Prof.Dr.Mehmet Çağlar

Prof.Dr.Evren Hınçal

Assoc.Prof.Dr.Harun Karslı

Assist.Prof.Dr.A.Mousa Othman

Assist.Prof.Dr.Murat Tezer

Assist.Prof.Dr.Burak Şekeroğlu

M 502 Topology II ( 3 0 3 )

M 503 Functional Analysis I ( 3 0 3 )

M 504 Functional Analysis ( 3 0 3 )

M 507 Partial Differential Equations and Boundary Value Problems ( 3 0 3 )

M 508 Partial Differential Equations ( 3 0 3 )

M 512 Real Analysis ( 3 0 3 )

M 519 Differential Equations ( 3 0 3 )

M 523 Algebra I ( 3 0 3 )

M 524 Algebra II ( 3 0 3 )

M 532 Differential Topology ( 3 0 3 )

M 536 Dynamical Systems ( 3 0 3 )

M 537 Fourier Analysis and Approximation (3 0 3)

M 551 Numerical Analysis and Optimization I ( 3 0 3 )

M 555 COMPLEX ANALYSIS I ( 3 0 3 )

M 556 COMPLEX ANALYSIS II ( 3 0 3 )

M 576 Applied Mathematics ( 3 0 3 )

M 583 Probability Theory I (3,0,3)

M 599 M.S. Thesis (Non-credit)

Course Content: 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.

M 502 Topology II ( 3 0 3 )

Course Content: 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

M 503 Functional Analysis I ( 3 0 3 )

Course Content: 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.

M 504 Functional Analysis ( 3 0 3 )

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

M 507 Partial Differential Equations and Boundary Value Problems ( 3 0 3 )

Course Content: 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.

M 508 Partial Differential Equations ( 3 0 3 )

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

M 512 Real Analysis ( 3 0 3 )

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

M 519 Differential Equations ( 3 0 3 )

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

M 523 Algebra I ( 3 0 3 )

Course Content: Proofs, sets, mappings, equivalence, induction, divisors and prime factorization, integers modules, permutations, an applications to cryptography, binary operations, groups, subgroups, cyclic groups and the order of an element, homomorphisms and isomorphisms, cosets and Lagrange’s theorem, groups of motion and symmetries, normal subgroups, factor groups, the isomorphism theorem, an application to binary codes.

M 524 Algebra II ( 3 0 3 )

Course Content: Examples of rings, basic properties of rings, integral domains and fields, ideals and factor rings, ring homeomorphisms, ordered integral domains, polynomials, factorization of polynomials over a field factor rings of polynomials, over a field, partieal fractions, symmetric polynomials, formal construction of polynomials, factorizing in integral domains, irreducible and unique factorization, principal ideal domains, vector spaces, algebraic extentions, splitting fields, finite fields, geometric constructions, the fundamental theorem of algebra, an applicatipn to cyclic and BCH codes.

M 532 Differential Topology ( 3 0 3 )

Course Content: 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.

M 536 Dynamical Systems ( 3 0 3 )

Course Content: 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.

M 537 Fourier Analysis and Approximation (3 0 3)

Course Content: 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.

M 551 Numerical Analysis and Optimization I ( 3 0 3 )

Course Content: Numerical Computations and Errors, Solving Nonlinear Equations, Solving sets of equations, Interpolation and Curve Fitting, Approximation of Functions, Numerical Integration Unconstrained and linearly constrained optimization, the Lagrange multipliers, general unconstrained minimization problems, Kuhn-Tucker theory, duality and its relation with classical optimization methods, one-dimensional search techniques, constrained gradient techniques, penalty-function and related some nonlinear programming methods, the solution of some estimation problems by computing algorithms, optimization techniques.

552 Numerical Methods in Linear Algebra (3 0 3)

1. Linear Systems of Equations.

2. Matrix Algebra with Mathlab.

3. LU, LQ and QR Decomposition.

4. Orthogonal Vectors and Matrices.

5. Norms, Vector Norms and Matris Norms.

6. Bases and Dimension.

7. Linear Systems Revisited.

8. The Solution of Linear Systems Ax=B with Mathlab.

9. Find the Eigenvalues and Eigenvectors with Mathlab.

10. Eigenvalues and Eigenvectors of symmetric and non-symmetric matrices.

11. Orthogonal Diagonalization.

12. The Singular Value Decomposition.

M 555 COMPLEX ANALYSIS I ( 3 0 3 )

Course Content: Fundamental concepts, elementary propertires of the complex numbers, further properties of the complex numbers, complex polynomials, holomorphic functions, the Cauch-Riemann equations, harmonic functions, real and holomorphic antiderivatives, complex line integrals, real and complex line integrals, complex differentiality and conformality, antiderivatives revisited, the Cauchy integral formula and the Cauchy integral theorem, Cauchy integral theorem and the Cauchy integral formula for more general curves, applications of the Cauchy integral, differentiability properties of holomorphic functions, complex power series, the power series expansion for a holomorphic function, the Cauchy estimates and the Liouville’s theorem, uniform limits of holomorphic functions, the zeros of a holomoırphic function, the behavior of a holomorphic function near an isolated singularity, expansion around singular poits, existence of Laurent expansions, the calculus of residues, applications of the calculus of residues to the calculation of definite integrals and sums, meromorphic functions and singularities at infinity.

M 556 COMPLEX ANALYSIS II ( 3 0 3 )

Course Content: The zeros of a holomorphic function, counting zeros and poles, the local geometry of holomorphic functions, further results on the zeros of holomorphic functions, the maximum modulus principle, the Schwarz lemma, holomorphic functions as geometric mappings, biholomorphic mappings of the complex plane to itself, biholomorphic mappings of the unit disc to itself, Möbius transformations, the Riemann mapping theorem, normal families, holomorphically simply connected domains, basic properties of harmonic functions, maximum principle and the mean value property, the Poisson integral formula, regularity of harmonic functions, the Schwarz reflection principle, Harnack’s principle, the Drichlet problem and subharmonic functions, the Perron method and the solution of the Drichlet problem, conformal mappings of annuli, basic concepts concerning infinite sums and products, the Weierstrass factorization theorem, the theorems of Weierstrass and Mittag-Lefler,applications of infinite sums and products, Jensen’s formula and an introduction to Blaschke products,the Hadamard gap theorem, entire functions of finite order.

M 576 Applied Mathematics ( 3 0 3 )

Course Content: 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.

M 583 Probability Theory I (3,0,3)

Course Content: Measure, integration, probability measures, measurable functions and random variables, probability measures in Rn, distribution functions, structure of conditional probability, structure of independence, limit theorems in Rn. Some probabilistic concepts, distiribution functions and their relationship, some moments and probability inequalities, the characteristic functions, moment generating functions and fractorial moment generating functions of some random vectors and related theorems, stochastic independence, transformations of random vectors, order statistics and related theorems, the multivariate normal distributions, Bayes and minimax estimations, quadratic forms.

M 599 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.

M 600 Seminar

**M 701 Advanced Abstract Algebra (3 0 3)**

**M 702 Math. Measure Theory ( 3 0 3)**

**M 703 Math. Integral Transforms ( 3 0 3)**

**M 704 Math. Spectral Theory of Linear Operators ( 3 0 3)**

**M 705 Math. Applied Functional Analysis ( 3 0 3)**

**M 706 Math. Dynamical Systems (3 0 3)**

**M 707 Math. Linear Positive Operators and Approximation I ( 3 0 3)**

**M 708 Math. Linear Positive Operators and Approximation II ( 3 0 3)**

**M 709 Math. Lie Algebras ( 3 0 3)**

**M 710 Math. Low Dimensional Topology ( 3 0 3)**

**M 711 Math. Integral Equations ( 3 0 3)**

**M 712 Math. Commutative Algebra (3 0 3)**

**M 713 Functional Analysis I (3,0,0)**

**M 714 Partial Differential Equations I (3,0,0)**

**M 715 Probability Theory I (3,0,0)**

**M 755 ADVANCED COMPLEX ANALYSIS (3,0,0) 3 credit**

**M 798 Seminar**

**M 799 Math. Ph.D. Thesis NC**

Modules, modules over a PID, p-groups and the Sylow theorems, products and factors, Cauchy’s theorem, group actions, the Sylow theorems, semidirect products, an application to combinatorics, series of groups, the Jordan-Hölder theorem, solvable groups, nilpotent groups, Galois groups and separability, the main theorem of Galois theory, insolvability of polynomials cyclotomic polynomials and Wedderburn’s theorem, the Wedderburn-Artin theorem

M 702 Math. Measure Theory ( 3 0 3 )

Fundamentals of measure and integration theory, Radon-Nikodym Theorem, Lp spaces, modes of convergence, product measures and integration over locally compact topological spaces.

M 703 Math. Integral Transforms ( 3 0 3 )

Fourier transforms, exponential, cosine and sine, application of Fourier transform to solve boundary value problems, Laplace transform, use of residue theorem and contour integration for the inverse of Laplace transform, application of Laplace transform to solve differential and integral equations,

Fourier-Bessel and Hankel transforms for circular regions, Abel transform for dual integral equations.

M 704 Math. 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.

M 705 Math. Applied Functional Analysis ( 3 0 3 )

Distributions, Review of Banach and Hilbert paces, Sobolev spaces, Semigroups,

Some techniques from nonlinear analysis.

M 706 Math. Dynamical Systems (3 0 3)

Local theory : Fundamental existence and uniqueness theorem, stable manifold theorem, Hartman-Grobman theorem, center manifold theorem, normal form theory. Global theory: global existence theorems, limit sets and attractors, periodic orbits, limit cycles and separatrix cycles, Poincare’ map, stable manifold theorem for periodic orbits, global phase portraits and separatrix configurations. Bifurcation theory: Structural stability and Peixoto’s theorem, bifurcations at nonhyperbolic equilibrium points, higher codimension bifurcations at nonhyperbolic equilibrium points, Hopf bifurcations and bifurcations of limit cycles from a multiple focus, global behavior of one-parameter families of periodic orbits, homoclinic bifurcations, Melnikov’s method.

M 707 Math. Linear Positive Operators and Approximation I ( 3 0 3 )

Best approximation, Modulus of continuity and smoothness, Peetre K- functional and its properties, Weierstrass approximation Theorem, Linear positive functionals and positive operators, Properties of C[a,b] and the space of integrable functions, Korovkin Theorem, Rate of convergence of linear positive operators, Voronovskaya type theorems, Divided difference

M 708 Math. Linear Positive Operators and Approximation II ( 3 0 3 )

Korovkin type theorems for functions defined on unbounded sets, Weighted function spaces and approximation problem, q- series and its generalizations, Some approximation of q-analog of the linear positive operators, Direct and inverse approximation theorems, Simultaneous approximation

M 709 Math. Lie Algebras ( 3 0 3 )

Basic concepts, semisimple Lie algebras, root systems, isomorphism and conjugacy theorems, existence theorem.

M 710 Math. 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.

M 711 Math. Integral Equations ( 3 0 3 )

Integral equations with separable kernels, method of successive approximations, classical fredholm theory, Applications to ordinary differential equations, applications to partial differential equations.

M 712 Math. Commutative Algebra (3 0 3)

Rings and ideals. Modules.Rings and modules of fractions. Primary decomposition. Integral dependence.

M 713 Functional Analysis I (3,0,0)

Course Content: Norm Spaces, Hahn-Banach Theorem, Linear Transformations, Close Graph Theorem, Applications of Banach Theorem to Differential Equations, Linear Operators, Spectral Theory, The properties of Resolvent ve Spektrum Theory, Banach Algebra and its properties.

Aim: At the end of the course the student,

i) Can solve the Cauchy Problems using contraction mapping.

ii) Can classified the functionals using Hahn-Banach

iii) Can define the general form the linear continuous functionals in Banach and Hilbert Spaces.

Pre-requisites:M401, M402

M 714 Partial Differential Equations I (3,0,0)

Course Content: Basic Definitions, Theorem of Cauchy-Kowalewsky, The classification of second order equations, canonical forms, Hyperbolic equations, Cauchy problem, Riemann method, Goursat problem.

Pre-requisites:M421, M422

Aim: The teaching of linear and nonlinear PDE, their classification,different problems posed for these equations, method of investigating such problems, and the applications of these methods.

.

M 715 Probability Theory I (3,0,0)

Course Content: Measure, integration, probability measures, measurable functions and random variables, probability measures in Rn, distribution functions, structure of conditional probability, structure of independence, limit theorems in Rn. Some probabilistic concepts, distiribution functions and their relationship, some moments and probability inequalities, the characteristic functions, moment generating functions and fractorial moment generating functions of some random vectors and related theorems, stochastic independence, transformations of random vectors, order statistics and related theorems, the multivariate normal distributions, Bayes and minimax estimations, quadratic forms.

Pre-requisites: M401, M402

Aim: The aim of the present course is to provide to the students the advance skills and tools of statistics and probability. Emphasis is directed on the application and the reasoning behind the application of these skills and tools for the purpose of enhancing engineering decision making.

It is expected that the students have prior knowledge on the subject of statistics and probability. The purpose of the present course is thus to ensure that the students will acquire during the course the required theoretical basis and technical skills such as to feel comfortable with the theory of basic statistics and probability. Moreover, in the present course as opposed to many standard courses on the same subject, the perspective is to focus on the use of the theory for the purpose of engineering model building and decision making.

M 755 ADVANCED COMPLEX ANALYSIS (3,0,0) 3 credit

Analytic continuation, analytic function element, analytic continuation along a curve, the monodromy theorem, the idea of a Riemann surface, the elliptic modular function and Picard’s theorem,elliptic functions, multiply connected domains, the Cauchy integral formula for multiply connected domains, holomorphic simple connectivity and topological simple connectivity, simple connectivity and connectedness of the complement, rational approximation theory, Runge’s theorem, Mergelyan’s theorem, some remarks about analytic capacity, special classes of holomorphic functions, Schliht functions and the Bieberbach conjecture, continuity to the boundary of conformal mappings, Hardy spaces, Hilbert spaces of holomorphic functions, the Bergman kernel, biholomorphic mappings, the geometry of Hilbert space, orthonormal systems in Hilbert space, special functions, the gamma and beta functions, the prime number theorem.

M 798 Seminar

M 799 Math. 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.