DESCRIPTION  OF GRADUATE  COURSES

MATH 500 M.S. Thesis (Non-credit)  Program of research leading to M.S. degree arranged between student and a faculty  member. Students register to this course in all semesters starting from the begining of their second semester while the research program or write-up of thesis is in progress.

MATH 501 Analysis  (3-0)3  General measure and integration theory. General convergence theorems. Decomposition of measures. Radon-Nikodym theorem. Outer measure. Carathe-odory extension theorem. Product measures. Fubini's theorem. Riesz representation theorem.  Prerequisite: Consent of the department.

MATH 502 Spectral Theory of Linear Operators (3-0)3    Compact operators, compact operators in Hilbert Spaces, Banach Algebras, The spectral theorem for normal operators, unbounded operators between Hilbert spaces, the spectral theorem for unbounded self-adjoint operators, self-adjoint operators, self-adjoint extensions.  Prerequisite: Consent of the department.

MATH 503 Algebra I  (3-0)3  Groups, quotient groups, isomorphism theorems, alternating and dihedral groups, direct products, free groups, generators and relations, free abelian groups, finitely generated abelian groups, actions. Sylow theorems, nilpotent and solvable groups, normal and subnormal series. Rings, ring homomorphisms, ideals, factorization in commutative rings, rings of quotients, localization, principle ideal domains, Euclidean domains, unique factorization domains, polynomials and formal power series, factorization in polynomial rings.   Prerequisite: Consent of the department.

MATH 504 Algebra II  (3-0)3  Modules, homomorphisms, exact sequences, projective and injective modules, free modules, vector spaces, tensor products, modules over a PID. Fields, field extensions, the fundamental theorem of Galois theory, splitting fields, algebraic closure and normality, the Galois group of a polynomial, finite fields. Prerequisite: Consent of the department.

MATH 505 Differentiable Manifolds (3-0)3  Differentiable manifolds, smooth mappings, tangent cotangent bundles, differential of a map, submanifolds, immersions, imbeddings, vector fields, tensor fields, differential forms, orientation on manifolds, integration on manifolds, Stokes’ theorem.  Prerequisite: Consent of the department.

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 511 Group Theory I   (3-0)3  Abelian groups; torsion, divisible, torsion-free groups, pure subgroups, finitely generated abelian groups. Solvable and nilpotent groups, Hall p - subgroups. Permutation groups. Representations. Fixed-point free automorphisms. Locally nilpotent groups, locally solvable groups. Finiteness properties. Infinite solvable groups. Prerequisite: Consent of the department

MATH 512 Group Theory II (3-0)3  Locally finite groups. Maximal and minimal condition on subgroups, Cernikov groups and automorphisms of Cernikov groups, direct limit inverse limit of groups, linear groups, locally finite simple groups, Hall universal group, centralizers of elements in simple locally finite groups. Prerequisite: Consent of the department.

MATH 513 Representation Theory of Finite Groups  (3-0)3  Ring theoretic preliminaries. Group representations and their characters. Characters, integrality and application to the structure theory of finite groups. Product of characters. Induced characters. Reduction and extension of characters. Brauer's theorem on characterization of characters.  Prerequisite: Consent of the department.

MATH 515 Commutative Algebra   (3-0)3  Rings and ideals. Modules.Rings and modules of fractions. Primary decomposition. Integral dependence.  Prerequisite: Consent of the department.

MATH 521 Finite Fields and    Applications  (3-0)3  Introduction to finite fields. Traces, norms and bases, factoring polynomials over finite fields, construction of irreducible polynomials, normal bases, optimal normal bases.  Prerequisite: Consent of the department

MATH 522 Coding Theory  (3-0)3 Basic concepts and examples, linear codes (Hamming, Golay, reed-Muller codes) bounds on codes, cyclic codes (BCH, RS; Quadratic Residue Codes), Goppa codes.  Prerequisite: Consent of the department

MATH 523 Algebraic Number Theory (3-0)3  Ring of integers of an algebraic number field. Integral bases. Norms and traces. The discriminant. Factorization into irreducibles. Euclidean domains. Dedekind domains. Prime factorization of ideals. Minkowski's theorem. Class-group and class number. Prerequisite: Consent of the department

MATH 524 Theory of Function Fields (3-0)3  Valuations. Divisors, repartitions, differentials. Riemann-Roch Theorem. Rational function fields, elliptic and hyperelliptic function fields. Congruence zeta function, the functional equation for the L-functions.  Prerequisite: Consent of the department

MATH 525 Analytic Number Theory (3-0)3  Dirichlet series, Dirichlet L-functions, Chebychev's y and q functions, prime number theorem, distribution of primes, functional equations.  Prerequisite: Consent of the department

MATH 526 Modular Functions (3-0)3  Elliptic functions, modular functions, Dedekind eta function, congruences for the coefficients of the modular function j, Rademacher's series for the par-tition function, modular forms with multiplicative coefficients,  Kronecker's theorem, general Dirichlet series and Bohr's equivalence theorem.  Prerequisite: Consent of the department

MATH 535 Topology   (3-0)3  Topological spaces. Neighborhoods. Basis. Subspace topology, product and quotient topologies. Compactness. Tychonoff's Theorem. Heine-Borel theorem. Separation properties. Urysohn's lemma and Tietze extension theorem. Stone-Cech compactification. Alexandroff one point compactification. Convergence of sequences and nets. Connectedness. Metrizability. Complete metric spaces. Baire's theorem.  Prerequisite: Consent of the department.

MATH 537 Algebraic Topology I (3-0)3 Fundamental group, Van Kampen's Theorem, covering spaces. Singular homology: Homotopy invariance, homology long exact sequence, Mayer-Vietoris sequence, excision. Cellular homology. Homology with coefficients. Simplicial homology and the equivalence of simplicial and singular homology. Axioms of homology. Homology and fundamental groups. Simplicial approximation. Applications of homology. Prerequisite: Consent of the department.

MATH 538 Algebraic Topology II (3-0)3  Cohomology groups, Universal Coefficient Theorem, cohomology of spaces. Products in cohomology, Kunneth formula. Poincare duality. Universal coefficient theorem for homology. Homotopy groups. Prerequisite: Consent of the department

MATH 541 Differential Topology (3-0)3  Manifolds and differentiable structures. Tangent space. Vector bundles. Immersions, submersions, embeddings. Transversality. Sard's theorem. Whitney embedding theorem. The exponential map and tubular neighborhoods. Manifolds with boundary. Thom's tranversality  theorem.  Prerequisite:  Consent of the department.

MATH 545 Differential Geometry I (3-0)3  Lie derivative of tensor fields. Connections, covariant differentiation of tensor fields, parallel translation, holonomy, curvature, torsion. Levi-Civita (or Riemannian) connection, geodesics, normal coordinates. Sectional curvature, Ricci curvature and scalar curvature, Schur's theorem. Jacobi Fields, conjugate points. Isometric immersions, the second fundamental form, formulae of Gauss and Weingarten. Equations of Gauss, Codazzi and Ricci. Metric and geodesic completeness, the Hopf-Rinow theorem. Variations of the energy functional. Prerequisite:  Consent of the department.

MATH 546 Differential Geometry II (3-0)3  Lie groups, principle fiber bundles, almost complex and complex manifolds, Hermitian and Kaehlerian geometry, symmetric spaces. Prerequisite: Math 545

MATH 551 Algebraic Geometry  (3-0)3  Theory of algebraic varieties: Affine and projective varieties, dimension, singular points, divisors, differentials, Bezout's theorem.  Prerequisite: Consent of the department.

MATH 555 Theory of Functions of a Complex Variable (3-0)3  Analytic functions. Singular points and zeros. The argument principle. Conformal mappings. Riemann mapping theorem. Mittag-Lefler theorem. Infinite products. Canonical products. Analytical continuation. Elementary Riemann surfaces.  Prerequisite: Consent of the department.

MATH 558 Introduction to Functions of Several Complex Variables (3-0)3  Holomorphic functions, comparison of one and several variables, domains of holomorphy, subharmonicity, pseudoconvexity,  invariant metrics, holomorphic maps, Stein and CR-manifolds, integral formulas, equation.  Prerequisite: Consent of the department

MATH 566 Positive Operators and Banach Lattices (3-0)3  Vector lattices. Positive operators and extension of positive operators. Order projections, order continuous operators, lattice homomorphisms. Banach lattices with order continuous norm, compactness and weak compactness in Banach lattices. Embedding Banach spaces. Banach lattices of operators. Compact operators and weakly compact operators on Banach lattices. Prerequisite: Consent of the department.

MATH 570 Functional Analysis  (3-0)3 Review of metric spaces, Normed Linear Spaces, Dual Spaces and Hahn-Banach Theorem, Bidual and Reflexivity, Baire’s Theorem, Dual Maps, Projections, Hubert Spaces, The spaces Lp(X,m),C(X), Locally Convex Vector Spaces, Duality Theory of lcs, Projective and Inductive topologies.  Prerequisite: Consent of the department

MATH 571 Topological Vector Spaces; (3-0)3 Introduction to topological Vector Spaces, locally convex topological Vector Spaces. Inductive and projective limits. Frechet Space. Montel, Schwartz, nuclear spaces. Bases in Frechet spaces and the quasi equivalance property. Köthe sequence spaces. Linear topological invariants. Prerequisite: Consent of the department

MATH 580 Applied Functional Analysis (3-0)3 Distributions, Review of Banach and Hilbert Spaces, Sobolev spaces, Semigroups, Some techniques from nonlinear analysis. Prerequisite: Consent of the department.

MATH 581 Numerical Analysis I (3-0)3  Error analysis. Solutions of linear systems: LU factorization and Gaussian elimination, QR factorization, condition numbers and numerical stability, computational cost. Least squares problems: the singular value decomposition (SVD), QR algorithm, numerical stability. Eigenvalue problems: Jordan canonical form and conditioning, Schur factorization, the power method, QR algorithm for eigenvalues. Iterative Methods: construction of Krylov subspace, the conjugate gradient and GMRES methods for linear systems, the Arnoldi and Lanczos method for eigenvalue problems. Prerequisite: Consent of the department.

MATH 582 Numerical Analysis II (3-0)3  Interpolation and approximation: Lagrange and Newton interpolation, Hermite interpolation, trigonometric interpolation and Fourier series. Spline interpolation B-splines and recursive algorithms. Numerical differentiation and quadrature: Newton-Cotes formulas, Gaussian integration rules. Extrapolation and Romberg integration, adaptive quadrature. Hierarchal and recursive quadrature formulas: Archimedes integration formula. Root finding methods. Prerequisite: Consent of the department

MATH 583 Partial Differential Equations I (3-0)3 Cauchy-Kowalevski Theorem. Linear and quasilinear first order equations. Existence and uniqueness theorems for second order elliptic, parabolic and hyperbolic equations. Correctly posed problems, Green's functions.  Prerequisite:  Consent of the department.

MATH 584 Partial Differential Equations II (3-0)3 Sobolev spaces:Weak derivatives, Approximation by smooth functions, Extentions, 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. Prerequisite: Consent of the department.

MATH 585 Nonlinear Problems of Applied   Mathematics 3-0)3  Initial and initial-boundary value problems for the first order nonlinear PDEs. Continuous solutions. Conservation laws and weak solutions. Burgers' equation. Quasi-linear hyperbolic systems. Riemann invariants. Nonlinear waves in gases and deformable solids. One parameter group transformation and similarity solutions of nonlinear problems for PDEs. Nonlinear waves in strings under transverse impact.  Prerequisite: Consent of the department.

MATH 586 Delay Differential Equations (3-0)3  General description of delay differential equations. Statement of the initial value problem. Classification. The method of steps. Existence and uniqueness theorems. Continuation of solutions. Integrable systems. Elemnets of functional differential equations. Linear systems. Stability theory: Direct Lyapunov's method, Razumikhin's theory. Periodic solutions. Special topics: Oscillations; Impulsive delay differential equations.

MATH 587 Ordinary Differential Equations I  (3-0)3  Initial Value Problem: Existence and Uniqueness of Solutions; Continuation of Solutions; Continuous and Differential Dependence of Solutions. Linear Systems: Linear Homogeneous And Nonhomogeneous Systems with Constant and Variable Coefficients; Structure of Solutions of Systems with Constant and Periodic Coefficients; Higher Order Linear Differential Equations; Sturmian Theory, Stability: Lyapunov Stability and Instability. Lyapunov Functions; Lyapunov's Second Method; Quasilinear Systems; Linearization; Stability of an Equilibrium and Stable Manifold Theorem for Nonautonomous Differential Equations.  Prerequisite: Consent of the department.

MATH 588 Ordinary Differential Equations II  (3-0)3  Nonlinear Periodic Systems: Limit Sets; Poincare-Bendixon Theorem. Linearization Near Periodic Orbits. Orbital stability. Bifurcation: Bifurcation of Fixed Points; The Saddle-Node Bifurcation; The Transcritical Bifurcation; The Pitchfork Bifurcation; Hopf Bifurcation; Boundary Value Problems: Linear Differential Operators; Boundary Conditions; Existence of Solutions of BVPs; Adjoint Problems; Eigenvalues and Eigenfunctions for Linear Differential Operators; Green's Function of a Linear Differential Operator.  Prerequisite: Consent of the department.

MATH 589 Impulsive Differential Equations (3-0)3  General description of impulsive differential equations: System with fixed moments of impulses; Systems with variable moments of impulses; Discontinuous dynamical systems. Linear systems: General properties of solutions; Periodic solutions, Floquet theory; Adjoint systems. Stability: Stability criterion based on linearization of systems; Direct Lyapunov method; B-equivalence; Stability of systems with variable time of impulses. Quasilinear systems: Bounded solutions; Periodic solutions; Quasiperiodic and Almost periodic solutions; Integral manifolds. Discontinuous dynamical systems and applications.

MATH 591 Graduate Seminar in Mathematics I (0-2)NC  Presentation involving current research given by graduate students and invited speakers.

MATH 592 Graduate Seminar in Mathematics II (0-2)NC  Presentation involving current research given by graduate students and invited speakers.

MATH 593 Numerical Solutions of  Partial   Differential Equations  (3-0)3  Finite difference method, stability, convergence and error analysis. Initial and boundary conditions, irregular boundaries. Parabolic equations; explicit and implicit methods, stability analysis, error reduction, variable coefficients, derivative boundary conditions, solution of tridiagonal systems. Elliptic equations, iterative methods, rates of convergence. Hyperbolic equations. The Lax-Wendroff method, systems of conservation laws, stability. Finite volume method.  Prerequisite: Consent of the department

MATH 594 Theory of Special Functions (3-0)3  Appell's symbol and hypergeometric series. The gamma function. The beta function. Dirichlet averages. Jacobi polynomials. Elliptic integrals.  Prerequisite: Consent of the department.

MATH 595 The  Boundary Element Method  and Application (3-0)3  Weighted residual methods, the boundary element  method for Laplace and  Poisson equations. The Dual reciprocity method, computer implementation.  Prerequisite: Consent of the department.

MATH 596 Computational Basis of Fluid Dynamics Equations  (3-0)3  Introduction to fluid behavior. Derivation of continuity, momentum and energy equations. Navies-Stokes equations. Stream function, vorticity.  Solutions of creeping, potential, laminar, boundary layer, turbulent flows. Solution of Navier-Stokes equations using finite difference methods in velocity-pressure , stream function-vorticity and stream function forms. Example solutions. Stability, convergence and error analysis.  Prerequisite: Consent of the department.

MATH 598 Fundamentals of Soliton Theory  (3-0)3  Solution of the Korteweg-de Vries Equation. Multi-soliton solution as Bergmann Potentials for Sturm-Liouville Equation. Topics in one-dimensional Scattering Theory. Associated Sturm-Liouville Equations. Inverse scattering problems. Evolution equations related to a linear system. A general class of solvable nonlinear evolution equations.  Prerequisite: Consent of the department.

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 800-899 Special Studies (4-2)Non-credit  M.S. Students choose and study a topic under the guidance of a faculty member, normally his/her advisor.

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.

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.