PhD. in Mathematics Department
MATH 506 Comprehensive Studies (0-4) NC
MATH 600 PhD Thesis NC
7 elective courses
Total minimum credit: 21
Number of courses with credit(min): 7
DESCRIPTION OF GRADUATE COURSES
MATH 506 Comprehensive Studies (0-4) NC
The aim of this course is to test the knowledge of the student in the basic areas of mathematics. For this purpose, a written exam is given in the following topics and subtopics: Algebra (A. Groups and Rings B. Modules and Fields), Analysis (A. Real Analysis B. Complex Analysis), Differential Equations (A. Ordinary DE B Partial DE), Geometry-Topology (A. Geometry B. Topology), Numerical Analysis (A Numerical Analysis I B. Numerical Analysis II). Each student is required to take the exam in 4 subtopics chosen from 3 distinct topics.
MATH 600 Ph.D. Thesis NC
Program of research leading to Ph.D.Degree arranged between student and a faculty member. Students register to this course in all semesters starting from the beginning of their second semester while the research programme or write-up of thesis is in progress.
MATH 606 The Theory of Algebras (3-0)3
Generalities on algebras over commutative rings. Group algebras. Morita duality and quasi-Frobenius algebras, Frobenius algebras. Polynomial identity algebras, Artin-Procesi Theorem.
Prerequisite: Consent of the department.
MATH 608 Geometric Algebra (3-0)3
Rings with involution, sesquilinear and Hermitian forms, products of Hermitian forms, Morita Theory for Hermitian modules. Construction of Clifford Algebras, structure of Clifford Algebras, the discriminant and the Arf Invariant, the Special Orthogonal Group and classical examples.
Prerequisite: Consent of the department.
MATH 615 Lie Algebras (3-0)3
Basic concepts, semisimple Lie algebras, root systems, isomorphism and conjugacy theorems, existence theorem.
Prerequisite: Consent of the department.
MATH 658 Elliptic Boundary Value Problems (3-0)3
Calculus of L2 derivatives, some inequalities. Elliptic operators, local existence and regularity of solutions of elliptic systems. Garding's inequality, global existence and regularity of solutions of strongly elliptic equations. Coerciveness results of Aroszajn and Smith, eigenvalue problems for elliptic equations.
Prerequisite: Consent of the department.
MATH 677 Numerical Methods in Ordinary Differential Equations (3-0)3
Introduction to Numerical Methods, Linear multistep methods. Runge-Kutta methods, stiffness and theory of stability. Numerical methods for Hamiltonian systems, iteration and differential equations.
Prerequisite: Consent of the department
MATH 688 Finite Element Solutions of Differential Equations (3-0)3
Calculus of variations. Weighted residual methods. Theory and derivation of interpolation functions. Higher order elements. Assembly procedure, insertion of boundary conditions. Finite element formulation of ordinary differential equations, some applications. Error and convergence analysis. Finite element formulation of non-linear differential equations. Time dependent problems. Convergence and error analysis. Solution of the resulting algebraic system of equations. Applications on steady and time dependent problems in applied continuum mechanics.
Prerequisite: Consent of the department
MATH 693 Directed Study in Mathematics I (1-0)1
Directed study in a selected area of mathematics. Term paper is required (The instructor, not to be the student's thesis supervisor writes a brief proposal for each topic which must be approved by the department head).
MATH 694 Directed Study in Mathematics II (1-0)1
Directed study in a selected area of mathematics. Term paper is required (The instructor, not to be the student's thesis supervisor writes a brief proposal for each topic which must be approved by the department head).
MATH 700-799 Special Topics in Mathematics (3-0)3
Courses not listed in catalogue. Contents vary from year to year according to interest of students and instructor in charge. Typical contents include contemporary developments in Algebra, Analysis, Geometry, Topology, Applied Mathematics.
MATH 900-999 Advanced Studies (4-0)Non-credit
Graduate students as a group or a Ph.D. student choose and study advanced topics under the guidance of a faculty member, normally his/her supervisor. MATH 710 Low Dimensional Topology (3-0) 3
- MATH 711 Impulsive Differential Equations (IDE) (3-0)3
- MATH 712 Large Cardinals and Combinatorial Principles in Set Theory (3-0)3
- MATH 736 Model Theory (3-0) 3
- MATH 738 Coding Theory (3-0) 3
- MATH 741 Analytic Function Spaces and Their Operators (3-0)3
- MATH 742 Topics in Partial Differential Equations (3-0)3
SPECIAL TOPICS IN MATHEMATICS
MATH 702 Initial Value Problems in the Space of Generalized Analytic Functions.(3-0)3
Initial value problems in Banach spaces, scales of Banach spaces, solution of IVP in scales of Banach spaces, the classical Cauchy-Kowalewski theorem, the Holmgren theorem, basic properties of generalized analytic functions, IVP with generalized analytic initial data.
MATH 710 Low Dimensional Topology (3-0)3
Preliminaries: Vector bundles, connections, characteristic classes, Hodge Theory. Spin Geometry of four-manifolds: Spin Structure, Dirac operator, Atiyah-Singer Index Theorem. Seiberg Written Module Space. Compactness of module space. Seiberg-Witten Invariants. Topology of four manifolds: Intersection forms of four manifolds, realizability of unimodular, symmetric bilinear forms as intersection forms.
MATH 711 Impulsive Differential Equations (IDE) (3-0)
General Description of IDE: Description of mathematical model. Systems with impulses at fixed times. Systems with impulses at variable times. Discontinuous dynamical systems. Impulsive oscillator. Linear Systems of IDE: General properties of solutions. Stability of solutions. Adjoint systems, Perron theorem. Linear Hamiltonian systems of IDE. Stability of Solutions of IDE: Stability criterion based on first order approximation. Stability in systems of IDE with variable times of impulsive effect. Direct Lyapunov method. Periodic and Almost Periodic Systems of IDE: Nonhomogeneous linear periodic systems. Nonlinear periodic systems. Almost periodic functions and sequences. Almost periodic IDE. Integral Sets of Systems of IDE: Bounded solutions of nonhomogeneous linear systems. Integral sets of quasilinear systems with hyperbolic linear part and with non-fixed moments of impulse actions.
MATH 712 Large Cardinals and Combinatorial Principles in Set Theory (3-0)3
Filter and ideals in partial orders, trees, Ramsey theory. Generalized Continuum Hypothesis, Martin's axiom. Closed unbounded sets, stationary sets. Principle, Suslin hypothesis, Kurepa hypothesis. Inaccessible, ineffable, compact and measurable cardinals.
MATH 738 Model Theory (3-0) 3
Propositional and first-order logic. The compactness theorem and consequences. Theories that are: complete, model-complete, quantifier-elliminable, categorical. Structures that are: prime, minimal, universal, saturated, stable.
MATH 738 Coding Theory (3-0) 3
Coding constructions, Bounds on the sizes of codes, sphere packing bound, Plotkin bound, Singleton bound, Griesmer bound, Johnson Bound, self-dual codes, codes over rings, codes and invariant theory, quasi-cyclic codes, finite geometry and coding theory, duality issues in coding theory, duality and product codes, covering radius of some classes of codes, orthogonal arrays and coding theory, decoding of codes, algebraic decoding and list decoding, complexity issues in coding theory, low density codes, turbo codes, frameproof codes, watermarking, sequences in coding theory and cryptology.
MATH 741 Analytic Function Spaces and Their Operators (3-0)3
Operators on Hilbert and Banach spaces, Bergman, Bloch, Besov, and Hardy spaces, functions of bounded mean oscillation, Carleson measures, duality, Berezin transform, Toeplitz, Hankel, and composition operators.
MATH 742 Topics in Partial Differential Equations (3-0)3
Sobolev spaces: Weak Derivatives, Approximation by Smooth functions, Extensions, Traces, Sobolev Inequalities, The Space H^-1. Second-Order Elliptic Equations: Weak Solutions, Lax-Milgram Theorem, Energy Estimates, Fredholm Alternative, Regularity, Maximum Principles, Eigenvalues and Eigenfunctions. Linear Evolution Equations: Second-order Parabolic equations (Weak Solutions, Regularity, Maximum principle), Second-order Hyperbolic Equations (Weak Solutions, Regularity, Propagation of disturbances), Hyperbolic Systems of First-order Equations, Semigroup theory.