic functions, Möbius transformations, zeros of an analytic functions, index, Cauchy's theorem and integral formula, singularities, residues, the maximum modulus theorem.

기초응용수학 1

이론

3

Basic Linear Algebras, The Differential and Directional Derivatives, The Chain Rue and Lagrange Multipliers, Maxima and Minima, Taylor's Formula for Several Variables, Newton's Method and Contractions, Inverse and Implicit Function Theorem, Higher Derivatives, Multiple Integrals, The Calculus of variations

기초응용수학 2

이론

3

Eigenvaluse, eigenvectors and similarity, Unitary equivalence, Svhur's Theorem, QR factorization and algorithm, Jordan canonical form and triangular factorizations, Hermitian matrices(variational characterization of eigenvalues, Congruence transformation, Matrix norms, Symmetric gauge functions and unitary invariant norms, Positive definite matrices/Inequalities, Perturbation of eigenvalues, Polar and Singular value decompositions

기하 및 위상수학 1

이론

3

General topology Review, Homotopy of paths, Fundamental group, covering space, retraction and fixed points, deformation retracts and homotopy type, fundamental group of circle, sphere and some surfaces, Van-Kampen Theorem, free products of groups, free groups, Classification of surface

기하 및 위상수학 2

이론

3

Surface, Gauss and mean curvature, Dupin indicatrix, rientation, Gauss -Bonnet Theorem and its application, Differentiable manifold, tangent vector field, Lie bracket, differential map, immersion and submersion, connection, parallelism, simple introduction of Riemannian metric and Riemannian manifold.

공업 및 물리수학 1

이론

3

Basic Linear Algebra, Ordinary Differential Equations, Partial Differential Equations, Fourier Transform, Laplace Transform, Integral Equations, Special Functions

공업 및 물리수학 2

이론

3

Calculus of Variations, Lagrangians and actions, Hamiltonians and Symplectic geometry, Hamilton-Jacobi Equations, Llectro- magnetic Theory, Maxwell Equation, Basic Quantum mechanics

과학적 계산 및

수치분석 1

이론

3

Gaussian Elimination, Variants of Gaussian Elimination, Direct Factorization Methods, Iterative Methods, Functional Iteration for a System of Equations, Special Method or Polynomials, The Power Method, Methods Base on Matrix Transformations, Iterative Linear Interpolation

과학적 계산 및

수치분석 2

이론

3

Interpolatory Quadrature, Roundoff Errors an Uniform oefficient Formulae, Euler-Cauchy Method, Multistep Methods Based on Quadrature Formulae, Laplace Equation in a ectangle, Solution of Laplace Difference Equations

수리적 확률 및

응용통계학 1

이론

3

Measure Theory, Random Variables, Central Limit Theorem, Martingales, Conditional Expectation, Markov Chains, Brown Motion, Itoh Integral, Stochastic Partial Differential Equations

수리적 확률 및

응용통계학 2

이론

3

Stationary Processes and Time Series Analysis, Ergodic Theorems, Covariance Stationary, Processes in the Frequency Domain, Time Series Analysis: Introduction, Stochastic Models for Time Series, Autoregressive Process, Moving Average Process, Mixed Autoregressive Moving Average Process, Autoregressive Integrated Moving Average Processes, Time Series Analysis in the Time Domain, Spectral Analysis of Time Series Data, Markov Chain Monte Carlo

군의 표현론 1

이론

3

Algebraic Foundations of Representation Theory, Fundamental Concepts of Group Theory, Fundamental Concepts and the Simplest Propositions of Representation Theory, Representations of Finite Groups, Basic Propositions of the Theory of Representations of Finite Groups, The Group Algebra of a Finite Group, Representations of the Symmetric Group, Induced Representations, Representations of the Group SL(2,Fq)

군의 표현론 2

이론

3

Representations of Compact Groups, Compact Topological Groups, Representations of Compact Groups, The Group Algebra of a Compact Group, Finite-Dimensional Representations of Connected Solvable Groups, Connected Topological Groups, Solvable and Nilpotent Groups, Lie's Theorem, Finite- Dimensional Representations of the Full Linear Group, Some Subgroups of the Group G, Irreducible Finite-Dimensional Representaitons of the Group GL(n,C)

환론 1

이론

3

rings,ring homo, prime ideals, rings of qotients, radicals and socles, direct sum and decomposition, classical isomorphism theorems

환론 2

이론

3

structure of ring theory, projective and injective modules, lifting idempotents, Faith-Utumi theorem, exact sequences, flat modules

가환대수학 1

이론

3

Minimal theory, primary decomposition, Hilbert Nullstellensatz, filter theory, flat families. lgebraic closed fields, ideal and modules, operator homomorphisms, tensor product rings, Dedekind domains

대수적정수론 1

이론

3

Places of a Field, valuation rings, Places, Places of the rational number field, Valuations of a field, Valuations of rank Valuations of the field of rational numbers, The approximation theorem, The product formula in fields of algebraic numbers and functions, The theorem of Gelfand-Tornheim, The perfect completion of a field with valuation, Normed vector spaces over complete fields, Extendability of an general valuation, Complete Non-Archimedean fields

대수적정수론 2

이론

3

Different of an extension field, Valuation-Theoretical Approach to Ideal Theory, The divisor Group, The norm of divisors, Different and Discriminant, The discriminant of a module, The discriminant of and ideal, field discriminant, The connection between different and discriminant, Factorization of Prime Ideals in Field Extensions, Divisor norm in algebraic number fields

호모로지대수학 1

이론

3

Complexes and the Exact Sequence, Categories and Functors, Additive and Abelian Categories, Homology Groups in Algebra and in Geometry, Cyclic Homology, -Non-commutative Differential Geometry, Homology of Discrete Groups, Derived Categories and Derived Functions, Structure of the Derived Category, Derived Functors, Sheaf Cohomology

호모로지대수학 2

이론

3

Triangulated Categories, Mixed Hodge Structures, The Category of Hodge Structures, Mixed Hodge Structures on Cohomology with Constant Coefficients, Hodge Structures on Homotopic Invariants, Perverse Sheaves, D -Modules, The Weyl Algebra, Algebraic D -Modules, Holonomic Modules, Equivalence of Categories

조합론 1

이론

3

Graph Models, Isomorphism, Edge Counting, Planar Graphs, Euler Cycles, Hamilton Circuits, Graph Coloring, Coloring Theorems, Properties of Trees, Search Trees and Spanning Trees, Traveling Salesperson Problem, Tree Analysis of Sorting Algorithms, Shortest Paths, Minimal Spanning, trees, Network Flows, Algorithmis Matching, Basic Counting Principles, Simple Arrangements and Selections, Arrangements and Selections with Repetitions

가환대수학 2

이론

3

dimension theory, modules of differentials, Gorenstein rings, free resolution. valuation theory, power series rings, local algebras, Zariski rings, regular local rings, Noetherian domains

비가환대수학 1

이론

3

Generalities on rings, Rings and ideals, Modules and Chain conditions, Gröbner basis computation algorithms, Left Poincaré-Birkhoff-Witt Rings, The Division Algorithm, Buchberger's Algorithm, Poincaré-Birkhoff-Witt Algebras, The Diamond Lemma, Poincaré-Birkhoff-Witt Theorems

비가환대수학 2

이론

3

Gröbner bases for modules, Gröbner bases for modules, Gröbner bases for subbimodules, Homogeneous Gröbner bases, Syzygies for modules, Schreyer's order, Free resolutions, The Gelfand- Kirillov dimension, The Gelfand-Kirillov dimension of PBW algebras, A primality test, rings of quantum spaces

계산선형대수

이론

3

Matrix-Vector Multiplication, Orthogonal Vectors and Matrices, The Singular Value Decomposition, QR Factorization, Gram Schmidt Orthogonalization, Householder Triangularization, Least Squares Problems, Floating Point Arithmetic, Stability of Householder Triangularization, Stability of Back Substitution, Conditioning of Least Squares Problems, Stability of Least Squares Algorithms

순서대수구조론

이론

3

l-Group, Lattice Ordered Group, Mathiak Valuations, Two-Generator One-Relator Groups, Ideal Theory, Embedding Theorem, Some applications

범주론

이론

3

Categories, Diagrams, Monomorphisms, Epimorphisms, and Isomorphisms, Initial and Terminal Objects, Products, Universal Constructions, Equalizers, Pullbacks, Functors, F-Algebras, Natural Transformations, Adjoints, Cartesian Closed Categories

그래프이론

이론

3

Graphs, degree of a vertex, Euler tours, Matching in bipartite graphs, Path covers, Planar Graphs, Colouring, Flows, Substructures in Dense Graphs, Substructures in Sparse Graphs, Ramsey Theory for Graphs, Hamilton Cycles, Random Graphs, Minors, Trees, and WQO

조합론 2

이론

3

Generating Functions Moels, Calculating Coeffients of Generatin Functions, Partitions, Exponential Generating Functions, Summation Method, Recurrence Relation Models, Divide and Conquer Relations, Solution of Linear Recurrence Relations, Solution of Inhomogeneous Recurrence Relations, Solution with Generating Functions, Counting with Venn Diagrams, Inclusion Exclusion Formula, Equivalence and Symmetry Groups, Burnside's Theorem, Cycle Index, Polya's Formula, rogressively Finite Games, Nim-Type Games

대수학연구세미나

실습

3

Seminar course: topics in algebra

유니버설대수

이론

3

Galois Connections and Closures, Homomorphisms and Isomorphisms, Direct and Subdirect Products, Terms, Trees, and Polynomials, Identities and Varieties, term Rewriting Systems, Algebraic Machines, Mal'cev-Type Conditions, Clones and Completeness, Tame Congruenece Theory, Term Condition and Commutator, Complete Sublattices, G-Clones and M-Solid Varieties, Hypersubstitutions and Machines

계산논리학

이론

3

Propositional Calculus, Normal Forms, Davis-Putnam Rules, Resolution, Predicate Calculus and Semantics, Herbrand's Theorem, Unification, Compactness and Completeness, Equational Logic and Term Rewriting System

응용그래프이론

이론

3

Graph Isomorphism, Subgraphs, Degrees, Indegrees, and Outdegrees, Paths, Circuit and Cycles, Connected Graphs and Digraphs, Tree and Spanning Trees, Eulerian Graphs and Digraph, Minimum Weight Spanning Trees, Maximum Weight Branchings, Minimum Weight Arborescences, Two Shortest Path Algorithms, The Steiner Network Problem, Flow in Networks and Menger's Theorem

수리적암호학

이론

3

fundamental theorem of algebra and number theory, fundametal concepts of signs, algebraic erro signs, system of linear signs, system of BCH signs, some applications

응용코딩이론

이론

3

BCH sign, MacWilliams equality, alebraic geometric signs, space-times signs. some applications, related topics

대수학특강

이론

3

Topics in algebra

계산대수기하학

이론

3

Ideals and Varieties, Noetherian Rings and the Hilbert basis Theorem, Projective Space and Graded Objects, Graded Rings and Modules, Hilbert Function and Series, Free Resolutions and Regular Sequences, Grobner Based and the Buchberger Algorithm, Monomial Ideals and Applications, Geometry of Points and the Hilbert Function, Snake Lemma, Derived Functors, Tor and Ext, Curves, Sheaves, and Cohomology, Projective Dimension, Cohen-Macaulay Modules

변분론

이론

3

Static Conjectural Variations Equilibria, Origin of the conjectural variations concept, Conjectural Variations Equilibria, Conjectures as Reduced Forms for Dynamic Interactions, Private provision of a public good, Oligopoly, Consistent Conjectures in Dynamic Settings, Conjectures for dynamic games, equilibria and consistency, The model of Basar, Turnovsky and d'Orey, Dynamic Conjectures, Incomplete Information and Learning, The model of Friedman and Mezzetti

다양체론

이론

3

Tensor algebra, Tensor, p-forms, Differentialbe manifolds, Vector and tensor field on a manifold, Vector fields on a manifold, Tensor field on a manifold, Exterior differential forms, Exterior differentiation, Maxwell's equations of the electromagnetic field, Differentiation on a manifold, The Lie derivative, Curvature and torsion forms, Pseudo-Riemannian and Riemannian manifolds, The metric tensor, Killing vertor fields

복소다양체론

이론

3

holotopic functions, fundamental theorems on complex manifolds, Dolbeault theorem, deformation theory, Hermitian differential geometry, Complex line bundles, Pseudodifferential operators, Elliptic complexes, Harmonic theory on complex manifold, Hodge manifolds, Kodaira Embedding theorem

게이지이론

이론

3

Group actions, fiber bundles, principal bundles, connections, gauge transformations, parallel transport, curvature, covariant derivatives, pseudo-Riemannian manifolds, Lagrangians, clifford algebras, spin bundles, and the Dirac operator.

텐서해석학

이론

3

Manifolds, vector bundles, vector fields and flows, differential operators, dynamical systems, exterior algebra, Hodge-de Rham Theorem, Some Applications

리만기하학

이론

3

Riemannian metrics, Affine connections; Riemannian connections, Affine connections, Geoesics; convex neighborhoods, Minimizing properties of geodesics, Curvature, Jacobi Fields, Isometric immersions, Complete Manifold, Hopf-Rinow and Hadamard Theorems, Space of constant curvature, Varitions of energy, The Rauch comparison theorem, The fundamental group of manifolds of negative curvature

의사리만기하학

이론

3

differential manifolds, pseudo-Riemannian metric, Curvature, Lorentz manifold, introduction to relativity Theory, Causality of Lorentz manifold, applications

곡선론

이론

3

Curves in Euclidean space, curvatures, curves of constant curvatures, curves on differential manifolds, geodesics, variations of curves, conjugate points, applications

극소곡면론

이론

3

first variation formula, minimal submanifolds in euclidean space and sphere, totally geodesic submanifolds, plateau problems, conjugate minimal surfaces, applications

기하학연구세미나

실습

3

Seminar course: topics in geometry

기하학특강

이론

3

Topics in geometry

미분기하학특강

이론

3

Topics in differential geometry

위상다양체론 1

이론

3

Topological Spaces, Topologies, Bases, Manifolds, New Spaces from old Subspaces, Product Spaces, Quotient Spaces, Group Actions, Connectedness and Compactness, Connectedness, Compactness, Locally compact Hausdorff Spaces, Simplicial Complexes, Euclidean Simplicial Complexes,

위상다양체론 2

이론

3

Abstract Simplicial Complexes, Triangulation Theorems, Orientations, Combinatorial Invariant, Curves and Surfaces, Homotopy and the Fundamental Group, Circles and Sphere, Some Group Theory, Free Products, Free Group, Free Abelian Group, The Seifert-Van Kampen Theorem, Covering Spaces, Classification of Coverings, Homotopy

위상변환군론 1

이론

3

Homeomorphisms, Basis, Continuous maps, Topological product, Compactness, Tychonoff theorem, Topological groups, Set products, Coset Spaces, Complete regularity, Homogeneous spaces, Local groups, Compact subsets of topological groups, Totally disconnected groups, Approximation by separable metric groups, Projective limits, Generating Sequences of sets, Smoothing functions, An analytic lemma, Groups without small subgroups

위상변환군론 2

이론

3

one-parameter factor groups, an extension theorem, Analytic coordinates, inverse sequences, Existence of n-cells in a groups, Simultaneous continuity of Derivatives, Differentiability with respect to group parameters, periodic and pointwise periodic homeomorphisms, Cross sections and zero-dimensional groups, Finite-dimensional groups and orbits, orbits which are manifolds

대수적위상수학 1

이론

3

Mayer-Vietoris sequence, the Universal Coefficient Theorems, Kunneth Theorems (product spaces and their homology and cohomology), fiber bundles, higher homotopy groups, Hurewicz' Theorem, Poincare and Alexander duality.

대수적위상수학 2

이론

3

obstruction theory, cohomology operations, homotopy theory, spectral sequences and computations, cohomology of groups, characteristic classes.

일반위상수학

이론

3

Set Theory and Logic, Cartesian Products, The Maximum Principle, Topological Spaces and Continuous Functions, Methric Topology, Quotient Topology, Connectedness and Compactness, Limit Point Compactness, Countability and Separation Axioms, Normal Spaces, The Tietze Extension Theorem, Imbeddings of manifolds, The Tychonoff Theorem, Metrization Theorems and Paracompactness, Complete Metric Spaces and Function Spaces

미분위상수학

이론

3

Submanifolds of R^(n+k), Differential Structures, Differentiable Maps and the Tangent Bundle, Embeddings and Immersions, Manifolds with Boundary, The weak and strong topologies on C^r(M,N), Approximations, Jets and the Baire Property, Analytic Approximations, The Morse-Sard Theorem, Transversality, Vector Bundles and Tubular Neighborhoods, Degrees, Euler Characteristic, Morse Theory, Isotropy

조합적위상수학

이론

3

Topological Spaces, The Jordan Theorem, Surfaces-Elementary curves and 1-complexes, Orientability of surfaces, Classification of a Closed surfaces, Triangulations, The body of a complexes, polyhedra, Sperner's Lemma and Its Corollaries, Sperner's lemma, The Fixed point theorem for an n-cell, Theorem on Essential mappings, The Sum Theorem, Sequences of subdivisions

기하적위상수학

이론

3

Topology of subsets of Euclidean space, Topologiacal surfaces, Simplicial surfaces, tangent, normal and binormal vectors, curvature and theorem of curves, plane curves, smooth surfaces, thagent and normal vectors, Curvature of smooth surfaces, the Weingarten map and the second fundamental form

위상수학연구

세미나

실습

3

Seminar course: topics in topology

위상군론

이론

3

Group theory, general topology, integration. Elementary properties, Haar measure, representation (Peter-Weyl), positive definite functions, Fourier transforms.

위상리군론

이론

3

Fundamental concepts on Lie groups, Lie algebras, fundamental theorems, representation of Lie groups, representation of Lie algebras, some structure theorem.

매듭이론

이론

3

A Century of Knot Theory, What is a Knot, Combinatiorial Techniques, Geometric Techniques, Algebraic Techniques, Geometry, Algebra, and The Alexander Polynomial, Numerical Invariants, Symmetries of Knots, High-Dimensional Knot Theory, New Combinatorial Techniques

리군론특강

이론

3

Topics in Lie theory

위상수학특강

이론

3

Topics in topology

대수적위상수학특강

이론

3

Topics in algebraic topology

기하적위상수학특강

이론

3

Topics in geometric topology

함수해석학 1

이론

3

Hilbert spaces, metrizations, quotient spaces, Banach spaces, Hahn-Banach theorem, Banach-Steinhaus theoorem, Brauder Fixed Point Theorem, Riesz representation theorem, weak topological space, weak* topological spaces, Locally convex, Banach algebraic structure

함수해석학 2

이론

3

C*-algebras, fundamental concepts, fundamental theorems on C*-algebras, Locally compact group, normal operators, Fredholm theory,, spectrums, operators on Banach spaces, unbounded operators, graphs, symmetric operators, Cayley transform, resolution of the identity, semigroups of operators, some applications to opertaor algebras

응용함수해석학 1

이론

3

some application on physics, moments, moment theory, moment sequences, Four Block Problem, Application on function spaces, Stieltjes moment problem, perturbation of point spectra, regular perturbation theory

응용함수해석학 2

이론

3

Some applications on functional analysis, recent Topics on functional analysis and applications, bound states of Schrodinger operators, Qualitative theory, dilation analytic potentials, Schrodinger operators with periodic potential

조화해석학 1

이론

3

Fourier coefficients, homogeous Banach spaces, Convergence of Fourier series, Sets of divergence, Conjugate function, Hardy spaces, Interpolations, Hausdorff-Young theorem, Lacunary series

조화해석학 2

이론

3

analytic classes, Fourier transform, Fourier-Stieltjes transform, distribution, periodic function, Paley-Wiener theorem, Kronecker theorem, Locally compact group, Maximal ideal spaces, Regular Banach spaces, Tensor product

프리에해석학 1

이론

3

Fourier series, several variable geometric concepts, Summable functions and Fourier series, Eigen function expansion, Fourier integrals, heisenberg's inequality, Time Limit function

프리에해석학 2

이론

3

Fundamental concepts of Function spaces, Hardy theorem, Hardy functions and filters, Wiener-Hopf Factorization, Milne's equation, Spitzer inequality, Series on Circle, Rotation group, Convolution algebra, Legendre inequality, SO(3), SL(2,R), Hyperbolic plane

작용소론 1

이론

3

Hilbert spaces, Normal operators, Hypernormal operators, dilation theory, unitary dilation, Projections on Hilbert spaces, compact operators, Projections, spectrum, Fredholm theory on Hilbert spaces, pre-Hilbert space, unbounded operators, closed operators

작용소론 2

이론

3

Invariant subspaces, Invariant subspace problem, Lomonosov theorem, hyper invariant subspaces, hyperinvariant subspace problem, compact operators, Shattern classes, normaloid, spectral theory of bounded operators, spectraloid operators, several kind of operators

복소변수함수론 1

이론

3

Analytic function spaces, Meromorphic function, Riemann mapping theorem, Sine function, Factorization, Gamma function, Runge theorem, Simple connectedness, Riemann surface, Monodromy theorem, Analytic manifold, Harmonic function, Dirichlet problem, Green function

복소변수함수론 2

이론

3

Harmonic and subharmonic funtions, Harmonic Functions, Maximal Theorems, Basic structure of Hp functions, Mean Convergence to Boundary Values, Harmonic Majorants, Cauchy and Cauchy-Stieltjes Integrals, Inequalities of Fejer-Riesz, Hilbert, and Hardy, Conjugate Funtions, Kolmogorov's Theorem, Trigonometric Series, Mean Growth and smoothness, Smoothness of the Boundary Function, Functions with Hp Derivative, Taylor coefficients, Hausdorff-Young lnequalities

편미분방정식론 1

이론

3

Small Vibrations of an Elastic String, Heat Conduction, Diffusion-Dispersion Phenomena, Saturated Flows Underground, Telegrapher's System, Partial Differential Equations of the First Order, Brownian Motion, Integral Equations, Green's Functions, and Eigenfunction Expansions, Green's Functions for Sturm-Liouville Problems, The Fredholm Alternative, Eigenvalues and Eigenfunctions

편미분방정식론 2

이론

3

Properties of Potentials, Integral Equations of Potential Theory, Parabolic Equations, Thermal Potentials, Hamilton's Principle, Hilbert Space and Strong and Weak Convergence, The Ritz- Galerkin Method, The Dirichlet Problem for the Laplace Equation, One-Dimensional Gas Dynamics, Discontinuous Solutions and Shocks, Free Boundary Value Problems, The One-Phase Stefan Problem, Free Boundary Value Problems for Hyperbolic Equations

수치해석 1

이론

3

Gaussian Elimination, Tridiagonal or Jacobi Matrices, Block-Tridiagonal Matrices, Gauss-seidel or Successive Iterations, Method of Residual Correction, Positive Definite Systems, Computation of Eigenvalues and Eigenvectors, Acceleration of the Power Method, Intermediate Eigenvalues and Eigenvectors, Pointwise Error in Interpolation Polynomials, Hermite or Osculating Interpolation, Weighted Least Squares Approximation, Some Properties of Orthogonal Polynomials

수치해석 2

이론

3

Branches of Nonsingular Solutions, Finite Element Schemes, Methods Using discontinuous Pressures, Formulation in Primitive Variables, Mixed Incompressible Methods, Numerical Analysis of Upwind Schemes, Upwinding in the Stream Function-Vorticity Scheme, Error Analysis of the Upwind Scheme, Approximating the Pressure with the Upwind Scheme, Numerical Algorithms, General Methods of Descent and Application to Gradient Methods, Least-Squares and Gradient Methods to Solve the Navier-Stokes Equations

힐버트공간론

이론

3

vector spaces, complex vector spaces, complete linear spaces, Projections, continuous function spaces, Riesz-Frech theorem, Unitary operator, Self-adjoint operator, invariant spaces

대역해석학

이론

3

Manifold, Linear differental equation, Numerical aspects, Geometry of C∞-function, Sobolev spaces, Elleptic operators on closed manifold, Poiasson principal, Elleptic case, Vector bundle, Index theorem, Pricipal G-bundle.

대수기하학1

이론

3

Affine and projective algebraic varieties, fundamental concepts and theorems, Morphisms and singularities, rational maps, nonsingular variety and nonsingular curve, some properties, projective morphism

대수기하학2

이론

3

Schemes and coherent sheaves. Derived Functoes, Sheaf cohomology and other related topics, Riemann-Roch Theorem, elliptic curves, intersection theory

계산기하학

이론

3

points, lines, polygons; subdivisions, arrangements, polytopes, cell complexes, segment tree, interval tree, range tree, priority search tree. Point Location, Plane-Sweep Algorithms, Convex Hulls, Proximity, Graph Drawing

다변수함수론

이론

3

integral formula, Subharmonic, applications, convexity, Hormander solution, Pseudoconvex domain, Singular function, Hefer lemma, Approximation problem, Sheave, zeros of holomorphic functions

추계적

편미분방정식

이론

3

Stochastic differential equations, Martin boundary, Brownian sheet, Numerical experiments with spdes, Contour processes of random trees, quasilinear stochastic differential equations, parabolic type, regular dependence of solutions, Fluctuations of a two-level critical branching system

응용수치

편미분방정식

이론

3

Weak Formulation of Boundary Value Problems, Ritz-Galerkin Approximation, Error Estimates, Sobolev Spaces, Variational Formulation of Elliptic Boundary Value Problems, Construction of a Finite Element Space, Polynomial Approximation Theory in Sobolev Spaces, Inverse Estimates, n-Dimensional Variational Problems, Finite Element Multigrid Methods, Additive Schwarz Preconditioners, Max-norm Estimates, Adaptive Meshes

작용소대수학

이론

3

C*-algebras, representation of C*-algebras, Von Neumann algebras, fundamental theorems, concepts, Factors, Decomposition of Von Neumann, dual operator algebras, continuity of operator algebras, predual of operator algebras

계산근사학

이론

3

Automata, Computability, and Complexity, Mathematical Nations and Terminology,Types of Proof ,Nondeterminaism ,Regular Expressions, Nonregular Language, Context-free Grammars, Pushdown Automata, Non-context-free Languages, The recursion Theorem, Decidability of logical theories, Turing Reducibility, Measuring complexity, The Class P, The Class NP, NP-completeness, Additional NP-complete Problems, Sabitch's Theorem, Hierarchy Theorems, Relativization, Circuit complexity

계산복잡도

이론

3

Algorithms, Union-Find Algorithms, Principle of Algorithms Analysis, Implementation and Empirical Analysis, Elementary Data Structures, Building Blocks, Arrays, Strings, Abstract Data Types, Pushdown Stack ADT, First class ADTs, Recursion and Trees, Quicksort, Merging and Mergesort, Priority Queues and Heapsort, Radix Sorting, Special-Purpose Sorts, Sorting Networks, Symbol Tables and BSTs, Radix Search, Digital Search trees, Patricia Trices

수치적최적이론

이론

3

Vector Spaces and Matrices, Transformations, Unconstrained Optimization, One-Dimensional Search Methods, Gradient Methods, Newton's Method, Simplex Method, Non-Simplex Methods, Problems with Equality Constraints, Convex Optimization Problems, Algorithms for Constrained Optimization

수리유체역학

이론

3

Navier-Stokes equation, Euler equation, Measure-valued solutions, Concentration-Cancellations, D'Alember's theorem, Kelvin's theorem, Conservation of angular momentum, Bjerknes' theorem, Rossby number, Ekman layer, Taylor-Proudman theorem

수리적

웨이브릿이론

이론

3

Fourier Transform, Poisson's Sum, Sampling Thoerem, Partial Sum and the Gibbs Phenomenon, Window Function, Short Time Fourier Transform, Discrete Short Time Fourier Transform, Discrete Gabor Representation, Continuous Wavelet Transform, Discrete Wavelet Transform, Wavelet Series, Wigner-Ville Distribution, Quadratic Superposition Principle, Ambiguity Function, Construction of Semiorthogonal Spline Wavelet, Decimation and Interpolation, Wavelet Decomposition Algorithm, Signal Reconstruction in Semiorthogonal Sobspaces

수리적

오토마타이론

이론

3

Additive Cellular Automata, CA-nased models, Biological applications of CA, CA as Parallel and Image Processing system, Polynomial Algebraic Characterization of CA Behavior, Probabilistic Analysis of CA Behavior, Fractional Dimensions in CA, Group Properties of CA, Cycle Set Characterization of Group CA, Characterization of Group CA with Inverse State-Trasition Function, Characterization of Nongroup CA, General Characterization of Linear Nongroup CA, Pseudoexhaustive Pattern Generation

수리적재정학

이론

3

Market efficiency, Diversifiable risk, Classifying market participants, Delta hedging, assumptions of mathematical finance, time value of money, Mathematically defining arbitrage, Multiple time steps, Many time steps, Putting interest rates in trading volatility, Alternate models, Transaction costs, Brownian motion, Ito's lemma, Justifying the derivation, Dividend-paying assets

수리생물학

이론

3

This course provides an introduction to the use of continuous and discrete differential equations in the biological sciences. Biological topics may include single species and interacting population dynamics, modeling infectious and dynamic diseases, regulation of cell function, molecular interactions and receptor-ligand binding, biological oscillators, and an introduction to biological pattern formation. Mathematical tools such as phase portraits, bifurcation diagrams, perturbaton theory, and parameter estimation techniques which are necessary to analyze and interpret biological models will also be covered.

수학적게임이론

이론

3

Choice of strategies, mim-max theorem for matrix games, General min-max theorem, Types of strict dominance, Characterization of extreme-point optimal strategies, Completely mised matrix games, Uniqueness of optimal strategies, Colonel Blotto game, Identification of friend and foe, Poker game, Formulation of the linear programming problem, linear programming problem and its dual

에르고딕이론

이론

3

Associated Isometries, Ergodicity The Ergodic Theorem, Isomorphism, Conjugacy, and Spectral Isomorphism, Measure-Preserving Transformations with Discrete Spectrum, Entropy, Topological Dynamics, Minimality, Non-Wandering Set, Topological Transitivity, Invariant measures, Invariant Measures for Continuous Transformations, Unique Ergodicity, Topological Entropy, Bowen's Definition, Calculation of Topological Entropy

추계적실험론

이론

3

Models, Simulation as Experimentation, Simulation in inference, Congruential Generators, Shift-Register Generators, Lattice Structure, Shufflings and Testing, General Principles, Discrete Distributions, Order statistics, Multivariate Distributions, Poisson processes and lifetimes, Markov processes, Gaussian processes, Metropolis' Method and Random Variables, Control and Antithetic Variates

해석학연구세미나

실습

3

Seminar course: Analysis on Topics

해석학특강

이론

3

Topics in analysis

유한요소 분석

이론

3

Cartesian Tensors, Variational Functions, Stiffness Matrix, Connectivity, Boundary Condition, Connectivity and Nodal Coordinates, Variational Functions, Triangular elements and Area Coordinates, Cylindrical Coordinates, Governing Differential Equation, The Displacement Method for Beam Analysis

응용제어이론

이론

3

Proportional-Derivative Control, Digital Control, Feedback Versus Precomputed Control, state-Space and Spectrum Assignment, Outputs and Dynamic Feedback, Dealing with Nonlinearity, Discrete-Time, Linear Discrete-Time Systems, Smooth Discrete-Time Systems, Continuous-Time, Linear Continuous-Time Systems, Linearizations Compute Differentials, Volterra Expansions, Reachability and Controllability, Time-Invariant Systems, Controllable Pairs of Matrices

응용통계열역학

이론

3

Laws of Thermodynamics, Distribution Laws, Distributions and Thermodynamics, Determination of Partition Functions, quilibria and Rates of Reaction, Heat Capacities, Low-Temperature, Phenomena, Thermodynamics and Spectroscopy, Computer Simulations, Solids and Biopolymers

추계적역학

이론

3

Stochastic bifurcation point, bifurcations of one-dimensional stochastic differential equations, p-bifurcations in the noisy Duffing-van der Pol Equation, The stochastic Brusselator, Numerical Approximation of Random Attractors, Random Hyperbolic Systems, Perturbed Systems, Lyapunov Exponents, Towards a Theory of Random Numerical Dynamics, Lévy Processes, Asymptotic Curvature, Stochastic Analysis on Product Manifolds, Evolutionary Dynamics in Random Environments

혼돈과프랙탈

이론

3

analysis of lineality, measure spaces, Hausdorff measure, dimensions, Packing measure, Fracters, stable problem, Poincare-Benixson theorem, Pertubation, mathematical biology, model theory, some related applications

수리생물학특강

이론

3

Topics in mathematical biology

석사학위 논문연구

이론

1

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