Web Site of the Department

Acting Head of Department : İlhan İkeda

Associate Department Heads: Fatih Ecevit, Müge Taşkın Aydın

Professors : Talin Budak, Sadık Değer, Ahmet Feyzioğlu, Burak Gürel, İlhan İkeda, Ferit Öztürk, Betül Tanbay, Cem Yalçın Yıldırım, Alp Bassa, Fatih Ecevit, Arzu Boysal

Associate Professor : Özlem Beyarslan, Ayhan Günaydın, Müge Taşkın, Ümit Işlak

Assistant Professors : Yasemin Kara

Instructors : Sinan Işik

Assistants :Bahri Fatih Barbaros, Barış Yeşiloğlu, Cem Yalım Özel, Ceren Ayşe Deral, Muhammed İkbal Ulvi, Mustafa Akdağ, Oğuz Yılmaz, Ömer Avcı, Şeyma Karadereli, Metin Ersin Arıcan, Berkay Kamil Taştan, Meleknaz Uzuner, Kerem Kubilay

•Adjunct

Mathematics is a rapidly developing and expanding field which, in addition to its traditional areas of application in the physical sciences, is continually being expanded into new areas of knowledge such as the biological and social sciences. In particular, the vast advances made in computer technology in the past few years have given rise to new mathematical disciplines. Considering these factors, the Department of Mathematics offers a Bachelor of Science program which is designed to prepare students for graduate study in mathematics or in related areas of the natural or social sciences or engineering. The program provides a good foundation for those who wish to pursue careers in teaching or in research, or in related areas of science, technology, business, or government where mathematics is important.

The department also offers double-major programs with other departments which lead simultaneously to a B.S. degree in mathematics and to a B.S. degree in the other major. Students are expected to complete successfully the first year of their major in order to join the double-major programs. The requirement for a double-major program is to complete the following mathematics courses, in addition to all the courses in the major field: MATH 101, 102, 201, 202 or MATH 131, 132, 231, 201, 202, followed by MATH 232, 321, 322, 331, 332, 431, plus two area electives with codes higher than 300 from the Department of Mathematics. Mathematics students can also participate in double-major programs with the Departments of Physics, Philosophy, Economics, and Molecular Biology and Genetics. In addition, there is a certificate in actuarial mathematics offered upon completion of the courses described under this heading.

UNDERGRADUATE PROGRAM

Total: 134 Credits/243/246 ECTS

*Students whos Proficiency Exam score is C are required to take AE 101 and AE 102, while students whose Proficiency Exam score is C+ are required to take AE 111 and AE 112. Students whose Proficiency Exam score is 1 or B are required to take an at least 3-credit HSS Elective course or MLE (Modern Language Elective. MLE courses should be selected from among any of the language courses of the same sequence.

**Physics courses are one of the 2 series:
- PHYS 101, PHYS 102, PHYS 201, PHYS 202; OR PHYS 121, PHYS 201, PHYS 202.

COURSE DESCRIPTIONS

MATH 101 Calculus I (4+2+0) 4 ECTS 6
(Analiz I)   
Functions, limits, continuity, differentiation and applications, integration, fundamental theorem of calculus, techniques and applications of integration, improper integrals and series, Taylor polynomials, power series, basic transcendental functions.

MATH 102 Calculus II (4+2+0) 4 ECTS 6
(Analiz II)    
Vector calculus, functions of several variables, directional derivatives, gradient, Lagrange multipliers, multiple integrals and their applications, change of variables, coordinate systems, line integrals, Green's theorem, divergence theorem, Stokes' theorem.
Prerequisite: MATH 101.

MATH 105 Introduction to Finite Mathematics   (4+2+0) 4 ECTS 6
(Sonlu Matematiğe Giriş)
Systems of linear equations and inequalities, matrices, determinants, inverses, Gaussian elimination; geometric approach to linear programming, basic combinatorics, binomial theorem, finite probability theory, conditional probability, Bayes' theorem, random variables, expected value, variance, decision theory.

MATH 106 Introduction to Calculus for Social Sciences   (4+2+0) 4 ECTS 6
(Sosyal Bilimler için Analize Giriş)
Functions of one variable, properties of quadratic, cubic, exponential and logarithmic functions, compound interest and annuities, limits, continuity and differentiation, applied maximum and minimum problems, basic integration techniques, sequences and series.

MATH 111 Introduction to Mathematical Structures  (4+2+0) 4 ECTS 8
(Matematiksel Yapılara Giriş)
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 well-ordering principle, recursive definitions, axiomatic systems, Euclid's postulates and non-Euclidean geometries.

MATH 131 Calculus of a single variable   (4+2+0) 4 ECTS 8
(Tek Değişkenli Analiz)
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  (4+2+0) 4 ECTS 8
(Çok Değişkenli Analiz)
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 (4+2+0) 4 ECTS 8
(Ayrık Matematik)  
Counting, the pigeonhole principle, permutations, combinations, binomial coefficients, generalized permutations and combinations, discrete probability, linear recurrence relations, generating functions, inclusion-exclusion, relations, closure of relations, equivalence relations, construction of integers and rationals, partial orderings, graphs.

MATH 201 Matrix Theory (4+2+0) 4 ECTS 5
(Matris Kuramı)   
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 (4+2+0) 4 ECTS 7
(Türevsel Denklemler) 
First-order differential equations, second-order linear equations, Wronskian, change of parameters, homogeneous and non-homogeneous equations, series solutions, Laplace transform, systems of first-order linear equations, boundary value problems, Fourier series.
Prerequisites: (MATH 101 or MATH 131) and (MATH 201 or MATH 221).

MATH 221 Linear Algebra (4+2+0) 4 ECTS 8
(Doğrusal Cebir I)  
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 or consent of the instructor.

MATH 222 Group Theory (4+2+0) 4 ECTS 8
(Gruplar Kuramı)  
Groups, subgroups, cyclic groups, generating sets, permutations, orbits, cycles, alternating groups, cosets, Lagrange's Theorem, direct products, finite abelian groups, homomorphism, normal subgroups, factor groups, simple groups, group actions, isomorphism theorems, Sylow's theorems.
Prerequisite: MATH 111.

MATH 231 Advanced Calculus I  (4+2+0) 4 ECTS 8
(İleri Analiz I)
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 or Math 102.

MATH 234 Advanced Calculus II  (4+2+0) 4 ECTS 8
(İleri Analiz II)
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 (4+2+0) 4 ECTS 8
(Halkalar, Cisimler ve Galois Kuramı)   
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 (3+2+0) 3 ECTS 6
(Sonlu Grupların Temsil Teorisi)
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 (3+0+2) 3 ECTS 6
(Matris Grupları)
General linear groups, closed subgroups of real and complex general linear groups, their topological properties, associated tangent spaces, exponential and logaritm functions, manifolds, maximal tori, homomorphisms.
Prerequisites: (MATH 102 or MATH 132) and MATH 222

MATH 327 Number Theory (3+2+0) 3 ECTS 6
(Sayılar Teorisi)  
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 (4+2+0) 4 ECTS 8
(Metrik Uzaylar)   
Topology, density, separability, convergence, compactness, connectedness, continuity, open and closed maps, equıcontinuity, Arzela-Ascoli theorem, contractions and fixed point theorems, completeness, Cantor's theorem, Blaire category theorem, completion.
Prerequisite: MATH 231.

MATH 332 Lebesgue Integration (3+2+0) 3 ECTS 6
(Lebesgue İntegrali)  
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.
Prerequisites: MATH 234 or consent of the instructor.

MATH 334 Analysis on Manifolds (3+2+0) 3 ECTS 6
(Çokkatmanlılarda Analiz)
Differentiation, inverse and implicit function theorems, integration, manifolds, differential forms, orientation, Stoke's theorem, Poincaré lemma, de Rham cohomology.
Prerequisite: MATH 221 and MATH 234.

MATH 336 Numerical Analysis (3+2+0) 3 ECTS 6
(Nümerik Analiz)   
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, QR-iteration, 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 (4+2+0) 4 ECTS 8
(Karmaşık Analiz I)
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 intgrals, 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 (3+2+0) 3 ECTS 6
(Olasılık ve İstatistiğe giriş)  
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.
Prerequisites: MATH 102 or MATH 132

MATH 345 Probability (3+2+0) 3 ECTS 6
(Olasılık)   
Axioms of probability, conditional probability, independence, discrete and continuous random variables, jointly distributed random variables, expectation, limit theorems.
Prerequisite: MATH 344 or condent of the instructor.

MATH 351 Qualitative Theory of Ordinary Differential Equations (3+2+0) 3 ECTS 6
(Sıradan Türevsel Denklemlerin Nitelik Kuramı)
Existence and uniqueness theorems, phase portraits in the plane, linear systems and canonical forms, nonlinear systems, linearization, stability of fixed points, limit cycles, Poincaré-Bendixson theorem.
Prerequisite: MATH 202.

MATH 352 Partial Differential Equations (3+2+0) 3 ECTS 6                          
(Kısmi Türevsel Denklemler)
Wave equation, heat equation, Laplace equation, classification of second order linear equations, initial value problems, boundary value problems, Fourier series, harmonic functions, Green's function.
Prerequisites: (MATH 132 and MATH 202) or (MATH 102 and MATH 202).

MATH 361 Combinatorics (3+2+0) 3 ECTS 6
(Kombinatorik)
Sieve methods, lattices, distributive lattices, incidence algebra, Mobius inversion formula, Mobius algebras, generating functions, exponential formula, Lagrange inversion formula, matrix tree theorem.
Prerequisites: MATH 201 or MATH 221

MATH 363 Graph Theory (3+2+0) 3 ECTS 6
(Çizgeler Kuramı)  
Basic definitions, trees, Cayley's formula, connectedness, Eulerian and Hamiltonian graphs, matchings, edge and vertex coloring, chromatic numbers, planar graphs, directed graphs, networks.
Prerequisites: MATH 221 or consent of the instructor.

MATH 401 History of Mathematics (3+2+0) 3 ECTS 6
(Matematik Tarihi)  
History of algebra, geometry, analytic geometry, calculus from Antiquity through the seventeenth century and more recent mathematical history.
Prerequisite: Consent of the instructor.

MATH 404 Computational Mathematics (3+2+0) 3 ECTS 6
(Hesaplamalı Matematik)
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 (3+2+0) 3 ECTS 6
(Matematiksel Mantık) 
Prepositional 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 (3+2+0) 3 ECTS 6
(Kümeler Kuramına Giriş)
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.
Prerequisites: MATH 111.

MATH 413 Model Theory (3+2+0) 3 ECTS 6
(Model Kuramı)
Language and Structure, theory, definable sets and interpretability, compactnees theorem, complete theories, Löwenheim-Skolem theorems, quantifier elimination, algebraic examples.
Prerequisites: MATH 111.

MATH 425 Introduction to Algebraic Geometry (3+2+0) 3 ECTS 6
(Cebirsel Geometriye Giriş)
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 (3+2+0) 3 ECTS 6
(Aritmetik Geometriye Giriş)
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 (3+2+0) 3 EC 6
(Temel Sayılar Kuramı II)
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 (3+2+0) 3 ECTS 6
(Karmaşık Analiz II)  
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 of conformal mapping.
Prerequisite: MATH 338.

MATH 433 Fourier Analysis (3+2+0) 3 ECTS 6
(Fourier Analizi)  
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 (3+2+0) 3 ECTS 7
(Fonksiyonel Analiz)  
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 (3+2+0) 3
(Eniyileme Kuramı) 
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  (3+2+0) 6
(Türevsel Denklemlerin Sayısal Çözümleri)
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.
Prerequisites: (MATH 102 or MATH 132) and MATH 202.

MATH 452 Dynamical Systems (3+2+0) 3 ECTS 6
(Dinamik Sistemler)
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 (3+2+0) 3 ECTS 6
(Varyasyonlar Hesabı)
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, principle of least action, conservation laws, Hamilton-Jacobi equation.
Prerequisite: MATH 202.

MATH 462 Cryptography (3+2+0) 3 ECTS 6
(Şifre Kuramı)  
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 (3+2+0) 3 ECTS 6
(Topoloji)  
Topological spaces, compactness, connectedness, continuity, separation axioms, homotopy, fundamental group.
Prerequisite: MATH 331.

MATH 472 Geometric Topology (3+2+0) 3 ECTS 6
(Geometrik Topoloji)
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 (3+2+0) 3 ECTS 6
(Genel Görelilik Kuramının Matematiksel Yönleri)
Review of special relativity, differentiable manifolds, tensors, Lie derivative, coveriant 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 (3+2+0) 3 ECTS 6
(Diferansiyel Geometri)
Fundamentals of Euclidean spaces, geometry of curves and surfaces in three-dimensional Euclidean space, Gauss map, first and 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 (3+2+0) 3 ECTS 6
(Diferansiyel Topoloji) 
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 (3+2+0) 3 ECTS 6
(İzdüşümsel Geometri) 
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 (3+2+0) 3 ECTS 6
(Gruplar ve Geometriler)
Plane Euclidean geometry and its group of isometries, affine transformations in the Euclidean plane, fundamental theorem of affine geometry, finite group of isometries of R2, Leonardo da Vinci's theorem, geometry on the sphere S2, motions of S2, orthogonal transformations of R3, Euler's theorem, right triangles in S2, projective plane, Desargues' theorem, the fundamental theorem of projective geometry.
Prerequisite: MATH 222 or consent of the instructor.

MATH 481-489, 491-499 Selected Topics in Mathematics (3+0+0) 3 ECTS 6
(Matematikten Seçilmiş Konular)
Selected topics in pure and applied mathematics.
Prerequisite: Consent of the instructor.

MATH 490 Project (1+0+4) 3 ECTS 6
(Proje) 
Individual research supervised by a member of the department.
Prerequisite: Consent of the instructor.