# Course Descriptions

Math 111
Introduction to Mathematical Structures
Matematiksel Yapılara Giriş
Credits4 ECTS8
Propositional logic, truth tables, equivalences, quantifiers, rules of inference, proof methods, sets, power sets, functions, sequences, countability, cardinality, divisibility, modular arithmetic, primes, mathematical induction, strong induction and wellordering principle, recursive definitions, axiomatic systems, Euclid's postulates and non-Euclidean geometries.
Prerequisite: None
Math 131
Calculus of a single variable
Credits4 ECTS8
Sequences, limits and continuity, differentiation and its applications, integration and its applications, fundamental theorem of calculus, transcendental functions, improper integrals.
Prerequisite: None
Math 132
Calculus of several variables
Credits4 ECTS8
Vectors and geometry in space, vector-valued functions and motion in space, functions of several variables, partial derivatives, multiple integrals, vector fields.
Prerequisite: Math 131
Math 162
Discrete Mathematics
Ayrık Matematik
Credits4 ECTS8
Counting, the pigeonhole principle, permutations, combinations, binomial coefficients, generalized permutations and combinations, discrete probability, linear recurrence relations, generating functions, inclusion-exclusion, relations, closures of relations, equivalence relations, construction of integers and rationals, partial orderings, graphs.
Prerequisite: None
Math 201
Matrix Theory
Matris Kuramı
Credits4 ECTS5
Systems of linear equations, Gaussian elimination, matrix algebra determinants, inverse of a matrix, Cramer's rule, rank and nullity, the eigenvalue problem, introduction to linear programming.
Prerequisite: None
Math 202
Differential Equations
Diferansiyel Denklemler
Credits4 ECTS7
First-order differential equations, linear equations, homogeneous and non-homogeneous, series solutions, the Laplace transform, systems of first-order linear equations, boundary value problems, Fourier series.
Prerequisite:   (Math 101 or Math 131) and (Math 201 or Math 221)
Math 221
Linear Algebra
Lineer Cebir I
Credits4 ECTS8
Vector spaces, bases, linear transformations, matrices, subspaces, systems of linear equations, echelon and reduced echelon forms, dimension, fundamental subspaces, rank, change of coordinates, determinants, cofactor expansion, minors, eigenvalues, eigenvectors, diagonalization, inner product spaces, orthogonality, Gram-Schmidt orthogonalization process, adjoint, unitary and orthogonal transformations, dual spaces.
Prerequisite: Math 111
Math 222
Group Theory
Credits4 ECTS8
Groups, subgroups, cyclic groups, generating sets, permutations, orbits, cycles, alternating groups, cosets, Lagrange's Theorem, direct products, finite abelian groups, homomorphisms, normal subgroups, factor groups, simple groups, group actions, isomorphism theorems, Sylow's theorems.
Prerequisite: Math 111
Math 231
Credits4 ECTS8
Sequences and functions, compact sets, continuity, uniform continuity, limits of functions, discontinuities, differentiation, derivatives for functions of several variables, differentiation of composite functions, Taylor's Theorem, definite integrals, substitution in multiple integrals, improper integrals.
Prerequisite: Math 132
Math 234
Credits4 ECTS8
Infinite series, conditionally convergent series, double series, uniform convergence, series and sequences of functions, power series, improper integrals with parameters, differentiation of transformations, linear functions, differentials and inverses of transformations, inverse and implicit function theorems.
Prerequisite: Math 231
Math 323
Rings, Fields and Galois Theory
Credits4 ECTS8
Rings, integral domains, field of fractions, polynomials, factorization, ideals, factor rings, homomorphisms, prime and maximal ideals, extension fields, algebraic extensions, finite fields, unique factorization domains, Euclidean domains, Gaussian integers, field automorphisms, splitting fields, Galois theory, insolvability of the quintic equations.
Prerequisite:  Math 222 or consent of the instructor
Math 324
Representation Theory of Finite Groups
Credits3 ECTS6
Representations, irreducibility, Maschke's theorem, semisimplicity, characters, character tables, orthogonality relations, induction and restriction of characters, Mackey decomposition theorem, algebraic integers, Burnside's p^aq^b-theorem, Frobenius' normal complement theorem.
Prerequisite:  Math 222 or consent of the instructor
Math 325
Matrix Groups
Credits3 ECTS6
General linear groups, closed subgroups of real and complex general linear groups, their topological properties, associated tangent spaces, exponential and logarithm functions, manifolds, maximal tori, homomorphisms.
Prerequisite:  (Math 102 or Math 132) and Math 222
Math 327
Number Theory
Sayılar Teorisi
Credits3 ECTS6
Divisibility theory, Euclidean algorithm, congruences, solutions of polynomial congruences, primitive roots, power residues, quadratic reciprocity law, arithmetical functions, distribution of prime numbers, Pell's equation, quadratic forms, some diophantine equations.
Prerequisite:  Math 111 or Math 162
Math 331
Metric Spaces
Credits4 ECTS8
Topology, density, separability, convergence, compactness, connectedness, continuity, open and closed maps, equicontinuity, Arzela-Ascoli theorem, contractions and fixed point theorems, completeness, Cantor's theorem, Baire category theorem, completion.
Prerequisite:  Math 231
Math 332
Lebesgue Integration
Credits3 ECTS6
Elementary measure theory, sets of measure zero, Lebesgue measure, Lebesgue measurable sets and functions, Lebesgue integral, convergence theorems, the space L^1, absolutely continuous functions, functions of bounded variation, Hilbert space L^2, Fourier series.
Prerequisite:  Math 234 or consent of the instructor
Math 334
Analysis on Manifolds
Credits3 ECTS6
Differentiation, inverse and implicit function theorems, integration, manifolds, differential forms, orientation, Stokes' theorem, Poincaré lemma, de Rham cohomology.
Prerequisite:  Math 221 and Math 234
Math 336
Numerical Analysis
Nümerik Analiz
Credits3 ECTS6
Solutions of nonlinear equations, bisection, Newton, and fixed point iterations, direct solutions of linear systems, Gaussian elimination with partial pivoting, LU and Cholesky factorizations, iterative solutions of linear systems, vector and matrix norms, Neumann series, Jacobi, Gauss-Seidel and SOR iterations, projection methods, steepest descents, conjugate-gradient and GMRES methods, matrix eigenvalue problem, power method, Givens rotations, Jacobi iteration, Hessenberg form, QRiteration, polynomial interpolation, Lagrange polynomials, Newton’s divided differences, Chebyshev polynomials, least squares, spline interpolation.
Prerequisite: (Math 101 or Math 131) and (Math 201 or Math 221)
Math 338
Complex Analysis I
Karmaşık Analiz I
Credits4 ECTS8
Complex numbers, exponential forms, roots of complex numbers, functions of a complex variable, limits, continuity, derivatives, Cauchy-Reimann Equations, polar coordinates, analytic functions, reflection principle, exponential and logarithmic functions, branches, trigonometric and hyperbolic functions, linear transformations, definite integrals, contour integrals, branch cuts, Cauchy-Goursat theorem, simply connected domains, Cauchy integral formula, Liouville's Theorem, maximum modulus principle, Taylor and Laurent series, residues and poles, Cauchy's residue theorem, residue at infinity.
Prerequisite: Math 132
Math 344
Introduction to Probability and Statistics
Credits3 ECTS6
Probability, conditional probability, Bayes’ theorem, independence, discrete and continuous probability distributions, expected value, estimation, confidence intervals, tests of hypothesis for one parameter, goodness of fit test, linear regression, analysis of variance.
Prerequisite: Math 102 or Math 132
Math 345
Probability
Olasılık
Credits3 ECTS6
Axioms of probability, conditional probability, independence, discrete and continuous random variables, jointly distributed random variables, expectation, limit theorems.
Prerequisite: Math 344 or consent of the instructor
Math 351
Qualitative Theory of Ordinary Differential Equations
Credits3 ECTS6
Existence and uniqueness theorems, phase portraits in the plane, linear systems and canonical forms, non-linear systems, linearization, stability of fixed points, limit cycles, Poincaré-Bendixson theorem.
Prerequisite: Math 202
Math 352
Partial Differential Equations
Kısmi Türevsel Denklemler
Credits3 ECTS6
Wave equation, heat equation, Laplace equation, classification of second order linear equations, initial value problems, boundary value problems, Fourier series, harmonic functions, Green's functions.
Prerequisite: (Math 132 and Math 202) or (Math 102 and Math 202)
Math 361
Combinatorics
Credits3 ECTS6
Sieve methods, lattices, distributive lattices, incidence algebra, Mobius inversion formula, Mobius algebras, generating functions, exponential formula, Lagrange inversion formula, matrix tree theorem.
Prerequisite: Math 201 or Math 221
Math 363
Graph Theory
Çizgeler Kuramı
Credits3 ECTS6
Basic definitions, trees, Cayley's formula, connectedness, Eulerian and Hamiltonian graphs, matchings, edge and vertex colouring, chromatic numbers, planar graphs, directed graphs, networks.
Prerequisite: Math 221 or consent of instructor
Math 401
History of Mathematics
Matematik Tarihi
Credits3 ECTS6
Selected topics in the history of mathematics and related fields.
Prerequisite: Consent of instructor
Math 404
Computational Mathematics
Credits3 ECTS6
Introduction to computational mathematics, basics of a mathematics software (Sage, Mathematica, Maple, MATLAB), solving systems of linear equations, interpolation, locating roots of equations, least squares problems, numerical integration, numerical differentiation and solution of ordinary differential equations.
Prerequisite: (Math 202 and Math 221) or consent of the instructor
Math 411
Mathematical Logic
Matematiksel Mantık
Credits3 ECTS6
Propositional and quantificational logic, formal grammar, semantical interpretation, formal deduction, completeness theorems, selected topics from model theory and proof theory.
Prerequisite: Math 111
Math 412
Introduction to Set Theory
Kümeler Kuramına Giriş
Credits3 ECTS6
Sets, relations, functions, order, set-theoretical paradoxes, axiom systems for set theory, axiom of choice and its consequences, transfinite induction, recursion, cardinal and ordinal numbers.
Prerequisite: Math 111
Math 413
Model Theory
Credits3 ECTS6
Language and structure, theory, definable sets and interpretability, compactnees theorem, complete theories, Löwenheim-Skolem theorems, quantifier elimination, algebraic examples.
Prerequisite: Math 111
Math 425
Introduction to Algebraic Geometry
Credits3 ECTS6
Affine varieties, Hilbert’s Nullstellensatz, projective varieties, rational functions and morphisms, smooth points, dimension of a variety.
Prerequisite: Math 323
Math 426
Introduction to Arithmetic Geometry
Credits3 ECTS6
Introduction to algebraic number theory and algebraic curves, geometric introduction to function fields of curves, affine and projective varieties, divisors on curves, Riemann-Roch theorem, basics of elliptic curves.
Prerequisite: Math 323 or consent of the instructor
Math 427
Elementary Number Theory II
Credits3 ECTS6
Quadratic Forms, quadratic number fields, factorization of ideals in quadratic number fields, ramification theory, ideal classes and units in quadratic number fields, elliptic curves over rationals.
Prerequisite: Math 162
Math 432
Complex Analysis II
Karmaşık Analiz II
Credits3 ECTS6
Convergent series of meromorphic functions, entire functions, Weierstrass' product theorem, partial fraction expansion theorem of Mittag-Leffler, gamma function, normal families, theorems of Montel and Vitali, Riemann mapping theorem, conformal mapping of simply connected domains, Schwarz-Christoffel formula, applications.
Prerequisite: Math 338
Math 433
Fourier Analysis
Credits3 ECTS6
Fourier series, Dirichlet and Poisson kernels, Cesàro and Abel summability. pointwise and mean-square convergence, Weyl's equidistribution theorem, Fourier transform on the real line and Schwartz space, inversion, Plancherel formula, application to partial differential equations, Poisson summation formula.
Prerequisite: Math 338 or consent of the instructor
Math 436
Functional Analysis
Fonksiyonel Analiz
Credits3 ECTS6
Review of vector spaces, normed vector spaces, lP and LP spaces, Banach and Hilbert spaces, duality, bounded linear operators and functionals.
Prerequisite: Math 331
Math 437
Optimization Theory
Eniyileme Kuramı
Credits3 ECTS6
Normed linear spaces, Hilbert spaces, least-squares estimation, dual spaces, geometric form of Hahn-Banach theorem, linear operators and their adjoints, optimization in Hilbert spaces, local and global theory of optimization of functionals, constrained and unconstrained cases.
Prerequisite: Math 331
Math 451
Numerical Solutions of Differential Equations
Diferansiyel Denklemlerin Sayısal Çözümleri
Credits3 ECTS6
Numerical solutions of initial value problems for ordinary differential equations (ODE), Picard-Lindelof theorem, single step methods including Runge-Kutta methods, examples and consistency, stability and convergence of multistep methods, numerical solution of boundary value problems for ODE’s, shooting, finite difference, and collocation methods, finite element methods, Riesz and Lax-Milgram lemmas, weak solutions, numerical solutions of partial differential equations, examples of finite difference methods and their consistency, stability, and convergence including Lax-Richtmeyer equivalence theorem, Courant-Friedrichs-Lewy condition, and von Neumann analysis, Galerkin methods, Galerkin orthogonality, Cea’s lemma, finite element methods for elliptic, parabolic and hyperbolic equations.
Prerequisite: (Math 102 or Math 132) and Math 202
Math 452
Dynamical Systems
Credits3 ECTS6
Dynamical systems with discrete and continuous time, differential equations on torus, invariant sets, topological dynamics, topological recurrence and entropy, expansive maps, homoemorphisms and diffeomorphisms of the circle, periodic orbits, hyperbolic dynamics, Grobman-Hartman and Hadamard-Perron theorems, geodesic flows, topological Markov chains, zeta functions, invariant measures and the ergodic theorem.
Prerequisite: Math 331 or consent of the instructor
Math 455
Calculus of Variations
Varyasyonlar Hesabı
Credits3 ECTS6
First variation of a functional, necessary conditions for an extremum of a functional, Euler's equation, fixed and moving endpoint problems, isoperimetric problems, problems with constraints, Legendre transformation, Noether's theorem, Jacobi's theorem, second variation of a functional, weak and strong extremum, sufficient conditions for an extremum, direct methods in calculus of variations, the principle of least action, conservation laws, Hamilton-Jacobi equation.
Prerequisite: Math 202
Math 462
Cryptography
Şifre Kuramı
Credits3 ECTS6
Simple crypto-systems, public key cryptography, discrete logarithms and Diffie-Hellman key exchange, primality, factoring and RSA, elliptic curve crypto-systems, lattice based crypto-systems.
Prerequisite: Math 221 or consent of the instructor
Math 471
Topology
Topoloji
Credits3 ECTS6
Topological spaces, compactness, connectedness, continuity, separation axioms, homotopy, fundamental group.
Prerequisite: Math 331
Math 472
Geometric Topology
Credits3 ECTS6
Basics of point set topology, quotient topology, CW complexes and their homology and fundamental group, classification of surfaces, introduction to knot theory, Seifert surfaces and Seifert forms, signature, Alexander polynomial, and Arf invariant of knots, introduction to Morse theory, Heegaard splittings of three manifolds, Dehn surgery, Lickorish-Wallace theorem.
Prerequisite: Math 331 or consent of the instructor
Math 474
Mathematical Aspects of General Relativity
Credits3 ECTS6
Review of special relativity, differentiable manifolds, tensors, Lie derivative, covariant derivative, parallel transport, geodesics, curvature, Einstein's field equations, Schwarzschild black hole, Cauchy problem, maximally symmetric spacetimes, singularity theorems.
Prerequisite: Consent of the instructor
Math 475
Differential Geometry
Diferansiyel Geometri
Credits3 ECTS6
Fundamentals of Euclidean spaces, geometry of curves and surfaces in three-dimensional Euclidean space, the Gauss map, the first and the second fundamental forms, theorema egregium, geodesics, Gauss-Bonnet theorem, introduction to differentiable manifolds.
Prerequisite: Math 234 or consent of the instructor
Math 476
Differential Topology
Diferansiyel Topoloji
Credits3 ECTS6
Smooth functions and smooth manifolds embedded in Euclidean space, tangent spaces, immersions, submersions, transversality, applications of the implicit function theorem, Morse functions, Sard's theorem, Whitney embedding theorem, intersection theory mod 2, Brouwer fixed point theorem, Borsuk-Ulam Theorem, and other related results.
Prerequisite: Math 331
Math 477
Projective Geometry
Projektif Geometri
Credits3 ECTS6
Projective spaces, homogeneous coordinates, dual spaces, the groups of affine and projective transformations and their properties, Desargues' theorem, Pascal's theorem, and other classical results, classification of conics, projective plane curves, singular points, intersection multiplicity, Bezout's Theorem, the group law on an elliptic curve, cross-ratio.
Prerequisite: Math 201 or Math 221
Math 478
Groups and Geometries
Gruplar and Geometriler
Credits3 ECTS6
Plane Euclidean geometry and its group of isometries, affine transformations in the Euclidean plane, fundamental theorem of affine geometry, finite group of isometries of R, Leonardo da Vinci's theorem, geometry on the sphere S, motions of S, orthogonal transformations of R, Euler's theorem, right triangles in S, projective plane, Desargues' theorem the fundamental theorem of projective geometry.
Prerequisite: Math 222 or consent of the instructor
Math 481-489
Selected Topics in Mathematics
Matematikten Seçilmiş Konular
Credits3 ECTS6
Selected topics in pure and applied mathematics.
Prerequisite: Consent of the instructor.
Math 490
Project
Proje
Credits3 ECTS6
Individual research supervised by a member of the department.
Prerequisite: Consent of the instructor.
Math 491-499
Selected Topics in Mathematics
Matematikten Seçilmiş Konular
Credits3 ECTS6
Selected topics in pure and applied mathematics.
Prerequisite: Consent of the instructor.
Phys 101 **
Physics I
Credits4
AE 111*/HSS/MLE*
Credits3
Unrest. Elect1
Credits3
PHYS** 102
Physics II
Credits4
AE 112*/HSS/MLE*
Credits3
Unrest. Elect2
Credits3
PHYS** 201
Physics III
Credits4
CMPE 150
Introduction to Computing
Credits3
TK 221
Turkish I
Credits2
TK 222
Turkish II
Credits2
HSS/MLE*
HSS/MLE
Credits3
HSS
HSS Elect.
Credits3
HTR 33
Hist. of Turk. Rep. I
Credits2
Unrest. Elective
Credits3
HSS
HSS Elect.
Credits3
HTR
Hist. of Turk. Rep. II
Credits2
Math
Elect.
Credits3
Math
Elect.
Credits3
Science Elect.
Credits3
Math
Elect.
Credits3
Math
Elect.
Credits3
Math
Elect.
Credits3
Unrest. Elect.
Credits3
HSS
HSS Elect.
Credits3
Math
Math. Elective
Credits3
Math
Math. Elective
Credits3
Unrest. Elect.
Credits3
Unrest. Elect.
Credits3
Science Elect.
Credits3