# Course Description

Prerequisite : MATH301

Recommended Prerequisite : MATH302

Structure of groups, Nilpotent group, Solvable group, Projective module and Injective module, Hom and duality, Tensor product, Fields, Galois Theory, Finite Fields, Separability, Cyclotomic Field

Prerequisite : MATH501

Recommended Prerequisite : MATH302

Structure of groups, Nilpotent group, Solvable group, Projective module and Injective module, Hom and duality, Tensor product, Fields, Galois Theory, Finite Fields, Separability, Cyclotomic Field

Recommended Prerequisite : MATH302

Rings and Ideals, Quotient ring, Module, Primary decomposition, Noetherian ring, Artinian ring, Discrete valuation ring, Dedekind domain, Completion, Dimension Theory

Recommended Prerequisite : MATH501, 503

Chain conditions, Prime ideals, Flatness, Completion and the Artin-Rees lemma, Valuation rings, Krull rings, Dimension Theory, Regular sequences, Cohen-Macaulay rings, Gorenstein rings, Regular rings, Derivations, Complete local rings

Prerequisite : MATH301

Recommended Prerequisite : MATH501

Arithmetic on number fields, Dirichlet unit Theorem, Ideal class group, Prime ideal decomposition, Hilbert Theory, Introductory class field Theory

Recommended Prerequisite : MATH505

Arithmetic of modular forms, Elliptic curves, Zeta function, L-series, Distribution of prime numbers

The sum of four squares, Polygonal number theorem, Hilbert-Waring problem, The Hardy-Littlewood method, Elementary properties of primes, Vinogradov’s theorem, The linear sieve, Chen’s theorem

Prerequisite : MATH501

We study algebraic varieties, the main objects in algebraic geometry, from scratch. In particular, the course covers affine, projective and quasi-projective varieties, coordinate rings, regular maps, functions fields, rational maps, bi-regular and bi-rational maps, singularities, blow-ups, divisors, canoncail divisors, intersections, and so forth. Also, many examples of algebraic curves and surfaces are dealt with.

Recommended Prerequisite : MATH301

Basic properties of finite groups, Group actions and Sylow Theorem, Free groups, The structure thoery, Classify the groups of special orders, p-groups, Solvable and nilpotent groups, Frattini subgroups, Fitting subgroups, Sylow basis for solvable groups

Recommended Prerequisite : MATH210

Analytic Function, Complex Integral, Singularity, Maximum Principle, Runge Theorem, Riemann Mapping Theorem, Analytic Continuation and Riemann Surface, Harmonic Function, Picard Theorem

Prerequisite : MATH311, 312

RecommendePrerequisite d : MATH311

Lebesgue Measure and Integral, Differentiation, Classical Banach space, Maximal Function, Measure Theory, Representation Theorem, Basic Theory of Functional Analysis

Prerequisite : MATH514

Recommended Prerequisite : MATH311

Lebesgue Measure and Integral, Differentiation, Classical Banach space, Maximal Function, Measure Theory, Representation Theorem, Basic Theory of Functional Analysis

Recommended Prerequisite : MATH413

Cauchy Problem, Laplace Equation, Hilbert Space Method, Sobolev space, Potential Method, Heat Equation, Wave Equation.

Recommended Prerequisite : MATH311

Topological Vector space, Banach space, Hahn-Banach Theorem, Operator Theory, Fredholm Theory, Hilbert space, Distribution, Fourier Transform, Banach Algebra

Recommended Prerequisite : MATH421

Differentiable Manifold & submanifold, Tangent Bundle, Vector Field, Frobenius Theorem, Tensor Theory, Differential form, Lie derivatives, Lie group &Lie algebra, Exponential Maps, Integration on manifolds

Prerequisite : MATH421

Immersion, Submersion, Transversality, Topological invariants

Recommended Prerequisite : MATH421

Simplicial complexes, Euler number, Homology Theory, CW complex, Lefschetz Fixed Point Theorem, Kunneth Formula, Cohomology Ring, Poincare duality, Intersection & Linking number

Recommended Prerequisite : MATH430

Decision problem, Neyman-Pearson Lemma, Likelihood ratio test, Uniformly most powerful test, Unbiased test, Sequential test, Non-parametric test, Contingency table, Baysian method

Recommended Prerequisite : MATH311, 431

Probability measure theory, Stochastic process, Brownian motion, Markov property, Weak convergence, Infinitely decomposable distribution, Martingale, Stochastic integral equation, Stochastic differential equation, Probability approximation

Recommended Prerequisite : MATH333, 430

Gauss-Markov theorem, Least squares method, Data analysis, Analysis of variance, Robust inference

Prerequisite : MATH230, 311

Basic Mathematical Theory for Evaluation of Financial Asset, Analysis, Portfolio Model, Stochastic Process and Probability Differential Equations

Recommended Prerequisite : MATH412

Method of image in PDE, Asymptotic expansion, Regular and Singular Regular & Singular perturbations, Surface layers, WKB Method, Green’s functions

Recommended Prerequisite : MATH413

Integral Equations, Volterra Equation, Fredholm Equation, Hilbert-Schmidt Theory, Wiener-Hopt method, PDE, (Distribution)

Recommended Prerequisite : MATH311

Variational principle in mathematics, Euler equation, Hamilton-Jacobi equation, Quasi-Convex function, Existence Theorem, Differentiability.

Prerequisite : Programming Experience

Recommended Prerequisite : MATH451

Interpolation by polynomials and trigonometric functions, Numerical integration and differentiation, System of linear equations, Data fitting

Recommended Prerequisite : MATH120, 261

Differential geometry of curves and surfaces, Computational algebraic geometry and topology of manifolds needed for computer vision and geometric design, Morphology for pattern recognition and wavelet and fractal geometry for image data compression

Prerequisite : Basic Knowledge in Groups, Matrices and Topology

Voltage graph, Group actions on graphs, Cayley graph, Embedding of graphs, Map Colorings, Genus of groups, Graph and matrices, Algorithm

Prerequisite : Basic Knowledge in Groups, Matrices and Topology Independent of MATH561

Combinatorial Enumerations, Polya Theory, Interconnection network, Block design, Finite geometry, Algorithm

Linear Codes, Nonlinear codes, Hadamard matrices, The Golay codes, Finite fields, Dual codes and their weight distribution, Codes and designs, Perfect codes, Cyclic codes, BCH codes, MDS codes, Reed-Muller codes, Bounds on the size of a code

Prerequisite : Knowledge of an introductory abstract algebra is required. (Knowledge of several areas of mathematics including number theory, groups, rings, fields and probability would help but not necessary.)

Recommended Prerequisite : MATH302

Public key crypto-algorithm, Cryptanalysis, Finite field Study public key cryptosytem based on a various number theoretical theories. We will discuss such as elliptic curve crypto system and RSA systems

Discrete geometry is intimately connected to computational geometry. This course will cover basic concepts of discrete geometry, including convexity, incidence problems, convex polytopes, arrangements of geometric objects, lower envelopes, crossing numbers. In addition, we will study how to design optimal algorithms for geometric problems, by exploiting these combinatorial and geometric properties.

Complex variety, Hilbert’s nullstellensatz, Riemann surfaces and algebraic curves, Residues, Quadric Line Complex

Recommended Prerequisite : MATH505

Algebraic varieties, Algebraic curves, Geometry on elliptic curves, Elliptic curves on local fields, Elliptic curves on global fields.

Modular form, Siegel modular form, Jacobi form, Quadratic form, L-function.

Recommended Prerequisite : MATH301

We study the basic concepts in homological algebra such as Hom, Tensor, Ext, Tor and show how these can be applied to solve purely algebraic problems.

Recommended Prerequisite : MATH510

Bergman Kernel & Integral Formula, Plurisubharmonic function, Pseudoconvexity, Domain of Holomorphy, problem, Levi Problem, Hardy Space

Recommended Prerequisite : MATH311

Basic Properties of Fourier Series, Uniform Convergence, Convergence & Divergence at a point, Hardy-Littlewood Maximal function, Fourier Transform on Lebesgue space

Prerequisite : MATH514

After a brief review of the theories of Fourier transforms, Schwartz space and oscillatory integrals, we will cover a selection of modern topics in harmonic analysis including restriction theorems, Bochner- Riesz operators, the Kakeya maximal operators, the spherical maximal theorem, convolution operators and their applications to partial differential equations and the theory of Besicovitch sets.

Basic sequences, Classical Banach spaces, Devoretsky-Rogers Theorem, Grodendick inequality, Choquet Integral Representation Theorem

Recommended Prerequisite : MATH520

Theory of connections, Manifolds, Riemann tensor, Sectional Ricci curvature, Scalar curvature, Jacobi fields, Geometric invariants, Gauss-Bonnet Theorem, Gauge transformation, Curvature & Topology

Prerequisite : 1. Commutative Algebra (Atiyah & McDonald’s book) 2. Basic Algebraic Geometry (Fulton’s Algebraic Curves, Hartshorne’s Algebraic Geometry – Chap 1) 3. Homological Algebra (Hu’s Introduction to homological algebra),

Recommended Prerequisite : MATH520

Sheaves, Cohomology, Infinitesimal Deformations, Geometry on Hermitian & Kaehler manifold

Recommended Prerequisite : MATH520

Manifold Embedding, Sard Theorem, Transversality, Vector Bundle Theory, Euler number, Hopf Degree, Morse Theory, Cobordism Theory

Recommended Prerequisite : MATH524

Universal coefficient Theorem, Poincare duality, Jordan-Brouwer Separation Theorem

Recommended Prerequisite : MATH520

Exponential Maps, Clifford algebra & Spinor group, Semi-simple Lie algebra, Representation Ring, Lie algebra representation, Peter-Weyl Theorem, Dynkin Diagram

Best uniform approximation, Condition numbers, Krylov method, Eigenvalue problems, Several Time Scale

Recommended Prerequisite : MATH413

Navier-Stokes Equations, Weak·Strong Solution, Vanishing viscosity limit, Euler Equation, Results of Kato & Ponce & Yudovich, Vortex Dunamics, Measure-valued Solutions, Singular Solutions of 3-D Euler Equations, Concentration-Cancellations

Recommended Prerequisite : MATH517

Schauder Theory, Fixed Point Theory, Harnack Inequality & Local Regularity or Fluid Equation, Existence & Uniqueness of the Solutions of Equations from Mathematical Physics

Prerequisite : Programming Experience

Recommended Prerequisite : MATH 551

Finding zeros and minimum points, Eigenvalue problems, Ordinary differential equations, Iterative methods for large system of linear equations

Recommended Prerequisite : MATH413, 651

Finite difference methods, Finite element methods, Parabolic problems, Hyperbolic problems, Elliptic problems, Error analysis in Sobolev spaces, Singularity

Prerequisite : Undergraduate Linear Algebra and Abstract Algebra

Recommended Prerequisite : MATH 464

The aim is to learn algebraic methods utilizing the well-developed matrix theory and group theory in the study of graph theory and its applications. It is also to learn algebraic aspect of discrete mathematics and give a mathematical foundation for related areas of combinatorics, such as, distance-regular graphs, association schemes and t-designs. Graphs and these combinatorial objects will be studied through an investigation of their structures, existence and constructions.

Recommended Prerequisite : MATH301, 421

Group presentations and Tietz transformations, Cayley graph, Graph coverings and related group theory, Maps on surfaces, Branched coverings of surfaces, Hurwitz numbers, Map colorings, Graph embedding invariants, Knots and spacial graphs

Recent research papers are studied independently under the guidance of a thesis supervisor. By giving a talk about them, each student improves his/her own research ability.

The understanding of the applications of the theories of mathematics to sciences and engineering is improved through the lectures of invited speakers.

The understanding of various major areas in mathematics is improved through the lectures of invited speakers

Recent research papers are studied independently under the guidance of a thesis supervisor. By giving a talk about them, each student improves his/her own research ability.