## Content of Courses for Mathematics MSc Program

**MAT701 Fuzzy Theory I (3 0 3) 6 ECTS**

The concept of fuzzy set; Classical Sets and their characteristic values; fuzzy set and Membership Degrees; the basic properties of fuzzy sets; fuzzy point; α-cuts sets; fuzzy interval; operations on fuzzy intervals; fuzzy normed linear spaces; operators on fuzzy normed linear spaces; fuzzy limit; fuzzy continuity.

**MAT702 Fuzzy Theory II****(3 0 3) 6 ECTS**

Fuzzy ring; fuzzy algebra, fuzzy α-algebra, characteristics of fuzzy α - algebras; Introduction to fuzzy measure theory; fuzzy integral theory and some important theorems.

**MAT711 Algebra I (3 0 3) 6 ECTS**

p-Groups, Cauchy's Theorem, The First Sylow Theorem, Sylow's group, the Second Sylow Theorem, Third Sylow Theorem, Applications of Sylow theory.

**MAT712 Algebra II (3 0 3) 6 ECTS**

R-Modules, Submodules and Direct Sums, R-Homomorphisms and Quotient Modules, Free Modules, Tensor Product, Complexes, Homology and Short Complete Sequences.

**MAT718 Advanced Statistical Methods (3 0 3) 6 ECTS**

Distributions, the binomial distributions, sampling theory, the smallest quadratic sum, curve fitting and regression.

**MAT721 Advanced Projective Geometry I (3 0 3) 6 ECTS**

Euclidean geometry and other geometries, affine planes, projective planes, other geometric structures, Dezarg planes, Pappus planes Fano planes.

**MAT722 Advanced Projective Geometry II (3 0 3 ) 6 ECTS**

Transformations in projective planes, Isomorphism and automorphism, perspective and projectivity, Central Collineations, Correlations, the algebraic structure of projective planes.** **

**MAT723 Finite Geometries I (3 0 3) 6 ECTS**

Finite structures, the geometry of finite vector spaces, combinatorial features, built-in and expansion, Automorfizim, affine and projective spaces, combinatorics of finite planes, Correlation and polarities.

**MAT724 Finite Geometries II (3 0 3) 6 ECTS**

Collineations of finite planes, Collineation groups, Central Collineation, the construction of finite planes, IV. type planes, Inverse planes combinations, Automorfizm.

**MAT725 n-dimensional Projective Spaces I (3 0 3) 6 ECTS**

Extension of Affine space to projective space, n-dimensional projective spaces, General projective coordinates, hyperplane coordinates, the principle of duality, dual rate, Projectivities, linear n-dimensional projective space onto itself Projectivities, Correlations.,

**MAT726 n-dimensional Projective Spaces II (3 0 3) 6 ECTS**

Second order Hypersurfaces, the projective classification of second order hypersurfaces and projective properties, affine and the metric classification of second order hypersurfaces.

**MAT731 Advanced Real Analysis I (3 0 3) 6 ECTS**

Basic Concepts, some main definitions and theorems on point sets, countability, Measurement Theory, Zero Measure Sets, Lusin Egerof and Lebesgue's Theorem, Not Measurable Sets, measurable functions, Riemann integral, Some Applications of the Lebesgue theory, Theory of General Measure and Integral.

**MAT732 Advanced Real Analysis II (3 0 3 ) 6 ECTS**

For multivariable the Lebesgue integral, Fubini Tonelli Hobson Theorem, Functions and Limited Variations.

**MAT735 Advanced Measurement Theory I (3 0 3 ) 6 ECTS**

measurable numerical functions, Borel sets, basic functions, normal representation of basic functions; integral of basic function, non-negative measurable functions and integrals, monotone convergence theorem, positive, negative part of function and their integrals, almost everywhere properties.

**MAT736 Advanced Measurement Theory II (3 0 3 ) 6 ECTS**

Convergence of integrable functions, the convergence of the measurement; Lp spaces, approximation of functions in Lp, Fourier series, absolute continuity, Lebesgue integral and its properties, its relationship with the Riemann integral, Lp spaces.

**MAT741 Topology I (3 0 3) 6 ECTS**

Metric Spaces, Topological Spaces, Interior, Closure, Boundary and Accumulation Points of a Set, Bases and Countability in Topological Spaces, Continuous Functions, Homeomorphism, Subspaces, Product and Quotient Spaces.

**MAT742 Topology II (3 0 3) 6 ECTS**

Convergence in Topological Spaces, Separation Axioms, Metrization, Connectedness, Compactness, Homotopy and Fundamental Groups, Simplicial Complexes, Homology.

**MAT743 Differential Topology I (3 0 3) 6 ECTS**

Linear Maps from IRn to IRm, Differentiable Maps from IRn to IRm, Properties of Differential, Inverse Function Theorem, Implicit Function Theorem, Local and Global Homeomorhisms, Brouwer Fixed Point Theorem, Maximum and Minimum Problems.

**MAT744 Differential Topology II (3 0 3) 6 ECTS**

Manifolds and Differentiable Structures, Vector Bundles, Immersion, Submersion and Embeddings, Transversality, Sard’s Theorem, Morse Lemma, Endpoint Map, Riemannian Manifolds, Vector Fields, Gradient Vector Fields, Manifolds with Boundary, Gluing Manifolds Together.

**MAT745 Topological Groups I (3 0 3) 6 ECTS**

Topological groups, Homomorphism, Homogen spaces.

**MAT746 Topological Groups II (3 0 3 ) 6 ECTS**

Topological groups, Lie Groups, Regular structures, Completeness.

**MAT749 Symbolic Programming Language I (3 0 3) 6 ECTS**

Programming techniques, Prolog programming language.

**MAT750 Computer AlgebraI (3 0 3) 6 ECTS**

Maple programming language, integral and differential equations.

**MAT751 Functional Analysis I (3 0 3) 6 ECTS**

Normed Spaces, Banach Spaces, Uniform Boundedness Principle, Hahn-Banach Theorem.

**MAT752 Functional Analysis II (3 0 3) 6 ECTS**

Linear Operators, Weak convergence, Self-adjoint operators.

**MAT753 Advanced Complex Analysis I (3 0 3) 6 ECTS**

Topological and Complex Regions, Power Series, Analytic functions, Mobius transformations, Cauchy's Theorem, Open Function Theorem, Maximum Modulus Theorem, Schwarz Lemma, Residue Theorem, Argument Rule, Rouche's theorem, normal families, Arzella - Ascoli's Theorem, Montel's Theorem, Riemann Mapping theorem.** **

**MAT754 Advanced Complex Analysis II (3 0 3) 6 ECTS**

Schwarz's rule, analytic continuation, Weierstrass factorization theorem, Runges Theorem, Theorem of Mittag-Leffers, Monodoromy Theorem, Harmonic Functions, Sub-Harmonic Functions.

**MAT755 Theory of Linear Positive Operators I (3 0 3) 6 ECTS**

Function spaces, Properties of C[a,b] and the space of integrable functions; Linear positive operators, Theorem, linear positive functionals; divided difference; Bernstein Polinomials, Korovkin Theorem; Weierstrass approximation; linear positive operators and convergence theorems.

**MAT756 Theory of Linear Positive Operators II (3 0 3) 6 ECTS**

Weighted function spaces; Korovkin type Theorems; Korovkin type theorems for functions defined on unbounded sets; analyticity in unit circle, k-positivity, Taylor coefficients, approximation in the unit circle the analytic functions space.

**MAT761 Advanced Numerical Analysis I (3 0 3) 6 ECTS**

Interpolation with divided differences, Lagrange method, finite difference, finite difference interpolation.

**MAT762 Advanced Numerical Analysis II (3 0 3) 6 ECTS**

Numerical solutions of equations, least squares method, various types of approximation methods, numerical solution of differential equations.

**MAT763 Applied Mathematical Analysis I****(3 0 3) 6 ECTS**

Laplace transform and its applications, Special functions and their applications, applications of special polynomials, inverse Laplace transform and its applications.

**MAT764 Applied Mathematical Analysis II****(3 0 3) 6 ECTS**

Fourier series and applications, Fourier integrals and applications, introduction to stability of the differential equations.** **

**MAT765 Well Posed Problems in Tikhonov Sense I (3 0 3) 6 ECTS**

The definition of in terms of Hadamard and Tikhonov well and bad posed problems. Tikhonov's theorem

**MAT766 Well Posed Problems in Tikhonov Sense II (3 0 3) 6 ECTS**

Regularizing operator, Stability Analysis, Operator Equations and Operator Equations with Variable Coefficients for the Stability Analysis, an example of the Cauchy problem for the heat equation, Integral Geometry Problems on examples, exercises.

**MAT769 Integral Transforms (3 0 3) 6 ECTS**

Laplace transforms, inverse Laplace transforms, special functions, Special Polynomials, Fourier transforms, application of boundary value problems.

**MAT771 Mathematics (3 0 3) 6 ECTS**

Linear transformations, vector analysis, multiple integrals, integral theorems, Fourier series, Curve fitting.** **

**MAT772 Advanced Math (3 0 3) 6 ECTS**

Curve integrals, surface integrals, integrals over hyperplanes, Divergence Theorem, Stokes Theorem, Green's Theorem.

**MAT773 Computer Package Programs****(3 0 3) 6 ECTS**

Basic filing systems, data-based programming, Clipper, and Fortran programming language.** **

**MAT782 Advanced Analysis II****(3 0 3) 6 ECTS**

Average functions, properties of average functions, convergence, absolute continuity, generalized derivatives and properties, compact operators, Fredholm operator, singular operators in the space of continuous functions.

**MAT783 Application of Matehematics I****(3 0 3) 6 ECTS**

Modeling, growth and parameters: exponential growth, modeling the growing limits, logistic equation, estimation of parameters, non-dimensionalization. Steady-state and linear stability analysis. Interacting Modelling Examples of modeling: predator-prey model, the Lotka-Volterra model, the mutualistic models, chemical kinetics problems.** **

**MAT784 Application of Matehematics II****(3 0 3) 6 ECTS**

Introduction to partial differential equation models: the reaction term, conservation law of balance, Transport equation. Diffusion: The initial and boundary values, the partial differential equation systems. The steady-state behaviors of the partial differential equations: Laplace's equation, the diffusion-chemotaxis model, the density-dependent diffusion.

**MAT797 M. Sc. Seminar (0 2 0 ) 3 ECTS**

Searching a specific topic based on multiple sources, making a written text in accord with spelling rules, and oral presentation of this to an audience. Acquisition of ability of discussion for the mutual exchange of information in verbal presentations.

**MAT798 M. Sc. Specialization (2 0 0) 15 ECTS**

Following the appointment of consultants for M.Sc. students, a theoric course which is opened in the semesters, midterm and summer vacations. Allows M.Sc. students to gain scientific ethic and discipline

**MAT799 M. Sc. Thesis Research (0 1 0) 15 ECTS**

Following the appointment of consultants for M.Sc. students, a practical course which is opened in the semesters, midterm and summer vacations.

## Content of Courses for Mathematics PhD Program

**MAT801 Approximation Theory I (3 0 3) 6 ECTS**

Radon measure, Some basic properties of probability, Co-semi groups, Cauchy problem, Determination of sub-spaces for a limited positive Radon measure, Determination of sub-spaces for a discrete Radon measure, Convergent sup-spaces, Determination of sub-spaces for the Dirac measures.

**MAT802 Approximation Theory II****(3 0 3) 6 ECTS**

Korovkin closure, Korovkin sub-spaces and their applications, Korovkin sub-spaces for finite positive operators, Korovkin-type theorems for unit operators.** **

**MAT803 Hilbert Spaces I (3 0 3) 6 ECTS**

Linear spaces, Unit Orthogonal expansion, Bilinear forms.

**MAT804 Hilbert Spaces II (3 0 3) 6****ECTS**

Linear Operators, Hilbert spaces, Dual spaces, Weak compact operators.

**MAT805 C*-Algebras I****(3 0 3) 6****ECTS**

Involute Algebras, C*-Algebras, Functional calculations, Approximately unit positive forms, K-theory for C*-Algebras.

**MAT806 C*-Algebras II****(3 0 3) 6****ECTS**

Second dual, Tensor product derivatives, Automorphism.

**MAT807 Hp-Spaces****(3 0 3) 6****ECTS**

Harmonic functions in z <1, introduction to Hp- spaces, Boundary values of analytic functions, Jensen formula, İnternal and external functions product, decomposition, Hp-sapces on the upper half plane.

**MAT808 Analytic Functions****(3 0 3) 6****ECTS**

Analytic Function spaces, Gamma and Riemann Zeta functions, analytic continuity.

**MAT809 Potential Theory and Subharmonic Functions (3 0 3) 6** **ECTS**

Dirichlet problem, Poisson kernel, Harnack theorem, the principle of maximum, potential theory, Green functions.

**MAT811 Differentiable Manifolds I****(3 0 3) 6****ECTS**

Differentiable Manifolds, Vector fields and forms.

**MAT812 Differentiable Manifolds II****(3 0 3) 6****ECTS**

Submonifolds theory, Whitney embedding theorems, flat embeddings.

**MAT813 Algebraic Topology I****(3 0 3) 6****ECTS**

Singular homology, cover spaces, cohomology, products, manifolds and dualite.

**MAT814 Algebraic Topology II****(3 0 3) 6****ECTS**

Vector bundles, Grassmanian manifolds, characteristic classes and numbers, bordism and cobordism.

**MAT821 Commutative Algebras****(3 0 3) 6****ECTS**

Commutative rings, Koszul complex, Cohen-Macaulay and smooth rings.

**MAT822 Homological Algebra****(3 0 3) 6****ECTS**

Cohomology groups, expansions, Spectral series.

**MAT831 Differential Geometry I****(3 0 3) 6****ECTS**

Riemmannian Manifolds and Riemannian Geometry.

**MAT832 Differential Geometry II****(3 0 3) 6****ECTS**

Curvature, Riemann submanifolds, Gauss-Bonnet theorem, Jacobi fields, curvature and topology.

**MAT833 New Topics in Geometry and Topology (3 0 3) 6****ECTS**

The Pushout spaces of manifolds.

**MAT841 Combinatorics****(3 0 3) 6****ECTS**

Binomial and Polya processings, Poset theory, Delta operators, Complexes and combinatorial properties.

**MAT851 Measurement of Topological Spaces I****(3 0 3) 6****ECTS**

Baire and Bore sets, measurements, smoothness of plish spaces, convergence of Baire measures.

**MAT852 Measurement of Topological Spaces II (3 0 3) 6****ECTS**

Fourier transformations of measurements, differentiables of Fourier transformations.

**MAT853 Advanced Fuzzy Measurement Theory I (3 0 3) 6 ECTS**

Fuzzy measurement, Fuzzy s-measurement, properties of Fuzzy s-measurement, fuzzy measuable functions, fuzzy integral theory.** **

**MAT854 Advanced Fuzzy Measurement Theory II (3 0 3) 6 ECTS**

Fuzzy integral, transformation theorems.

**MAT855 Harmonic Analysis****I (3 0 3) 6 ECTS**

Local compact spaces, integral spaces.

**MAT856 Harmonic Analysis II****(3 0 3) 6 ECTS**

Convolution, representation, character group.** **

**MAT861 Well-Posed Problems in terms of Hadamard I****(3 0 3)****6 ECTS**

Well-posed and ill-posed problems in terms of Hadamard, Hadamard example, Hadamard well-posed problems in terms of reduced physical examples, Some example and applications, Cauchy problems.

**MAT862 Well-Posed Problems in terms of Hadamard II****(3 0 3)****6 ECTS**

Regülatör, fort he operatör equation of the first type of regulatory, regulatory fort he problem of integral geometry.

**MAT863 Integral Geometry Problems I****(3 0 3)****6 ECTS**

Radon problem, some integral geometry problems.

**MAT864 Integral Geometry Problems II****(3 0 3)****6 ECTS**

Riemann metrics, Nonlinear IGP, IGP and invers problems of the scattering theory.

**MAT865 Self-Adjoint Compact Operator Theory I (3 0 3) 6 ECTS**

Basic properties of Hilbert spaces, Linear operators, ortonormal sets, compact operators.

**MAT866 Self-Adjoint Compact Operator Theory II****(3 0 3) 6 ECTS**

Spectrum, Spectral analysis of compact operators, Sturm-Liouville Problem, spectral analysis of self-adjoint operators.** **

**MAT867 Inverse Problems for the Differential Equations and Its Applications I (3 0 3) 6 ECTS**

Direct and inverse problems fort he differential equations, characteristic properties of inverse problems, general methods researching for inverse problems.

**MAT868 Inverse Problems for the Differential Equations and Its Applications II (3 0 3) 6 ECTS**

Inverse problems for transport equations, Inverse problems for kinetic equations and ıts applications on integral geometry.

**MAT871 Deductive Program Designing****(3 0 3) 6 ECTS**

Propositional logic, Knuth-Bendix algorithm, Herbrand theorem, deductive questioning.

**MAT872 Logic Programming****(3 0 3) 6 ECTS**

Finite programs, Prolog applications.

**MAT881 Nonlinear Functional Analysis****(3 0 3) 6 ECTS**

Analysis of Banach Spaces, Degrees of Brouwer function, Degrees of Leray-Schauder function, monoton operators, Navier-Stokes equation.

**MAT882 Banach Algebras (3 0 3) 6 ECTS**

Ideals and sub algebras, spectral transformation theorem, Maximum regular ideals, Gelfand topology, semi-simple Banach algebras.

**MAT884 Linear Operators Theory I****(3 0 3) 6 ECTS**

Continuity and boundedness of linear operators, products and sums of operators, inverse and adjoint operators, differential and integral operators.

**MAT885 Generalized Functions Theory I****(3 0 3) 6 ECTS**

Continuous linear functionals, Weak topology and weak continuity, test space and test functions, generalized functions.

**MAT886 Generalized Functions Theory II****(3 0 3) 6 ECTS**

Calculations of generalized functions, generalized derivatives, differential equations and generalized functions.

**MAT897 PhD. Seminar (0 2 0) 3 ECTS**

Searching a specific topic based on multiple sources, making a written text in accord with spelling rules, and oral presentation of this to an audience. Acquisition of ability of discussion for the mutual exchange of information in verbal presentations.

**MAT898 PhD. Field of Specialization****(3 0 0) 15 ECTS**

Following the appointment of consultants for PhD students, a theoric course which is opened in the semesters, midterm and summer vacations. Allows PhD students to gain scientific ethic and discipline.

**MAT899 PhD. Thesis Research (0 1 0) 15 ECTS**

Following the appointment of consultants for PhD students, a practical course which is opened in the semesters, midterm and summer vacations.