Ek

1b

yuksek

lisans
Matematik Anabilim Dalı Yüksek Lisans Dersleri ve İçerikleri
6106007001995
M.S
c
. 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.
6106005011995
Complex Analysis I
( 3 0 3 )
Conform transformations, Riemann transformation theorem, Analytical
continuatio
n, monodomy theorem, Picard theorem
6106005221995
Complex Analysis II
( 3 0 3 )
Compactness and Convergence in the Space of Analytic Functions, Space of
Analytic Functions, Spaces of Meromorphic Functions, The Riemann Mapping
Theorem, Weirstrass Factoriz
ation Theorem, Runge’s Theorem, Simple
Connectedness, Analytic Continuation along a Path, Harmonic Functions, The
Dirichlet Problem, Green’s Functions, Entire Functions.
6106005321995
Differential Topology
( 3 0 3 )
Differential Manifolds: Differentiabl
e 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 s
pace, 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 integratio
n 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.
6106005331995
Computer Algebra
( 3 0 3 )
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 G
eometry Theorem
Proving, The Inverse Kinematics Problem in Robotic.
6106005341995
Perturbation Methods in Applied Mathematics
( 3 0 3 )
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, Singula
r 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 Bo
undary Value Problems.
6106005351995
Variational Methods of Approximation
( 3 0 3 )
Existence and Uniqueness of Solutions, Linear Algebraic Equations, Linear
Operator Equations, The Contraction Mapping Theorem, Variational Boundary
Value Problems, Regul
arity 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.
6106005361995
Dynamical Systems
( 3 0 3 )
Second order differential equations in phase plane, linear systems and exponential
operators, Canonical forms, Stability for
ms, Lyapunov functions, The existence of
periodic solutions, Applications to various fields, Oscillation theory.
6106005411995
Algebra I
( 3 0 3 )
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.
6106005421995
Algebra II
( 3
0 3 )
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 t
heory. Homological methods: The functors Ext, Tot, (co

) homology,
Derived categories, Derived functors, Stable categories, Applications to co
homology of groups, Schemes.
6106005511995
Topology I
( 3 0 3 )
Euler’s Theorem, Topological equivalence, surfa
ces, 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,
Ident
ification spaces, the torus, the cone construction, glueing lemma, Projective
spaces attaching maps.
6106005521995
Topology II
( 3 0 3 )
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, Bor
suk

Ulam theorem,
The Lifschets fixed

point theorem, Dimension, knots and covering spaces,
Examples of knots, Group covering spaces, Alexander polynomials
6106005531995
Topological Groups
( 3 0 3 )
Topological Groups and continuous homogeneous spaces,
Lie groups and
Differentiable manifolds, Tangent spaces, Adjoint representations, Lie algebra,
Root decompositions, Weyl Groups and Dynkin diagrams
6106005551995
Algebraic Topology
( 3 0 3 )
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.
6106005571995
Partial Differential Equations and Boundary Value Problems
( 3 0 3 )
Nonhomogeneous Problems, Heat Flow with Sources Method of Eigen function,
Expansion with homogeneous Boundary Conditions, Method of Ei
gen 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.
6106005581995
Applied Functional Analysis and Variati
onal Methods in
Engineering
( 3 0 3 )
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 Sp
aces, 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 soluti
on, 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 Constrain
ts, Eigenvalue Problems,
Variational Methods of Approximation, The Ritz Method, The Weighted Residual
Method, Time Dependent Problems.
6106005601995
Rings and Modules
( 3 0 3 )
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.
6106005651995
Exa
mples of Group I
(3 0
3)
The Ascending Central Series and Nilpotent Groups
,
The Derived Series and
Solvable Groups
,
Constructions
,
Free Groups and Presantations
,
Matrices
,
Finite
Groups
6106005661995 Examples of Groups II
(3 0 3)
Finite Groups
,
Infinite abelian groups
,
Infinite Nonabelian Groups
,
The group
PSL(2,Q)
6106005681995 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
6106005691995
Combinatorial Commutat
ive Algebra and Toric Varieties
(3 0 3)
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 localiz
ations of toric ideals.
6106005701995
I
ntroduction to Operator Theory
(3 0 3)
1.
Spectral Properties of Bounded Linear Operators.
2.
Use of Complex Analysis in Spectral Theory.
3.
Compact Linear Operators on Normed Spaces.
4.
Spectral Properties of Compact
Linear Operators on Normed Spaces.
5.
Spectral Prtoperties of Bounded Self

Adjoints Linear Operatorts.
6.
Positive Operators.
7.
.
Projection Operators
8.
Unbounded Linear Operators and their Hilbert
–
Adjoint Operators.
9.
Hilbert

Adjoint Operators , Symmetric and Self

Adjoint Linear Operators.
10.
Closed Linear Operators and Closeres.
11.
Spectral Properties of Self

Adjoint Linear Operators.
Spectral
Representation of Unitary Operators
6106005711995
Numer
ical Methods in Linear Algebra
(3 0 3)
1.
Linear Systems o
f 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.
6106005731995
Real Analysis
( 3 0 3 )
Introduction to measure theory,
Abstract Integral, Positive Borel Measures, Riesz
representation theorem, Lebesque measure.
6106005741995 Harmonic Analysis
(3 0 3)
Orthogonal systems, Orthogonal series, Fourier series, Some summability
techniques, fundamental properties of harmonic f
unctions, 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,
6106005751995
Fou
rier Analysis and Approximation
(3 0 3)
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 an
d 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 characteri
stic points, Convergence rate of
integral operators on characteristic points.
6106005761995
Advanced Differential Geometry
(3 0 3)
Differentiable manifolds, Multilinear algebra, Exterior differential calculus,
Connections, Riemannian geometry, Lie grou
ps and moving frames, Complex
manifolds, The geometry of the gauss map, The intricsic geometry of surfaces,
Global differential geometry.
6106005771995 Homological Algebra I
(3 0 3)
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.
6106005
781995 Homological Algebra II
(3 0 3)
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 Le
mma
and covariant

contravariant right derived functors, Tor , Ext, Dimension, Covers
and Envelopes.
6106005792009
Process Dynamics and Control I
(3 0 3)
Laws and languages of process control, time

domain dynamics and control,
conventional control sys
tems, advanced control systems, Laplace

domain dynamics
and control, frequency

domain dynamics and control, process identification.
6106005801995
Functional Analysis
( 3 0 3 )
Normed spaces, Linear and multilinear transformations, The product of normed
spaces, Series concept in normed spaces, Hilbert spaces.
6106005812009
Introduction to Digital Signal Processing (3 0 3)
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
6106005822009
Statistical Signal Processing
(3 0 3)
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.
6106005832009
Mathematical M
ethods for Signal Processing (3 0 3)
Linear operators and Matrix Inverses; Eigenvalues and Eigenvectors; Singular
Value Decomposition; Some Special Matrices and Their Applications; Detection
Theory Algorithms; Estimation Theory Algorithms; Optimal Filter
ing Algorithms;
Methods of Iterative Algorithms; Least Mean Square Adaptive Filtering; Neural
Networks; Clustering; Methods of Optimization.
6106005842009
Process Dynamics and Control II
(3 0 3)
Multivariable process, analysis of multivariable
systems, design of controllers for
multivariable processes, sampled

data control systems, stability analysis of
sampled

data systems, applications.
6106005911995
Differential Equations
( 3 0 3 )
Existence and Uniqueness theorems, linear equation with co
nstant coefficients,
nonlinear equations, classification of points solving equations using
transformations.
6106005921995
Partial Differential Equations
( 3 0 3 )
Diffusion equation, Wave equation, classification of second order Particular
Differential
equations, Cauchy problems solution using Fourier series, separation of
variables
6106005981995
Algebraic Geometry
( 3 0 3 )
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
Varie
ties, Hilbert Function and Dimension of a Variety, Elementary Properties of
Dimension, Dimension and Algebraic Independence, Dimension and
Nonsingularity.
6106007051995
Seminar
6106007111995
Special Studies
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