### Courses

#### Core Courses

Course No. | Course Name | Credit | Description |
---|---|---|---|

MTH 501 | Algebra I | 3 | Basic Theory of Groups, Rings, Fields, and Vector Spaces, Structure of Rings, Theory of Abelian Groups. |

MTH 502 | Real Analysis I | 3 | Lebesgue Measure, Lebesgue Measurable Functions, Lebesgue Integration, Differentiation and Integration Spaces of Measurable Functions. |

MTH 503 | Topology I | 3 | Fundamental Group, Van Kampen Theorem, Covering Space, Group Actions. |

MTH 505 | Probability I | 3 | Relation between Measure Theory and Probability, Fundamental Concepts of Probability, Law of Large Number, Conditional Expectation, Martingale, Ergodic Theory. |

MTH 506 | Applied Mathematics I | 3 | Distribution Functions, Theory of Distribution, Green Functions and Boundary Value Problems, Fourier Transformation |

MTH 507 | Complex Analysis | 3 |
Holomorphic Functions of One Complex Variable, Harmonic Functions, Boundary Behavior, Maximum Principle, Approximation by Rational Functions. Conformal Mapping, Analytic Continuation, Zeros of Holomorphic Functions. |

MTH 514 | Geometry I | 3 |
Holomorphic Functions of One Complex Variable, Harmonic Functions, Boundary Behavior, Maximum Principle, Approximation by Rational Functions. Conformal Mapping, Analytic Continuation, Zeros of Holomorphic Functions. |

MTH 550 | Basic Numerical Analysis I | 3 | Numerical errors, Numerical linear algebra, Interpolation, Root-finding, Numerical integration and differentiation, Newton’s method |

#### Major Courses

Course No. | Course Name | Credit | Description |
---|---|---|---|

MTH 601 | Algebra II | 3 | Noetherian Rings and Abelian Groups, Primary Decomposition, Localization and Tensor Products, Local Rings, Completeness. |

MTH 602 | Homological Algebra | 3 | Commutative Categories, Derived Functors, Spectral Sequences, Tor Functor, Ext Functor. |

MTH 603 | Real Analysis II | 3 | Complex Measures, Lebesgue Integration, Radon-Nikodym Theorem, Decomposition Theorems. Measure and Topology, Functions on Lp-Spaces |

MTH 604 | Functional Analysis I | 3 | Basic Notions of Topological Vector Spaces, Normed Linear Spaces, Hanh-Banach Theorems, Uniform Boundedness Principle, Open Mapping Theorem, Closed GraphTheorem, Dual Spaces, Weak Topology, Weak Convergence, Bounded and Compact Operators. |

MTH 605 | Topology II | 3 | Simplicial Homology, Eilenberg-Steenrod Axioms, Cech Homology Theory, Singular Homology, CW-Complexes. |

MTH 606 | Algebraic Topology I | 3 | Simplicial Homology, Eilenberg-Steenrod Axioms, Cech Homology Theory, Singular Homology, CW-Complexes. |

MTH 607 | Differential Geometry I | 3 | Theory of Connections, Linear and Affine Connections, Riemannian Connections, Curvature and Space Forms, etc. |

MTH 608 | Differentiable Manifold | 3 | Differential Manifolds, Vector Bundles, Differential Forms, Frobenius' Theorem, Riemannian Metric.. |

MTH 609 | Probability II | 3 | Continuity Theorem of Characteristic Functions, Central Limit Theorem, Renewal Theory, Local Limit Theorem, Invariance Principle Theorem. |

MTH 610 | Stochastic Modeling | 3 | Poisson Process, Renewal Process, Discrete Time Markov Chain, Continuous Time Markov Chain, Elementary Queueing Theory and their Applications. |

MTH 611 | Advanced Complex Analysis | 3 | Subharmonic Functions, Maximum Principle, Hardy Spaces, Bergman Spaces, Bloch Spaces, BMOA Space, Duality, Carleson Measures. |

MTH 612 | Applied Mathematics II | 3 | Hilbert Space, Operator Theory, Integral Equation. |

MTH 613 | Mathematical Logic | 3 | Predicate Calculus, Completeness Theorem, Lowenheim Skolem Theorem, Compactness Arguments, Gödel Incompleteness Theorem, Elements of Model Theory. |

MTH 614 | Numerical Analysis | 3 | Numerical ODE, Euler method, Higher-order Taylor method, Runge-Kutta method, System of ODEs, Spectral method |

MTH 618 | Numerical Partial Differential Equation | 3 | F.D.M for parabolic, Elliptic, Hyperbolic P.D.E |

MTH 620 | Analytical Mechanics | 3 | Vector Field, Integral Curve, Hamiltonian Equation, Variational Method, Differential Form and Stoke's Theorem, Hamilton-Jacobi Theory, Hamiltonian System. |

MTH 621 | GeometryⅡ | 3 | Riemannian metrics, Levi-Civita connections, curvature and Jacobi fields, second variation formula, comparison theorems, shortest path, homogeneous spaces, Morse theory and closed geodesics, sphere theorem, finiteness theorem, non-negatively curved spaces, non-positively curves spaces. |

MTH 632 | Algebraic Number Theory I | 3 | Quadratic reciprocity law, p-adic fields, Hilbert symbols, quadratic forms. |

MTH 633 | Analytic Number Theory I | 3 | Dirichelt Series, Riemann Zeta function, prime number theorem, modular forms. |

MTH 701 | Representation Theory of Groups | 3 | Character Theory, Induced Representations, Compact Groups, Brauer's Theorem. |

MTH 704 | Quadratic Forms | 3 | P-adic Numbers, Quadratic Forms over Local Fields, Quadratic Forms over Integral Domains, Composition of Binary Quadratic Forms. |

MTH 705 | Algebraic Geometry I | 3 | Affine and Projective Algebraic Varieties over Any Fields, Schemes and Sheaves. |

MTH 706 | Algebraic Geometry II | 3 | Riemann-Rock Theorem on Curves, Riemann Hypothesis on Curves over Finite Fields, Deformation Theory. |

MTH 707 | Lie Groups and Lie Algebras | 3 | Topological Structures of Classical Linear Groups, Structures of Lie Algebras, Campbell-Hausdorff Formula. |

MTH 708 | Functional Analysis II | 3 | Banach Algebra, Spectral Theory, Commutative Banach Algebra, Operators on Hilbert Spaces. |

MTH 709 | Hilbert Space Theory | 3 | Bounded Operators, Closed Operators, Finite Rank Operators, Compact Operators, Symmetric Operators, Adjoint Operators, Self-adjoint Operators, Spectral Theory. |

MTH 710 | Potential Theory | 3 | Harmonic Functions, Bounded Harmonic Functions, Nonnegative Harmonic Functions, Singularities, Kelvin Transform, Spherical Harmonies, Boundary Behavior, Decomposition Theorem Hardy and Bergman Spaces. |

MTH 711 | Convex Analysis | 3 | Basic Notions, Dual Correspondences, Representation Theorems, Inequalities, Differentiation Theory, Extreme Value Problems, Saddle Functions, Minimax Problems, Convex Algebra. |

MTH 712 | Algebraic Topology II | 3 | Homology Theory, Fundamental Group, Dual Homotopy Theory, Blacker-Massey Theorem. |

MTH 713 | Differential Topology | 3 | Manifold, Function Space, Transversality, Vector Bundle, Tubular Neighborhood, Morse Theory, Cobordism. |

MTH 714 | Homotopy Theory | 3 | Homotopy Group, Homotopy Theory of CW Complexes, Postnikov System, CW-homology Group. |

MTH 715 | Topological Transformation Group I | 3 | Transformation Group, Minimal Set, Proximal, Distal Extension. |

MTH 716 | Topological Transformation Group II | 3 | General Theory of G-spaces, Homology Theory of Finite Group Action, Locally Smooth Actions on Manifolds. |

MTH 717 | Fixed Point Theory | 3 | Fixed Point Classes and the Nielson Number, Calculation of Nielson Number, Fixed Point Class Functor, Fixed Point Classes of a Fiber Map. |

MTH 718 | Differential Geometry II | 3 | Submanifolds, Variational Theory of Length Integral, Complex Manifolds, Homogeneous Spaces, Symmetric Spaces. |

MTH 719 | Theory of Complex Manifold | 3 | Almost Complex Structure, Hermitian Metric, Kaehler Metric, Holomorphic Vector Bundles, Line Bundles, Connection, Chern Classe, Hodge Theory, Vanishing Theorems, Bochner Technique, Kodaira Embedding Theorem. |

MTH 720 | Global Analysis | 3 | Jet Bundles, Differential Operators, Derivative Functors, Dual Functors, Vector Bundle Neighborhoods, Nonlinear Differential Operators, Polynomial Differential Operators, Index Theorem, Calculus of Variations. |

MTH 722 | Geometric Function Theory | 3 | Ahlfors-Schwarz Lemma, Intrinsic Pseudo-distances, Hyperbolicity, Complete Hyperbolicity, Bergman Metric, Normal Family, Currents, Nevanlinna Theory, Distribution of Curves. |

MTH 723 | Stochastic Process I | 3 | Markov Process, Random Walk, Gaussian Process, Brownian Motion, Invariance Theorem. |

MTH 724 | Stochastic Process II | 3 | Markov Process, Martingale, Levy Process, Jump Process, Diffusion Process, Branching Process. |

MTH 725 | Stochastic Analysis | 3 | Stochastic Integral, Stochastic Differential Equations, Stochastic Integral Equations, Stochastic Calculus. |

MTH 726 | Probabilistic Potential Theory | 3 | Stochastic Integral, Stochastic Differential Equations, Stochastic Integral Equations, Stochastic Calculus. |

MTH 727 | Topic in Random Walks | 3 | Stochastic Processes, Random Walks. |

MTH 728 | Theory of Differential Equations | 3 | Perturbation, Stability, Hamiltonian. |

MTH 729 | Operator Calculus | 3 | Laplace Transformation, Fourier Transformation, Distribution Function. |

MTH 730 | Nonlinear Programming | 3 | Convex Programming, Kuhn-Tucher Constraint Qualification, Duality, Generalized Subdifferentials, Calculus of Variations |

MTH 731 | Theory of Partial Differential Equations I | 3 | 2nd Order Partial Differential Equations, Boundary Value Problems of Parabolic, Elliptic Equations, Initial Condition, Existence and Uniqueness of Solution, Fourier Transform. |

MTH 732 | Several Complex Variables | 3 | Power Series, Holomorphic Functions of Several Complex Variables, Holomorphic Spaces, Global Property, Local Property, Representation by Integrals. |

MTH 733 | Theory of Combinatorics | 3 | Three Fundamental Principles, Möbius Formula, Polya's Enumeration Theorem, Polymatroids, Operations on Matroids. |

MTH 734 | Theory of Models | 3 | Elementary Equivalence, Elementary Extensions, Saturated Models, Automorphism of Models Categoricity in Power, Element Types, Model Completeness, Elimination of Qualtifiers, Applications to Theories of Fields, Nonstandard Analysis |

MTH 735 | Advanced Numerical Analysis | 3 | Mathematical Modeling and Applications of its Algorithm. |

MTH 736 | Fluid Dynamics | 3 | Ideal Fluid, Newtonian Fluid, Influence of Compress Ability. |

MTH 737 | Theory of PartialDifferential Equations II | 3 | Variational Method, Mountain-Path Theorem. |

MTH 738 | Commutative Algebra | 3 | Abelian Groups, Tensor Product, Noetherian Rings, Dedekind Domains, Classical Ideal Theory, Valuation Theory, Graded Rings. |

MTH 739 | Theory of Ring Structure | 3 | Wedderburn's Structure Theorem, Radical, Semi-Simple Algebra, Simple Algebras, Galois Cohomology, Brauer Groups, Division Algebras, Norms. |

MTH 740 | K - theory | 3 | K0 of Rings, K1 of Rings, K0 and K1 of Categories, Homologies of Commutative Rings, Steinberg Group, Milnor's K2, K2 of Fields and Number Theory, High-Dimensional K-Theory. |

MTH 741 | Introduction to Cryptology | 3 | Mathematical Background, Number Theoretic Problems, String Code, Block Code, Public-Key Encryption. |

MTH 742 | Advanced Cryptology | 3 | Hash Functions and Data Integrality Authentication, Digital Signatures, Key Establishment and Management Protocols. |

MTH 743 | 3-manifold theory | 3 | In this course, We will discuss about Basic 3-Manifolds, Handle Bodies, Heegard Splittings, Dehn's Lemma, Incompressible Surface and Seifert Manifolds. |

MTH 744 | Hyperbolic geometry and Klein groups | 3 | In this course, We will Study 3-Dimensional Hyperbolic Manifolds, Kleinian Groups, Geometrically Finite Kleinian Groups, Finitely Generated Kleinian Groups, and Deformation Structures. |

MTH 745 | Metric Differential Geometry I | 3 | Inner Metric Spaces (Length Spaces), Curvature in Metric Spaces, Alexandrov Spaces, Dilatations, Hausdorff Distance, Collapsing, Convergence Theorems, Finiteness Theorems, Compactness Theorems, Diffeomorphism Theorem, Margulis Lemma, Etc. |

MTH 746 | Metric Differential Geometry 2 | 3 | See MTH 745. |

MTH 747 | Communications in Mathematicla Sciences | 3 | Computer Typesetting in Mathematics, Symbols and Mathematical Computations, Mathematical Communications using Internet and WWW, Presentation Methods of Mathematics. |

MTH 748 | Function Theoretic Operator Theory | 3 | Harmonic Function Spaces, Holomorphic Function Spaces, Toeplitz Operators, Hankel Operators, Composition Operators, Boundedness, Compactness, Commuting Property, Schatten Classes. |

MTH 749 | Harmonic Analysis | 3 | Hardy-Littlewood Maximal Function, Nontangential Maximal Function, Singular Integral Theory, Interpolation of Operators, Ap-measures, Weighted Normed Inequalities, Carleson Measures, BMO, Hp-space. |

MTH 750 | Queueing Theory and its applications | 3 | M/M/1, M/G/1, G/M/1, Priority Queue, Vacation Queue, MMPP, MAP, MMPP/M/1, Traffic Modeling and Analysis of Telecommunication System. |

MTH 751 | Applied Stochastic Process | 3 | Fundamental Concepts of Probability, Poisson Process, Renewal Process, Discrete Time Markov Chain, Continuous Time Markov Chain, Brownian Motions. Applications to Communication System. |

MTH 752 | Nonlinear Partial Differential Equation | 3 | Index Theory, Mountain Pass Theorem and its Applications. |

MTH 753 | Cryptographic Protocol I | 3 | Authentication, Identification, Zero-Knowledge Interactive Protocol, Digital Signature, Secret Sharing Scheme, Cryptographic Primitive. |

MTH 754 | Cryptographic torotocol II | 3 | Electronic Cash, Electronic Election, Key Management, Multi-Party Protocol, Electronic Commerce, Information Hiding, Key Establishment Protocol. |

MTH 755 | Cryptosystem | 3 | Substitution-Permutation Network, Stream Ciphers, Public-Key Encryption, Hash Functions, Mode of Operations. |

MTH 756 | Elliptic Curves | 3 | Operations on Elliptic Curves, Theory of Elliptic Curves on Various Fields-Finite Fields, Complex Number Field, Local Fields and Global Fields, Integral Points on Elliptic Curves, Computations of Mordell-Weil Group. |

MTH 757 | Automorphic Representation Theory | 3 | Lie Groups, Representations of Algebraic Groups, Spectral Theorem, Whittaker Model, Theory of Automorphic Forms, Eisenstein Series, Rankin-Selberg Method, Langlands Program. |

MTH 758 | Computational NumberTheory | 3 | Linear Algebra and Algorithms of Lattice Structure, Algorithms of Polynomials, Algebraic Number Theory and Algorithms, Algorithms of Elliptic Curves, Methods of Factorizations. |

MTH 759 | Algebraic Number Theory Ⅱ | 3 | Adeles, Ideles, L-series, Reciprocity Law, Class Field Theory, Brauer-Siegel Theorem. |

MTH 760 | Analytic Number Theory Ⅱ | 3 | Poisson Summation Formula, Mellin Transform, Gamma Function |

MTH 761 | Theory of Discrete Groups | 3 | Cayley graph, R-Tree, Gromov hyperbolic group, Rips complex, automatic group, Kleinian group. |

MTH 762 | Geometric Group Theory | 3 | Free groups and free products, finite generating groups, finite representation groups and their growth properties, polynomial and exponential growth of groups. |

MTH 763 | Knot TheoryⅠ | 3 | Basics of knot theory, invariants of knots and links, compositions, Seifert surfaces, Alexander polynomials and Jones polynomials. |

MTH 764 | Knot TheoryⅠ | 3 | Heegard splitting, Dehn surgery of 3-manifolds, Poincare conjecture, foliation theory, fibrations. |

MTH 765 | Scientific Computing I | 3 | Practical introduction to computational problem solving, Numerical optimization, Fourier transform, Multigrid method |

MTH 766 | Scientific Computing II | 3 | Conditioning of problems and stability of algorithms, Floating point arithmetic, Scientific visualization, Applied approximation theory, Eigenvalue problems, Monte-Carlo simulation |

MTH 781 | Biomathematics I | 3 | Discrete and differential equations of biological systems, Population models, Models for interacting population, Reaction Kinetics, Dynamics of infectious diseases, Molecular and cellular biology, Population genetics and evolution |

MTH 782 | Biomathematics II | 3 | Spatial Pattern formation with reaction diffusion systems, Pattern formation in growing domains, Bacterial pattern and chemotaxis, Epidermal wound healing, Growth and control of tumors, Geographic spread and control of epidemics |

MTH 785 | Mathematical Modeling | 3 | Mathematical models for real world problems and their errors, Analytical methods in computations |

MTH 786 | Mathematical Modeling | 3 | Mathematical models for real world problems and their errors, Analytical methods in computations |

MTH 787 | Computational Fluid Dynamics | 3 | Numerical techniques for solving PDEs with a focus on fluid dynamics, Compressible and incompressible Navier-Stokes equations, Conservation laws |

MTH 790 | Finite difference method for derivatives | 3 | The course covers finite difference methods to the solution of partial differential equations arising in financial applications, their accuracy and stability, and related topics such as sensitivities of derivatives of the solution, i.e., Greeks. The course deals with the numerical solution of European and American options, and algorithms for exotic options, such as Equity-linked securities (ELS) on multi-asset products. We also study numerical estimations for volatility surfaces based on market data. |

MTH 801 | Research in Algebra I | 3 | |

MTH 802 | Research in Algebra II | 3 | |

MTH 803 | Research in Analysis I | 3 | |

MTH 804 | Research in Analysis II | 3 | |

MTH 805 | Research in Topology I | 3 | |

MTH 806 | Research in Topology II | 3 | |

MTH 807 | Research in Geometry I | 3 | |

MTH 808 | Research in Geometry II | 3 | |

MTH 809 | Research in Probability I | 3 | |

MTH 810 | Research in Probability II | 3 | |

MTH 811 | Research in Applied Mathematics I | 3 | |

MTH 812 | Research in Applied Mathematics II | 3 | |

MTH 813 | Research in Algebra III | 3 | |

MTH 814 | Research in Analysis III | 3 | |

MTH 815 | Research in Topology III | 3 | |

MTH 816 | Research in Geometry III | 3 | |

MTH 817 | Research in Probability III | 3 | |

MTH 818 | Research in Applied Mathematics III | 3 | |

MTH 831 | Seminar I | 3 | |

MTH 832 | Seminar II | 3 |