Matematik Anabilim Dalı Yüksek Lisans Dersleri ve İçerikleri

MATH 500 M.S. Thesis (Non-credit) özel ders,yuksek lisans,ödev,tez

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.

501 Complex Analysis I ( 3 0 3 )

özel ders,yuksek lisans,ödev,tez

Conform transformations, Riemann transformation theorem, Analytical continuation, monodomy theorem, Picard theorem

502 Topology II ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

The Fundamental group, homotopic maps, Construction of the fundamental group. Calculations. Homotopy type, The Brouwer fixed-point Theorem, The simplical complex surfaces. Classification, Orientation, Euler characteristic, Surgery momology theory, Cyeles and boundaries, The calculation of momology groups, degree and Litsehetz number, Euler Roinear’e formula, Borsuk-Ulam theorem, The Lifschets fixed- point theorem, Dimension, knots and covering spaces, Examples of knots, Group covering spaces, Alexander polynomials

522 Complex Analysis II ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

Compactness and Convergence in the Space of Analytic Functions, Space of Analytic Functions, Spaces of Meromorphic Functions, The Riemann Mapping Theorem, Weirstrass Factorization Theorem, Runge’s Theorem, Simple Connectedness, Analytic Continuation along a Path, Harmonic Functions, The Dirichlet Problem, Green’s Functions, Entire Functions.

532 Differential Topology ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

Differential Manifolds: Differentiable manifolds, local coordinates, Induced structures and examples, germs, tangent vectors and differentials Sard’s theorem and regular values, Local properties of immersions and submersions, Vector fields and flows tangent bundles, Embedding in Euclidean space, Tubular neighborhoods and approximations, Classical Lie groups, Fiber bundles induced bundles, Vector bundles and Whitney sums, Transversality. Cohomology: Multilinear algebra and tensors, Differential forms, Volume element and Orientation integration on forms, Stokes theorem, Relationship to singular homology, de Rham’s theorem and singular co homology, Products and duality: Cross product and the Kunneth theorem, co homology cross product, Cup and cap product, Orientation, Bundle Duality on compact manifolds, Intersection theory: The Euler class, Lefchetz numbers and vector fields, Gysin map and Stiefel Whitney classes, Cobordism and bordism: cobordism and orientable cobordism, Thom space and Thom homomorphism, bordism of a topological space.

533 Computer Algebra ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

The introduction to Gröbner Basis, Symbolic recipes for polynomial equations, factorisation of polynomials, Groebner Basis and Integer programming, Groebner Basis for codes, Groebner Basis for decoding, Automatic Geometry Theorem Proving, The Inverse Kinematics Problem in Robotic.

534 Perturbation Methods in Applied Mathematics ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

Ordering, Asymptotic Sequences and Expansions, Limit Process, Expansions, Matching, Regular Perturbation, Singular Problem, Singular Perturbation Problems with Variable Coefficients, Theorem of Erdelyi, Nonlinear Example for Singular Perturbation, Singular Boundary Problems, Method of Strained, Coordinates for Periodic Solutions, Two Variable Expansions Procedure, Weakly Nonlinear Systems, Strongly Nonlinear Oscillations, Limit Process Expansions for Second Order Partial Differential Equations, Singular Boundary Value Problems.

535 Variational Methods of Approximation ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

Existence and Uniqueness of Solutions, Linear Algebraic Equations, Linear Operator Equations, The Contraction Mapping Theorem, Variational Boundary Value Problems, Regularity of the Solutions The Lax MilgramTheorem, The Neumann Boundary Value Problems, The Boundary Value Problems with Equality Constraints, The Ritz Method Description Convergence and Stability, The Weighted Residual Method, The Bubnov-Galerkin Method, The Method of Least Squares, Collocation and Sub domain Methods, Time Dependent Problems.

536 Dynamical Systems ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

Second order differential equations in phase plane, linear systems and exponential operators, Canonical forms, Stability forms, Lyapunov functions, The existence of periodic solutions, Applications to various fields, Oscillation theory.

541 Algebra I ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

Groups: Generalities groups acting on a set, Sylow Theorems, free group, direct product and sums. Rings: Generalities, commutative ring, principal ideal domains, unique factorization domains. Euclidean domains, Noetherian rings, Hilbert’s Euclidean domains. Noetherian rings Hilbert’s Theorem, Field of fractions, Localization.

542 Algebra II ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

Galois Theory: Categories and functors, Generalities, Additive abelian categories, Yoneda’s lemma. Module categories: Definitions, Projective, injective modules, Semi-simple rings, Modules over Noetherian rings and principal ideal domains, Morita theory. Homological methods: The functors Ext, Tot, (co-) homology, Derived categories, Derived functors, Stable categories, Applications to co homology of groups, Schemes.

551 Topology I ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

Euler’s Theorem, Topological equivalence, surfaces, Abstract spaces, The Classification Theorem, Topological Invariants continuity, open and closed sets, cont. Functions, Peano curve, The Tietze ext. Theorem, Compactness and Connectedness, Heine-Borel Theorem, Product spaces , Path connectedness, Identification spaces, the torus, the cone construction, glueing lemma, Projective spaces attaching maps.

553 Topological Groups ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

Topological Groups and continuous homogeneous spaces, Lie groups and Differentiable manifolds, Tangent spaces, Adjoint representations, Lie algebra, Root decompositions, Weyl Groups and Dynkin diagrams

555 Algebraic Topology ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

General Topology, Group Theory, Modules, Euclidean spaces, Categories, Functors, Chain complexes, Chain homotopy, Singular homology, Exactness, Mayer-Vietuoris sequences, Some applications of homology, Axiomatic characterization of homology, Homology with coefficients, Universal coefficient theorem for homology, The Künnetth formula, Cohomology cup end cop products, Universal coefficient theorem for co homology of fiber bundle, The co homology algebra and the Steenrod squaring operations, Hurewietz homomorphism, CW complex spectral sequences.

557 Partial Differential Equations and Boundary Value Problems ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

Nonhomogeneous Problems, Heat Flow with Sources Method of Eigen function, Expansion with homogeneous Boundary Conditions, Method of Eigen function, Expansion Using Green’s Formula, Poisson’s Equation, Green’sFunctions for time – independent problems, Fredholm Alternative and Modified Green’s Functions, Green’s Functions for Poisson’s Equations, Perturbed Eigenvalue Problem, Infinite Domain Problems, Complex Form of Fourier Series, Fourier Transform and the Heat Equation, Fourier Sine and Cosine Transforms, Green’s Functions for Time-Dependent Problems, Green’s Functions for the Wave Equation, Green’s Functions for the Heat Equation, The Method of Characteristics for Linear and Quasi Linear Wave Equations, Characteristics for First Order Wave Equations ,The method of Characteristics for Quasi-Linear Partial Differential Equations, The Laplace Transform Solution of Partial Differential Equations, Elementary Properties of the Laplace Transform, Green’s Functions, Initial Value Problems for Ordinary Differential Equations, Inversion of Laplace Transforms Using Contour Integrals.

558 Applied Functional Analysis and Variational Methods in Engineering ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

General Concepts and Formulas, A Review of the Field Equations of Engineering Kinematics, Kinetics and Mechanical Balance Laws, Thermodynamics Principles, Constitutive Laws, Concepts from Functional Analysis, Vector Spaces, Linear Transformations and Functional Theory of Normed Spaces, Theory of Inner Product Spaces, Variational Formulations of Boundary Value Problems, Linear Functionals and Operators on Hilbert Spaces, Soboley Spaces and Concept of Generalized solution, The Minimum of a Quadratic Functional Problems with Equality Constraints, Existence and Uniqueness of Solutions, Linear Algebraic Equations, Linear Operator Equations, Variational Boundary Value Problems, Boundary Value Problems with Equality Constraints, Eigenvalue Problems, Variational Methods of Approximation, The Ritz Method, The Weighted Residual Method, Time Dependent Problems.

560 Rings and Modules ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

Rings: Ideals, Factorization in Commutative Rings, Rings of Quotients and Localization. Modules: Modules, Homomorphisms and Exact Sequences, Free Modules and Vector Spaces ,Projective and Injective Modules, Commutative Rings and Modules, Chain Conditions, Prime and Primary Ideals, Noetherian Rings and Modules, Dedekind Domains, The Structure of Rings, Simple and Primitive Rings, The Jacobson Radical, Semi-Simple Rings, The Prime Radical, Prime and Semi-Prime Rings.

565 Examples of Group I. (3 0 3)özel ders,yuksek lisans,ödev,tez

The Ascending Central Series and Nilpotent Groups, The Derived Series and Solvable Groups, Constructions, Free Groups and Presantations, Matrices, Finite Groups

566 Examples of Groups II (3 0 3)

Finite Groups, Infinite abelian groups, Infinite Nonabelian Groups , The group PSL(2,Q)

04 568 Generating Functions (3 0 3)

1. Classical Ortogonal Polynomials

2. Generating Functions ( Linear, Bilinear, Bilateral )

3. Obtaining Generating Functions

3.1. Series Rearrangement Technique

3.2. Decomposition Technique

3.3. Operational Technique

3.4. Lie Algebraic Technique

4. Lagrange’s Expansion and Gould’s Identity

569 Combinatorial Commutative Algebra and Toric Varieties (3 0 3)özel ders,yuksek lisans,ödev,tez

Convex Bodies. Combinatorial Theory of Polytopes and polyhedral sets. Polyhedral spheres. Minkowski sum and mixed volume. Lattice Polytopes and fans. Toric Varieties. Enumeration, sampling and integer programming, Grobner Bases, Betti numbers and localizations of toric ideals.

570 Introduction to Operator Theory (3 0 3)

Spectral Properties of Bounded Linear Operators.

Use of Complex Analysis in Spectral Theory.

Compact Linear Operators on Normed Spaces.

Spectral Properties of Compact Linear Operators on Normed Spaces.

Spectral Prtoperties of Bounded Self-Adjoints Linear Operatorts.

Positive Operators.

.Projection Operators

Unbounded Linear Operators and their Hilbert – Adjoint Operators.

Hilbert-Adjoint Operators , Symmetric and Self-Adjoint Linear Operators.

Closed Linear Operators and Closeres.

Spectral Properties of Self-Adjoint Linear Operators. Spectral Representation of Unitary Operators

04 571 Numerical Methods in Linear Algebra (3 0 3)

Linear Systems of Equations.

Matrix Algebra with Mathlab.

LU, LQ and QR Decomposition.

Orthogonal Vectors and Matrices.

Norms, Vector Norms and Matris Norms.

Bases and Dimension.

Linear Systems Revisited.

The Solution of Linear Systems Ax=B with Mathlab.

Find the Eigenvalues and Eigenvectors with Mathlab.

Eigenvalues and Eigenvectors of symmetric and non-symmetric matrices.

Orthogonal Diagonalization.

The Singular Value Decomposition.

573 Real Analysis ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

Introduction to measure theory, Abstract Integral, Positive Borel Measures, Riesz representation theorem, Lebesque measure.

04 574 Harmonic Analysis (3 0 3)özel ders,yuksek lisans,ödev,tez

Orthogonal systems, Orthogonal series, Fourier series, Some summability techniques, fundamental properties of harmonic functions, Convolution, Poisson’s integral, properties of the kernels of Abel-Poisson, Poisson, Weierstrass and Fejer integrals, Fejer type integral operator. Integral operators with positive kernel, Approximation properties of family of integral operator,

04 575 Fourier Analysis and Approximation (3 0 3)özel ders,yuksek lisans,ödev,tez

Fourier series, Fourier transformations in L1 space, Sequences of integral operators, Kernel function and their properties, Characteristic points of functions in L1 and Lp, Weierstrass theorem and approximation properties, Approximation properties of family of integral operator, Modulus of contiuous of functions in L1 and thier properties, Modulus of contiuous of functions in Lp and their properties, Convergence of integral operators on characteristic points, Convergence rate of integral operators on characteristic points.

04 576 Advanced Differential Geometry (3 0 3)özel ders,yuksek lisans,ödev,tez

Differentiable manifolds, Multilinear algebra, Exterior differential calculus, Connections, Riemannian geometry, Lie groups and moving frames, Complex manifolds, The geometry of the gauss map, The intricsic geometry of surfaces, Global differential geometry.

04 577 Homological Algebra I (3 0 3)özel ders,yuksek lisans,ödev,tez

Modules, Categories and Functors, Hom functor, Free Modules, Tensor Product, Adjoint Isomorphisms, Projective Modules, Injective Modules, Flat Modules, Purity, Specific Rings, Localization, Pullback and Pushout Systems, Direct and Inverse Limit.

04 578 Homological Algebra II. (3 0 3)özel ders,yuksek lisans,ödev,tez

Abelian Categories and Complexes, homology functors, Projective – injective resolutions and nth homology, Long exact sequence and Snake Lemma and homotopy, Comparision Theorem and Left Derived Functors, Horseshoe Lemma and covariant – contravariant right derived functors, Tor , Ext, Dimension, Covers and Envelopes.

04 579 Process Dynamics and Control I (3 0 3)özel ders,yuksek lisans,ödev,tez

Laws and languages of process control, time-domain dynamics and control, conventional control systems, advanced control systems, Laplace-domain dynamics and control, frequency-domain dynamics and control, process identification.

580 Functional Analysis ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

Normed spaces, Linear and multilinear transformations, The product of normed spaces, Series concept in normed spaces, Hilbert spaces.

04 581 Introduction to Digital Signal Processing (3 0 3)özel ders,yuksek lisans,ödev,tez

Discrete-Time Signals and Systems; The z-Transform; Frequency Analysis of Signals and Systems; Discrete Fourier Transform; Filter Structures; Filter Design Techniques; Applications of Digital Signal Processing

04 582 Statistical Signal Processing (3 0 3)özel ders,yuksek lisans,ödev,tez

Introduction to Random Processes; Correlation Function and Power Spectral Density of Stationary Processes; Noise Mechanisms, the Gaussian and Poisson Processes; Introduction to Signal Detection Theory; Linear Mean Square Filtering; Wiener and Kalman Filtering; Least Square Filtering; Signal Modeling and Parametric Spectral Estimation.

04 583 Mathematical Methods for Signal Processing (3 0 3)özel ders,yuksek lisans,ödev,tez

Linear operators and Matrix Inverses; Eigenvalues and Eigenvectors; Singular Value Decomposition; Some Special Matrices and Their Applications; Detection Theory Algorithms; Estimation Theory Algorithms; Optimal Filtering Algorithms; Methods of Iterative Algorithms; Least Mean Square Adaptive Filtering; Neural Networks; Clustering; Methods of Optimization.

04 584 Process Dynamics and Control II (3 0 3)özel ders,yuksek lisans,ödev,tez

Multivariable process, analysis of multivariable systems, design of controllers for multivariable processes, sampled-data control systems, stability analysis of sampled-data systems, applications.

591 Differential Equations ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

Existence and Uniqueness theorems, linear equation with constant coefficients, nonlinear equations, classification of points solving equations using transformations.

592 Partial Differential Equations ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

Diffusion equation, Wave equation, classification of second order Particular Differential equations, Cauchy problems solution using Fourier series, separation of variables

598 Algebraic Geometry ( 3 0 3 )özel ders,yuksek lisans,ödev,tez

Introduction, Plane Curves and Conics, Bezout’s Theorem, Affine and Projective Varieties, Noetherian Rings, Hilbert Basis Theorem, Nullstellenzats, Zariski Topology, Noether Normalization, Projective Space and Projective Varieties, Projective Closure of Affine Varieties, Rational Maps, Rational Map and Birational Isomorphism, Degree of a Rational Map, Blow-ups, Dimension of Varieties, Hilbert Function and Dimension of a Variety, Elementary Properties of Dimension, Dimension and Algebraic Independence, Dimension and Nonsingularity.

705 Seminar

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