# 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.

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.

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.

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.

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.

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.

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.

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.

Math 231

Advanced Calculus I

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.

Math 234

Advanced Calculus II

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Math 345

Probability

Olasılık

Credits3 ECTS6

Axioms of probability, conditional probability, independence, discrete and continuous random variables, jointly distributed random variables, expectation, limit theorems.

Math 351

Qualitative Theory of Ordinary Differential Equations

Sıradan Türevsel Denklemlerin Nitelik Kuramı

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.

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.

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.

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.

Math 401

History of Mathematics

Matematik Tarihi

Credits3 ECTS6

Selected topics in the history of mathematics and related fields.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Math 471

Topology

Topoloji

Credits3 ECTS6

Topological spaces, compactness, connectedness, continuity, separation axioms, homotopy, fundamental group.

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.

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.

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.

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.

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.

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.

Math 481-489

Selected Topics in Mathematics

Matematikten Seçilmiş Konular

Credits3 ECTS6

Selected topics in pure and applied mathematics.

Math 490

Project

Proje

Credits3 ECTS6

Individual research supervised by a member of the department.

Math 491-499

Selected Topics in Mathematics

Matematikten Seçilmiş Konular

Credits3 ECTS6

Selected topics in pure and applied mathematics.

Phys 101 **

Physics I

Credits4 ECTS8

AE 101/111*/HSS/MLE*

Adv. Eng./HSS/MLE

Credits3 ECTS3/5

Unrest. Elect1

Credits3 ECTS5

PHYS** 102

Physics II

Credits4 ECTS8

AE 102/112*/HSS/MLE*

Adv. Eng./HSS/MLE

Credits3 ECTS4/5

Unrest. Elect2

Credits3 ECTS5

PHYS** 201

Physics III

Credits4 ECTS6

CMPE 150

Introduction to Computing

Credits3 ECTS5

TK 221

Turkish I

Credits2 ECTS3

TK 222

Turkish II

Credits2 ECTS3

HSS/MLE*

HSS/MLE

Credits3 ECTS5

HSS

HSS Elect.

Credits3 ECTS5

HTR 33

Hist. of Turk. Rep. I

Credits2 ECTS3

Unrest. Elective

Credits3 ECTS5

HSS

HSS Elect.

Credits3 ECTS5

HTR

Hist. of Turk. Rep. II

Credits2 ECTS3

Math

Elect.

Credits3 ECTS6

Math

Elect.

Credits3 ECTS6

Science Elect.

Credits3 ECTS5

Math

Elect.

Credits3 ECTS6

Math

Elect.

Credits3 ECTS6

Math

Elect.

Credits3 ECTS6

Unrest. Elect.

Credits3 ECTS5

HSS

HSS Elect.

Credits3 ECTS5

Math

Math. Elective

Credits3 ECTS6

Math

Math. Elective

Credits3 ECTS6

Unrest. Elect.

Credits3 ECTS5

Unrest. Elect.

Credits3 ECTS5

Science Elect.

Credits3 ECTS5