Required Courses

Course Code	Course Title	Credits	ECTS
MAT 101	Mathematics I	4 Credits	ECTS: 7
Functions and graphs, limit and continuity of functions, derivative of functions, geometric meaning of derivative, rules of differentiation, chain rule, maximum-minimum problems, derivatives of trigonometric functions, implicit differentiation, mean value theorem, graphing and asymptotes, Riemann sums and definite integrals, the fundamental theorem of calculus, area between two curves, improper integrals.
MAT 102	Mathematics II	4 Credits	ECTS: 7
Volumes and surface areas of revolutions, evaluating arc length, polar coordinates, sequences and series, convergence tests, power series, Taylor and Maclaurin series, limits and continuity of multivariable functions, partial derivatives and chain rule, differentials, directional derivatives and tangent planes, maximum-minimum problems for multivariable functions, double and triple integrals, cylindrical and spherical coordinates.
MAT 111	Mathematical Logic	3 Credits	ECTS: 6
Propositions, proof techniques, sets and set operations, relation and its properties, kinds of relations, functions, inverse relation and inverse function, operation, Mathematical concepts.
MAT 201	Linear Algebra I	3 Credits	ECTS: 6
Systems of linear equations, matrix, algebraic properties of matrix operations, determinant function and its properties, vector spaces, sub vector spaces, base and dimension, inner product spaces, linear mappings and matrix representations, eigenvalues and eigenvectors, diagonalization, diagonalization of symmetric matrices.
MAT 202	Differential Equations	3 Credits	ECTS: 6
First order differential equations, high order differential equations, general solutions of homogeneous and non-homogenous differential equations, systems of linear differential equations, solutions by power series, Laplace transformations, existence and uniqueness theorems, boundary-value problems, introduction to partial differential equations.
MAT 209	Advanced Analysis I	3 Credits	ECTS: 7
Topology of the spaces R, R2 and R3 , concept of compactness, multivariable functions, limit, continuity and uniform continuity, partial derivatives, directional derivatives, differentiation and tangent planes, implicit function theorem, invers function theorem, maximum and minimum for multivariable functions, vector valued functions.
MAT 210	Advanced Analysis II	3 Credits	ECTS: 7
Function sequences and series, uniform and pointwise convergence, double integrals, change of variables in double integrals, triple integrals, cylindrical and spherical coordinates, evaluating area and volume in multiple integrals, region transformations, line integrals, Green’s theorem, surface integrals, Divergence and Stokes theorems and applications.
MAT 212	Linear Algebra II	3 Credits	ECTS: 6
Systems of linear equations, matrices and algebraic properties of matrix operations, matrix factors and block matrices, determinant function and its properties, vector spaces, subspaces, base and dimension, inner product spaces, eigenvalues and eigenvectors, similar matrices and diagonalization, conics, quadratic forms.
MAT 309A	Algebra	4 Credits	ECTS: 7
Divisibility in integers and its properties, prime factors, integer congruence, introduction to groups, subgroups, normal subgroups, divisible groups, isomorphism theorems for groups, inner direct product, outer direct product, introduction to rings, ideals, divisible rings, homomorphism and isomorphism in rings, finite permutation groups, Cayley’s theorem.
MAT 311	Complex Functions Theory I	3 Credits	ECTS: 7
Complex numbers and its topology, complex powers and roots, functions with complex variables, limit and continuity, analytic functions, differentiable and analyticity, Cauchy-Riemann equations, elementary functions, complex integrals, Cauchy Goursat theorem, Cauchy integral formulas, Liouville theorem, Cauchy inequality, fundamental theorem of algebra, Taylor series, Laurent series, zeros and polar points, residues, residue theorem.
MAT 312A	Functional Analysis	3 Credits	ECTS: 6
Metric spaces, open and closed balls, convergence, Cauchy sequence and completeness, examples of complete spaces, linear spaces, normed spaces, Banach spaces, spaces with finite dimension, linear operators, bounded and continuous operators, dual spaces, inner product spaces, Hilbert spaces, orthogonality in inner product spaces, direct summation, convex sets, closed and convex subspaces.
MAT 395	Numerical Analysis	3 Credits	ECTS: 6
The definition of numerical analysis and error analysis, numerical solutions of nonlinear algebraic equations, finding roots, interpolation and approximation to functions, numerical derivative and numerical integral, numerical solutions of systems of linear equations.
MAT 396	Partial Differential Equations	3 Credits	ECTS: 6
The first order linear, quasilinear, nonlinear partial differential equations, characteristic surfaces, Cauchy-Kowalevski theorem, the second order linear partial differential equations and their classifications, heat equation, wave equation, Green identities, Laplace equation, the maximum principle, Poisson equation, the methods of Fourier series.
MAT 495	Project Course I	3 Credits	ECTS: 6
It is a course for math students having an undergraduate background on a specific area under an academic staff in the Mathematics Department.
MAT 496	Project Course II	3 Credits	ECTS: 6
It is a course for math students having an undergraduate background on a specific area under an academic staff in the Mathematics Department.

Elective Courses

MAT 205	Mathematics III	3 Credits	ECTS: 6
Coordinate systems in 3-dimensional space, vectors, scalar and vector products, line and plane equations, conic equations, vector functions, derivative and integrals, arc length and curvature, parametric surfaces, vector fields, line integrals, fundamental theorem for line integrals, Green’s theorem, rotations and divergences, surface integrals, Stokes’ theorem, divergence theorem.
MAT 310A	Differential Geometry	4 Credits	ECTS: 6
Curves in the Euclidean space, curvature and curl of curves, Frenet formulae, concepts of curves in the plae and space, fundamental theorem of local curves, equivalence of curves, natural equations, concept of a surface, parametrization of curves, tangent plane, concept of differentiation, vertor fields, the first fundamental form, normal and essential curvatures, parallel translation, covariant derivative, geodesic, Gauss-Bonnet theorem.
MAT 411	Measure Theory	3 Credits	ECTS: 6
Sequences of sets, the concept of sigma algebra, Borel algebra, measures and outer measures, Lebesgue outer measure and Lebesgue measure, measurable functions, integrals of simple functions, integrals of positive functions, integrable functions, the relations between Lebesgue integral and Riemann integral, L_p space and its applications.
MAT 412	Transformations in Complex Analysis	3 Credits	ECTS: 6
Linear transformations, special power functions, exponential and logarithmic functions, trigonometric and hyperbolic functions, conformal mappings, fractional linear mappings, Scwarz-Christoffel mappings, Poisson integral formulas and various applications.
MAT 413	Mathematical Analysis	3 Credits	ECTS: 6
Limit, continuity and derivative of functions, uniform continuity, pointwise and uniform convergence of function sequences, replacement of limits, Weierstrass approximation theorems, function series and convergence tests, power series and Taylor theorem.
MAT 414	Approximation Theory	3 Credits	ECTS: 6
Positive linear functionals, positive linear operators, approximation to functions by algebraic polynomials, approximation to functions by trigonometric polynomials, the Korovkin theorem, convergence conditions of positive linear operators sequences, modulus of continuity and rate of convergence.
MAT 415	History of Mathematics	3 Credits	ECTS: 6
The historical developments of algebra, geometry and analytic geometry and the life of the famous mathematicians.
MAT 421	Introduction to Cryptography	3 Credits	ECTS: 6
The basic cryptography systems, general principles, systems with single alphabet and multiple alphabets, simple analysis techniques, general properties of systems with open keys, general properties of systems with block and flow crypto, general structure of Boole functions, squeeze functions, confirmation codes.
MAT 422	Introduction to Coding Theory	3 Credits	ECTS: 6
Introduction to coding theory and basic concepts, Galois field and its arithmetic, linear codes, boundaries of code parameters, BCH, RS and Quadratic residue codes, Goppa codes, joining codes, Decoding algorithms.
MAT 423	Introduction to Finite Fields	3 Credits	ECTS: 6
Groups, rings and fields, polynomials, field extensions, characterizations of finite fields, roots of irreducible polynomials, trace, norm and bases, roots of identity element, orders of polynomials and primitive polynomials, irreducible polynomials.
MAT 424	Reduced Sequences and Combinatorial Properties	3 Credits	ECTS: 6
Reduced sequences, Pascal triangle and Pascal-type triangles, various applications, generating functions, continuous fractions, normal sums of reduced sequences, generating matrices and applications to the reduced sequences, matrices and determinant problems, special polynomials constructed by sequences and their classifications.
MAT 425	Numbers Theory	3 Credits	ECTS: 6
Divisibility, the least common multiple, Euclidean algorithm, primes, prim number theorem, canonical factors, fundamental theorem of arithmetic, congruences, equivalence relation, linear congruences, Çin remainder theorem, modular arithmetic and Fermat’s theorem, arithmetic functions, primitive roots, quadratic congruences.
MAT 431	Geometries and Transformations	3 Credits	ECTS: 6
The geometry as an axiomatic science, non-Euclidean geometries, relations between other geometries and Euclidean geometry, invariant properties.
MAT 432	Topology	3 Credits	ECTS: 6
Set theory, metric toplogy, topological spaces, the concept of base, subspaces, product and division topology, compactness, connectedness and its applications.
MAT 433	Algebraic Topology Topoloji	3 Credits	ECTS: 6
Euler’s theorem, topological equivalence, surfaces, abstract spaces, topological invariants, continuity, compactness and connectedness, identification spaces, essential group and the concept of homotopy, triangularization, surfaces and the concepts of simplex and homology, Euler-Poincare formula, Borsuk-Ulam theorem, Brouwer and Lefschetz fixed point theorems.
MAT 434	Variational Analysis	3 Credits	ECTS: 6
Concept of functional, necessity conditions for minimization of integrals by Euler, Legendre, Weierstrass and Jacobi, variational problems and eigen value problems, Hamiltan-Jacobi equations, applications.
MAT 441	Ordinary Differential Equations Theory	3 Credits	ECTS: 6
Existence and uniqueness of solutions of ordinary differential equations, phase-portraits of linear systems, linear and non-linear systems, linearization and stability analysis, Lyapunov’s method, eigen value problems, Sturm-Liouville problems.
MAT 442	Numerical Solutions of Differential Equations	3 Credits	ECTS: 6
Numerical solutions of ordinary differential equations, Euler, Backward Euler, Trapezoidal, Taylor, Runge Kutta and multi-step methods, boundary-value problems and numerical solutions of equation systems, method of finite differences, numerical solutions of Laplace, heat and wave equations.
MAT 443	Mathematical Modeling	3 Credits	ECTS: 6
Classification of models, continuous and discrete models, stability analysis in continuous and discrete models, phase-portraits of models, stability analysis.
MAT 444	Financial Mathematics	3 Credits	ECTS: 6
Discrete and continuous time models, non-total financial market, Ito’s lemma, Black Scholes model, evaluating different investment tools.
MAT 445	Applied Mathematics	3 Credits	ECTS: 6
Modeling by ordinary and partial differential equations, initial and boundary-value problems, Dirichlet ve Neumann problems, Euler inequality, orthogonal and orthonormal functions, integral functions.
MAT 446	Mathematical Biology	3 Credits	ECTS: 6
Partial differential equations, dynamical systems, systems of differential equations, stability analysis, continuous and discrete equations, bifurcation analysis, Reaction-diffusion equations, diffusion and random walks, turing, pattern formation.
MAT 447	Dynamical Systems	3 Credits	ECTS: 6
Linear systems, the second order differential equations, systems of linear and non-linear equations, linearization, stability analysis, phase-portraits, Lyapunov functions, Poincare-Bendixon theorem and periodic solutions, population dynamics and models.
MAT 448	Special Functions in Applied Mathematics	3 Credits	ECTS: 6
Gamma and Beta functions, Bessel equations and Bessel functions, orthogonal functions and their properties, Legendre polynomials, Tchebycheff polynomials, Fourier series and its analysis.
MAT 449	Introduction to Difference Equations	3 Credits	ECTS: 6
Systems of difference equations, homogenous system, non-homogenous systems and their solutions, control of difference systems, stability theory for difference equations, the first and second Lypunov stability methods, applications of difference equations to geometric, dynamical, electrical, probabilistic and economics.
MAT 450	Introduction to Control Theory	3 Credits	ECTS: 6
Control of continuous and discrete linear systems, mathematical models of dynamical systems, time-domain analysis, mappings and block diagrams, state-space formulations, feedback and non-feedback systems.
MAT 452	Fuzzy Logic and Set Theory	3 Credits	ECTS: 6
Fuzzy logic, properties of fuzzy sets, fuzzy functions and their derivative and integrals, fuzzy linear programming, probability and its fuzzy sense.
MAT 499	Independent Study Course	3 Credits	ECTS: 6
It is a course for an active student studying and investigating on a specific topic under an academic staff in the Mathematics Department. It is expected (if possible) a scientific material such as paper, technical note, proceeding and etc.

Mathematics

Required Courses

Elective Courses