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Matematik Doktora Programı

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Hakkında yorumlar Matematik Doktora Programı - Kurumda - Urla - İzmir

  • Program tanımları

    PhD. in Mathematics Department

    Math 600 Ph.D. Thesis (0-1) N.C.
    Math 597 Comprehensive Studies (0-2) N.C.
    In addition to above courses, 7 courses for students with M.S. degree (14 courses for students with B.S. degree) must be taken from the list of Graduate Courses.

    Total minimum credit: 21 for students with M.S. degree, 42 for students with B.S. degree.
    Number of courses with credit (min): 7 for students with M.S. degree, 14 for students with B.S. degree.

    DERS İÇERİKLERİ

    MATH 501 Advanced Mathematics
    Derivative. Integral. Multivariable Functions. Infinite Series and Products. Vector Analysis. Gradient, Divergence, Curl operations. Stokes’ Theorem. Potential Theory. Dirac Delta Function. Analytic Functions. Cauchy-Riemann Conditions. Cauchy’s Integral Theorem. Laurent Series. Calculus of Residues. Ordinary Differential Equations. Separable, Exact and Homogeneous Equations. Partial Differential Equations. Green’s Functions.

    MATH 503 Introduction to Pure Mathematics
    Sets. Functions. Zorn’s Lemma. Groups. Lagrange’s Theorem. Factor Groups. Isomorphism Theorems. Finitely Generated Abelian Groups. Rings. Modules. Vector Spaces. Linear Functions. Jordan Form of the Linear Operators. Symmetric and Orthogonal Operators. Metric Spaces. Topological Spaces. Continuous Functions. Connected. Spaces. Compact Spaces. Applications to Analysis.

    MATH 510 Hilbert Space Theory with Applications
    The Lebesque Integral. Hibert spaces. Linear operators. Fredholm integral equation. Voltera integral equation. Applications to ordinary differential equations. Sturm- Liouville systems. Inverse differential operators and Green's functions. Application of Fourier transform to ordinary differential and integral equations. Generalized functions. Fundamental solutions and Green's function for partial differential equations. Week solutions of elliptc boundary value problems. Applications of Fourier transform to partial differential equations. Miscellaneous applications to equations of mathematical physics.

    MATH 511 Data Analysis with Mathematica
    Introduction with Mathematica, Random variables, Distributions, Expected values, Survey sampling, Estimation of parameters and fitting of probability distribution. Testing hypothesis, summarizing data, comparing two samples, the analysis of variances, the analysis of categorical data, linear least squares, decision theory.

    MATH 513 Mathematical Methods of Fluid Mechanics
    Euler’s Equations, rotation and vorticity. The Navier- Stokes Equations. Potential Flow. Boundary Layers. Vortex Sheets. Stability and Bifurcation. Characteristics. Shocks. The Riemann Problem. Combustion Waves.

    MATH 515 Real Analysis
    Lebesgue measure and Lebesgue integration. The Lebesgue spaces, General measure and integration, Decomposition of measures, Radon-Nikodym theorem, Extension of measures, Product measures and Fubini's theorem.

    MATH 516 Complex Analysis
    Analytic functions. Cauchy-Riemann equations. Harmonic functions. Elementary functions: the exponential function, trigonometric functions, hyperbolic functions. The logarithmic function and its branches. Contour Integrals and Cauchy’s theorem. Cauchy integral formula. Liouville’s theorem and the fundamental theorem of algebra. Maximum moduli of functions. Incompressible and irrotational flow. Complex potential. Laurent’s series and classification of singularities. Sources and vortices as singular points of potential flow. Calculus of residues. Conformal mappings. Fractional linear transformations. Applications of conformal mappings. Laplace’s equation. Electrostatic potential. Elements of elliptic functions. Analytic continuation and elementary Riemann surfaces.

    MATH 517 Advanced Linear Algebra
    Analytic functions. Cauchy-Riemann equations. Harmonic functions. Elementary functions: the exponential function, trigonometric functions, hyperbolic functions. The logarithmic function and its branches. Contour Integrals and Cauchy’s theorem. Cauchy integral formula. Liouville’s theorem and the fundamental theorem of algebra. Maximum moduli of functions. Incompressible and irrotational flow. Complex potential. Laurent’s series and classification of singularities. Sources and vortices as singular points of potential flow. Calculus of residues. Conformal mappings. Fractional linear transformations. Applications of conformal mappings. Laplace’s equation. Electrostatic potential. Elements of elliptic functions. Analytic continuation and elementary Riemann surfaces.

     MATH 518 Numerical Linear Algebra
    Solution of linear equations, eigenvector and eigenvalue calculation, matrix error analysis, reduction by orthogonal transformation, iterative methods.

    MATH 519 Methods of Mathematical Physics
    Vector and Tensor Analysis, Potential Theory and Dirac Delta Function Matrices and Groups, Continuous Groups , Distributions, Hilbert Spaces , Differential Equations, Nonhomogeneous Equations , The Special Functions I, The Special Functions II, Fourier Series and Integral Transform, Laplace, Mellin and Hankel Transforms , .Calculus of Variations , Integral Equations.

    MATH 521 Module and Ring Theory I
    Category of Modules. Products. Coproduct. Generators and Cogenerators. Injective and Projective Modules. Essential Extensions. Injective hulls. Superfluous epimorphisms. Projective Covers. Semisimple Modules and Rings. Socle and Radical of Modules and Rings. Radical of
    Endomorphism Rings. Co-Semisimple Modules and Rings.

    MATH 522 Module and Ring Theory II
    Finitely Presented Modules. Coherent Modules and Rings. Noetherian Modules and Rings. Finitely Copresented Modules. Artinian and co-Noetherian Modules. Flat Modules. Regular modules and Rings. (Semi)hereditary Modules and Rings. Supplemented Modules. (Semi)perfect Modules and Rings.

    MATH 523 Algebraic Topology I
    Topological Spaces. Separation Axioms. Continuous Functions.
    (Path)connected Spaces. Compact Spaces. Homotopy. Fundamental Group of Topological Spaces. Homotopy Groups. Exact Sequences for Homotopy Groups.

    MATH 524 Algebraic Topology II
    Standard Simplexes. Complexes of Simplexes. Boundaries. Singular Simplexes. Chain Complexes. Singular Homology group of Topologic Space. Homology groups of couples. Complete sequence of couple. Computation of Homology group. Relations between Homotopy and Homology groups.

    MATH 525 Introduction to Homological Algebra
    Modules. Isomorphism Theorems. Category and Functor. Exact Séquences. 5-Lemma.3x3-Lemma. Pullback and Pushout Diagrams. Functor Hom. Injective Modules. Projective Modules. Tensor Product. Flat Modules. Relation Between Hom and Tensor Product. Complexes and Homology. Injective and Projective Resolutions. Derived Functors. Exact Sequences for Derived Functors.

    MATH 527 Basic Abstract Algebra
    Integers. Sets, Linear Algebra. Groups. Subgroups, Factor Groups. Isomorphism Theorems. Finitely Generated Abelian Groups. Rings. Ideals. Maximal, Prime Ideals. PID. Irreducible Polynomials. Fields. Algebraic Extensions. Modules. Exact Sequences.

    MATH 529 Abelian Groups
    Abelian Groups. Direct Sum and Direct Product. Free and Divisible Groups. Direct Summands. Pullback and Pushout Diagrams. Direct and Inverse Limits. Topological Groups. Completeness. Pure subgroups. Basic subgroups.

    MATH 530 Quantum Calculus
    q-Derivative and h-Derivative, Generalized Taylor’s Formula for Polynomials Gauss’s Binomial Formula, q-Binomial Coefficients and Linear Algebra over Finite Fields, Two Euler’s Identities, Jacobi’s Triple Product Identity, q-Hypergeometric Functions Ramanujan Product Formula, q-Antiderivative, q-Gamma and q-Beta Functions, Bernoulli Polynomials and Bernoulli Numbers, Applications in Number Theory and Combinatorial Analysis, Applications in Physics, Applications in Statistics and Engineering, Non-linear Resonance Theory of Particles

    MATH 531 Numerical Solutions of Ordinary Differential Equations
    Initial-value problems: Runge-Kutta, extrapolation and multistep methods. Stable methods for stiff problems. Boundary-value problems: Shooting and multiple shooting. Difference schemes, collocation. Analysis. Conditioning of boundary value problems. Consistency, stability and convergence for both initial and boundary value problems. Fourier transform techniques. Fourier analysis, Fourier spectral methods. Geometric integrators. Lie group methods, symplectic methods, Magnus series method.

    MATH 533 Ordinary Differential Equations
    This course develops techniques for solving ordinary differential equations. Topics covered include: introduction to First-Order Linear Differential Equations; Second-Order Differential Equations, existence and uniqueness theory for first order equations, power series solutions, nonlinear systems of equations and stability theory, perturbation methods, asymptotic analysis, confluent hyper geometric functions. Mathieu functions. Hill's equation.

    MATH 534 Partial Differential Equations
    General theory of partial differential equations; first order equations; classification of second order equations; theory and methods of solution of elliptic, parabolic, and hyperbolic types of equations; maximum principles; Green's functions; potential theory; and miscellaneous special topics.

    MATH 535 Perturbation Method For Differential Equations
    Dimensional analysis, scaling argument, asymptotic series, Regular and singular perturbation methods for algebraic Equation and linear ordinary differential equation, nonlinear oscillation and two timing, WKP method, Laplace`s method, Stationary phases, steepest descent, boundary layer theory.

    MATH 539 Numerical Analysis
    Error analysis, direct and iterative methods for linear systems of equations, solution of nonlinear equations, and systems of nonlinear equations. Interpolation and approximation theory, numerical differentiation and integration.

    MATH 540 Numerical Solution of Partial Differential Equations
    Finite difference schemes for parabolic, hyperbolic, elliptic equations. Order of the Accurancy of finite difference schemes. Stability of and convergence for difference schemes. Leapfrog, Lax-Wendroff, implicit, ADI methods, SOR, direct methods for partial differential equations

    MATH 541 Graph Theory
    Graphs, varieties of graphs, connectedness, extremal graphs, blocks, trees, partitions, line graphs, planarity, Kuratowsky's theorem, colourability, chromatic numbers, five color theorem, four color conjecture

    MATH 543 Commutative Algebra
    Commutative rings. Prime ideals and maximal ideals. Nilradical and Jacobsob radical. Operation on ideals. Modules over commutative rings. Nakayama Lemma. Tensor product of modules. Restriction and extension of scalars. Exactness property of tensor product. Rings and modules of tractions. Local properties. Extended and contracted ideals in rings of tractions.

    MATH 544 Introduction to Commutative Algebra
    Commutative rings, prime and maximal ideals. Primary decomposition. Modules. over commutative rings. Chain conditions on modules. Modules over principal ideal domains. Integral dependence on subrings. Dimension theory.

    MATH 546 Advanced Module Theory
    Category of Modules.Generators and Cogenerators.M-generated Modules. Category ?(M). Generators in ?(M). M-injective Modules. Self-injective Modules. M-projective Modules. Local Rings. Finitely Presented Modules. Inverse Limits. Finitely Copresented Modules. Finite Uniform Dimension. Complements and Uniform Dimension. Extending Modules.Locally Noetherian Extending Modules.Locally Artinian Modules.

    MATH 551 Probability Theory I
    Axioms of Probability. Combinatorial methods. Conditional Probability and Independence. Distribution Functions. Random variables (Discrete and Continuous). Joint Distributions. Sum of Random Variables. Expectations and Variances. Limit Theorems. Notions of measure theory, measurable functions and integration.

    MATH 552 Probability Theory II
    Finitely Presented Modules. Coherent Modules and Rings. Noetherian Modules and Rings. Finitely Copresented Modules. Artinian and co-Noetherian Modules. Flat Modules. Regular modules and Rings. (Semi)hereditary Modules and Rings. Supplemented Modules. (Semi)perfect Modules and Rings.

    MATH 553 Stochastic Processes and Their Applications
    Probability Spaces and Random variables. Expectation and Independence. Bernoulli Processes and Sums of Independent Random Variables. Poisson process. Markov Chains. Limiting Behavior and Applications of Markov chains. Potentials, Excessive functions, and Optimal Stopping of Markov Chains. Markov Processes. Elements of Brownian motion, Gaussian process.

    MATH 554 Brownian Motion and Schrödinger's Equation
    Basic concepts, killed Brownian motion, Schrodinger's operator, Stopped Keynman-Kac functional, Conditional Brownian motion and conditional gauge, Green functions, Condition gauge and q-Green function.

    MATH 559 Mathematics and Technology for High School Teachers
    Sets and functions. Limit, derivative and their applications. Linear algebra with Mathematica, Scientific Work Places and Graphic calculators.

    MATH 560 Computer Assisted Problem Solving
    Fundamental concepts in computer assisted problem solving. Solution algorithms. Some software programs. Project from mathematical topics.

    MATH 563 Introduction to Finite Elements
    Variational formulation of Elliptic Boundary Value Problems. Galerkin-Ritz approximation. Finite element interpolation in Sobolev spaces. Error estimates. Computer implementation of Finite Element Method(FEM). Stabilized FEMs.

    MATH 564 Functional Analysis
    Linear metric and normed spaces, duality, weak topology, spaces of functions, generalized derivatives and distributions, Sobolev spaces, linear operators.

    MATH 565 Introduction to Spectral Theory
    Hilbert Spaces. Spectral theory in finite dimensional spaces. Spectral properties of bounded ad compact linear operators. Spectral theorem of bounded normal operators. Spectral representation of bounded self-adjoit operators. Unbounded linear operators and their adjoints. Closed operators. Spectral representation of unitary operators. Spectral representation of unbounded self-adjoint operators. Regular Sturm-Liouville operators. Linear operators in quantum mechanics.

    MATH 566 Mathematical Foundations of Finite Element Method
    Theoretical foundations of finite element method for elliptic boundary value problems, Sobolev spaces, interpolation in Sobolev spaces, variational formulation of elliptic boundary value problems, basic error estimates, applications to fluid dynamics, practical aspects of the finite element method.

    MATH 567 Mathematical Methods of Quantum Mechanics I
    The Basic Concepts of Quantum Mechanics. Schrodinger's equation. The theory of Symmetry Selected Applications.

    MATH 568 Mathematical Methods of Quantum Mechanics II
    The Basic Concepts of Quantum Mechanics. Schrodinger's equation. The theory of Symmetry Selected Applications.

    MATH 571 Mathematical Methods of Classical Mechanics I
    Basic concepts of analytical mechanics. Generalized coordinates. Variational principles of mechanics. Hamilton’s principle of least action. Euler-Lagrange equations. Lagrangian for a system of particles. Conservation laws. Energy, momentum, angular momentum. Mechanical similarity and virial theorem. Systems with one and two degrees of freedom. Phase flow. Motion in a central field. Kepler’s problem. Elastic collisions. Rutherford’s formula. Small oscillations. Normal coordinates. A chain of coupled oscillators. Rigid body motion. The Euler angles. The Cayley-Klein parameters. Inertia tensor. The Euler top. The Lagrange top.

    MATH 572 Mathematical Methods of Classical Mechanics II
    Hamiltonian mechanics. Legendre’s transformation. Hamilton’s equations. Hamilton’s function and energy. Cyclic coordinates. Routh’s function. Variational principle. The action as a function of coordinates. Maupertui’s and Fermat’s principles. Poisson brackets. Momentum space. Hamiltonian dynamics in rotating frame. Canonical transformations. Geometrical theory of the phase space. Symplectic structure. Infinitesimal canonical transformations. Conservation theorems and Poisson brackets. Hamilton’s mechanics in arbitrary variables. Hamilton-Jacobi equation. Principal and characteristic functions. Separation of variables. Central force problem. Action-angle variables.

    MATH 573 Modern Geometry I
    Groups of transformations of Euclidean and pseudo-Euclidean spaces. The theory of curves. The theory of surfaces in three-dimensional space. The Riemannian metric. The second fundamental form. The Poincare model of Lobachevsky’s geometry.The complex geometry. Surfaces in complex space. The conformal form of the metric on a surface. Isothermal co-ordinates. Gaussian curvature in terms of conformal co-ordinates. Surfaces of constant curvature. The Fundamental Theorem of Surfaces. Gauss-Weingarten equations. Theorema Egregium of Gauss. Surfaces of constant negative curvature and the “Sine-Gordon” equation. Minimal surfaces. The Concept of a Manifold and the simplest Examples.

    MATH 574 Modern Geometry II
    Tensors. Algebraic Theory and transformation rules. Skew-symmetrical tensors. Differential forms. Tensors in Riemannian and pseudo-Riemannian spaces. Vector fields and Lie algebras. The Lie derivative. The fundamental matrix Lie algebras. The exterior derivative and integration of differential forms. The general Stokes formula. Differential forms on complex spaces. The Kahlerian metrics. The curvature form. Covariant differentiation and the metric. Parallel transport of vector fields. Geodesics. The Riemann curvature tensor. The general curvature tensor. The symmetries of the curvature tensor. Examples of the curvature tensor in spaces of dimensions 2 and 3. The simplest concepts of the general theory of relativity.

    MATH 575 Integral Equations
    Classification of integral equations. Integral equations solvable by integral transforms. Fredholm's theory of the linear integral equation of the second kind. Volterra's integral equation of the second kind. Volterra's integral equation and its solution by Lioville's iteration method. Study of linear integral equations by Schmidt's method.

    MATH 576 Introduction to Soliton Theory
    IVP for Burgers’ equation and Cole-Hopf transformation. Shock solitons and their dynamics. Backlund transformation. General solution of the Liouville equation. The Sine-Gordon equation. Topological soliton. Bianchi permutability theorem and nonlinear superposition principle. Multi-soliton solutions. Collisions and bound states of solitons. From Riccati equation to the inverse scattering transform. Zero curvature and Lax representations. Zakharov-Shabat problem. Hirota direct method in soliton theory. KdV equation and the Schrodinger spectral problem. Elements of quantum scattering theory. The inverse problem. Gel’fand-Levitan-Marchenko equation and N-soliton solution. Analytic properties of the scattering amplitude. Integration of the KdV equation. Infinite hierarchy of integrals of motion. Current developments in soliton theory.

    MATH 577 Supersymmetric Quantum Mechanics
    Review of Schrodinger equation. Factorization method. Construction of the hierarchy of Hamiltonians. Partner Hamiltonians. Examples- Harmonic oscillator and Morse potential. Supersymmetry and the radilal Problem. Isotropic oscillator. Breaking of SUSY in Quantum Mechanics. Supersymmetric WKB Approximation. Witten’s index. Examples: Electron in magnetic field.

    MATH 578 SU(3)-symmetry and Quarks
    Symmetries in classical and quantum mechanics. Isospin Operators for multinucleon systems. Multiplets. Hypercharge. Generators and Lie algebra of SU(3). Hadron states classification and quarks. The transfrmation properties of quark states. Colour. Construction of SU(3) Multiplets from the elementary representations. Meson multiplets.
    Quark models with inner degrees of freedom. Confinement. Particles with charm and SU(4). Beauty and Truth.

    MATH 581 Topology I
    Fundamental concepts; subbasis, neighborhoods, etc. Continuous functions. Subspaces, product spaces and quotient spaces. Weak topologies and embedding theorem. Convergence by nets and filters. Separation and countability. Compactness, local compactness and compactifications. Connectedness. Paracompactness. Metrization and Baire category. Uniform spaces and function spaces. Stone-Weierstrass theorem.

    MATH 582 Topology II
    Continuation of MATH 581
    Prerequisite : MATH 581

    MATH 585 Symmetries and Groups
    Basic group theory and representations. Symmetry groups in physics. Groups and differential equations.

    MATH 586 Hilbert spaces and Quantum theory
    The basic concepts of classical and quantum mechanics. Quantum field theory. Classical topology and Quantum states. Supersymmetric Quantum Theory and the Index Theorem

    MATH 588 Fractal Geometries
    The discovery of fractal geometry, Measures of dimensions. Derivatives of non-integral order, Compositions of fractal geometries. Measures and uncertainty. Fractal morphogenesis.

    MATH 590 Time Scales

    MATH 596 Graduate Seminar
    Oral presentations on topics dealing with current research and technical literature. Includes presentation of latest research results by quest lecturers, staff and advanced students.

    MATH 597 Comprehensive Studies
    Students must complete four projects in the basic areas of mathematics and then must pass a written exam in each project. Four projects can be taken in the following topics: Algebra, Real Analysis, Complex Analysis, Functional Analysis, Ordinary Differential Equations, Partial Differential Equations, Geometry, Topology, Numerical Analysis

    MATH 598 Selected Topics in Applied Mathematics
    Selected topics in mathematical problems arising from various applied fields such as mechanics, economics, etc. Prerequisite : Consent of instructor.

    MATH 599 Master Thesis
    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 600 Ph.D. Thesis
    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 9XX Special Topics
    Graduate students as a group or PhD choose and study advanced topics under the guidance of faculty member, normally his/her advisor.

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