Web Site of the Department

Head of Department: Talin Budak

Professors: Talin Budak, Alp Eden, Ahmet Feyzioglu, Zerrin Gokturk, Nilgun Isik, Haluk Oral, Ercument Ortacgil, John Pym•, Ayse Soysal, Betul Tanbay, Yalcin Yildirim

Associate Professor: Sadik Deger

Assistant Professors: Arzu Boysal, Olcay Coskun, Burak Gurel, Muge Kanuni, Muge Taskin Aydin

Instructors: Ozlem Beyarslan, Fatih Ecevit, Dr. Gulay Oke, Dr. Ferit Ozturk, Gulnihal Yucel

•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: 141/143 credits

*Students who pass the preparatory year with an English proficiency level of C are required to take AE 111 and AE 112.

**Students with storng background can take (PHYS 121-PHYS 201-PHYS 202) instead of (PHYS 101-PHYS 130-PHYS 201)

CERTIFICATE IN ACTUARIAL MATHEMATICS

Undergraduates who successfully complete the following 10 courses may receive a "Certificate in Actuarial Mathematics" at the time of their graduation:

MATH 101 Calculus I or MATH 131 Cal. for Math. Stud. I 4

MATH 102 Calculus II or MATH 132 Cal. for Math. Stud. II 4

MATH 343 Probability 4

MATH 336 Numerical Analysis 4

MATH 342 Life Insurance Math. 3

MATH 344 Statistics or (EC 231 + EC 232) 3 or (3+3)

MATH 442 Risk Analysis 3

EC 101 Int. to Econ. I or EC 201 Econ. for Engineers I 3

EC 102 Int. to Econ. II or EC 202 Econ. for Engineers II 3

IE 310 Operations Research 3

COURSE DESCRIPTIONS

MATH 101 Calculus I (Analiz I) (4+2+0) 4
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 (Analiz II) (4+2+0) 4
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
(Sonlu Matematige Giris)
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
(Sosyal Bilimler icin Analize Giris)
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
(Matematiksel Yapilara Giris)
Prepositional logic, quantification, methods of proof, sets, relations, functions, operations, equivalence relations, cardinality, introduction to algebraic structures.

MATH 131 Calculus for Mathematics Students I (4+2+0) 4
(Matematik Ogrencileri Icin Analiz I)
Fundamental properties of real numbers, sequences and subsequences, Bolzano-Weierstrass theorem, limits of functions, continuity, intermediate and extreme value theorems, differentiation and its applications, mean value theorems.

MATH 132 Calculus for Mathematics Students II (4+2+0) 4
(Matematik Ogrencileri Icin Analiz II)
Riemann integration, fundamental theorem of calculus, techniques and applications of integration, improper integrals, basic transcendental functions, infinite series, convergence tests, Taylor polynomials, power series.
Prerequisite: MATH 131.

MATH 162 Discrete Mathematics (Ayirtik Matematik) (4+2+0) 4
Introduction to basic problems, sums and recurrences, elementary number theory, properties of binomial coefficients, special numbers, discrete probability theory, generating functions.

MATH 201 Matrix Theory (Matris Kurami) (4+2+0) 4
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 (Turevsel Denklemler) (4+2+0) 4
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 132, MATH 201.

MATH 224 Linear Algebra (Lineer Cebir I) (3+2+0) 3
Vector spaces, linear transformations, rank and nullity, change of basis, canonical forms, Euclidien spaces, Gram-Schmidt orthogonalization method.
Prerequisites: MATH 111 and MATH 201.

MATH 231 Calculus for Mathematics Students III (4+2+0) 4
(Matematik Ogrencileri Icin Analiz III)
Vector calculus, functions of several variables, directional derivatives, gradient, vector-valued functions, divergence and curl, Taylor's theorem, Lagrange multipliers, multiple integrals, change of variables, line integrals, Green's theorem.
Prerequisite: MATH 132.

MATH 232 Introduction to Complex Analysis (Karmasik Analize Giris) (3+2+0) 3
The field of complex numbers, the extended complex plane and its topological properties, series of complex functions, M-test, power series, analytic functions, elementary functions and their mapping properties.
Prerequisites: MATH 101 or MATH 131, MATH 111.

MATH 321 Algebra I (Cebir I) (4+2+0) 4
Introduction to group theory, subgroups, Lagrange's theorem, factor groups, permutation groups, group homomorphisms, isomorphism theorems, introduction to ring theory, ideals, ring homomorphisms, divisibility, polynomial rings, field of rational functions.
Prerequisites: MATH 111 and MATH 201.

MATH 322 Algebra II (Cebir II) (4+2+0) 4
Vector spaces over an arbitrary field, linear independence and bases, linear transformations and matrices, fields, field extensions, algebraic extensions, Kronecker's theorem, finite fields.
Prerequisite: MATH 321 or consent of instructor.

MATH 327 Number Theory (Sayilar Teorisi) (3+2+0) 3
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 Real Analysis I (Gercel Analiz I) (4+2+0) 4
Metric spaces, convergence, completeness, continuity, compactness, connectedness, contraction mapping principle.
Prerequisite: MATH 232.

MATH 332 Real Analysis II (Gercel Analiz II) (4+2+0) 4
Sequences and series of functions, Arzelà-Ascoli theorem, Stone-Weierstrass theorem, Fourier series, inverse and implicit function theorems, integration.
Prerequisites: MATH 331 or consent of instructor.

MATH 333 Fourier Series (Fourier Serileri) (3+2+0) 3
Topics from the theory of integration, Fourier series, Dirichlet kernel and convergence tests, orthogonal families, convergence in the mean, Parseval's equation, Fejér and Poisson kernels, applications.
Prerequisite: MATH 232.

MATH 336 Numerical Analysis (Sayisal Analiz) (4+2+0) 4
Solutions of nonlinear equations, Newton's method, fixed points and functional iterations, LU factorization, pivoting, norms, analysis of errors, orthogonal factorization and least square problems, polynomial interpolation, spline interpolation, numerical differentiation, Richardson extrapolation, numerical integration, Gaussian quadratures, error analysis.
Prerequisite: MATH 201.

MATH 342 Life Insurance Mathematics (4+2+0) 4
(Hayat Sigortalari Matematigi)
Introduction to the theory of interest, survival distribution and life tables, life insurance and life annuities, commutation functions, fully discrete and continuous premiums, net premium reserves, expense factors, modified reserve methods, nonforfeiture benefits and dividends.

MATH 343 Probability (Olasiliklar Hesabi) (4+2+0) 4
Sets and counting, probability and relative frequency, conditional probability, Bayes' theorem, independence, discrete and continuous random variables, binomial, Poisson and normal distributions, functions of random variables, law of large numbers, generating functions, characteristic functions, moments, compound distributions, central limit theorems, Markov chains and their limiting probabilities.
Prerequisite: MATH 101 or MATH 132.

MATH 344 Statistics (Istatistik) (3+2+0) 3
Methods of data analysis and data presentation, sampling distributions, point estimation and properties of estimators, Cramer-Rao inequality, parameter estimation, maximum likelihood and moment matching, interval estimation, hypothesis testing, the Newman-Pearson lemma, likelihood ratio test, goodness of fit tests, linear regression, analysis of variance, nonparametric tests.

MATH 351 Qualitative Theory of Ordinary Differential Equations (3+2+0) 3
(Siradan Turevsel Denklemlerin Nitelik Kurami)
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 (Kismi Turevsel Denklemler) (3+2+0) 3
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 102 and MATH 202) or (MATH 202 and MATH 231).

MATH 363 Graph Theory (Çizgeler Kurami) (3+2+0) 3
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 162 and MATH 224.

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

MATH 404 Mathematica® (Mathematica®) (3+2+0) 3
Mathematica® as an interactive symbolic calculator, graphics, algebra and calculus, solving equations, solving differential equations, lists, matrices, transformation rules, functional operations and pure functions, introduction to programming, Mathematica® packages such as discrete mathematics, linear algebra, number theory, numerical mathematics, statistics.
Prerequisites: Two MATH courses and consent of instructor.

MATH 411 Mathematical Logic (Matematiksel Mantik) (3+2+0) 3
Prepositional and quantificational logic, formal grammar, semantical interpretation, formal deduction, completeness theorems, selected topics from model theory and proof theory.
Prerequisite: Consent of instructor.

MATH 412 Introduction to Set Theory (Kümeler Kuramina Giris) (3+2+0) 3
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 and consent of instructor.

MATH 421 Algebra III (Cebir III) (3+2+0) 3
Additional topics in the theory of groups, separability, Galois theory, solvability of an algebraic equation by radicals, applications to geometrical constructions by ruler and compass.
Prerequisite: MATH 322.

MATH 424 Linear Algebra II (Lineer Cebir II) (3+2+0) 3
Vector spaces over an arbitrary field, linear independence, bases and dimension, matrices associated with homomorphisms, diagonalization, modules over a principal ideal domain, elementary divisors, canonical forms of matrices, linear, quadratic and bilinear forms, duality, inner product and unitary spaces, adjoint operator, normal, unitary and hermitian operators, spectral theorem.
Prerequisites: MATH 224 and MATH 322.

MATH 431 Complex Analysis I (Karmasik Analiz I) (4+2+0) 4
Complex differentiation, Cauchy-Riemann equations, holomorphic functions, conformal mappings, contour integration, Cauchy's theorem, Taylor and Laurent series, open mapping theorem, maximum modulus principle, applications of the residue theorem.
Prerequisite: MATH 232.

MATH 432 Complex Analysis II (Karmasik Analiz II) (3+2+0) 3
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 431.

MATH 436 Functional Analysis (Fonksiyonel Analiz) (3+2+0) 3
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 Kurami) (3+2+0) 3
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 442 Risk Analysis (Risk Analizi) (3+2+0) 3
Definition of risk, risk treatment, measures of risk and risk aversion, modeling of loss distributions, basic ratemaking and reserving techniques, reinsurance, individual and collective risk theory, credibility theory and ruin theory.
Prerequisite: Consent of instructor.

MATH 443 Basic Pension Mathematics (Temel Emeklilik Matematigi) (3+2+0) 3
Multiple life and multiple decrement models, actuarial functions, design and financing of retirement plans.
Prerequisites: MATH 342 and MATH 343.

MATH 444 Ratemaking Models (Fiyatlandirma Modelleri) (3+2+0) 3
Advanced statistical methods in nonlife insurance, ratemaking, loss reserving methods, reinsurance models, insurer's solvency, simulation models in insurance, the insurance firm as a financial institution.
Prerequisites: MATH 442 and consent of instructor.

MATH 447 Game Theory (Oyunlar Kurami) (3+2+0) 3
Definition of a game, two-person zero-sum games, min-max theorem, computation of optimal strategies, n-person games, other topics.
Prerequisites: MATH 201 and MATH 343.

MATH 451 Numerical Solutions of Differential Equations (3+2+0) 3
(Diferansiyel Denklemlerin Sayisal Cozumleri)
Runge-Kutta methods for ordinary differential equations, multi-step methods, error analysis, stability, finite difference methods for boundary value problems, collocation method, explicit and implicit methods for solving parabolic partial differential equations, finite difference methods, Galerkin method, solution methods for hyperbolic equations.
Prerequisites: MATH 336 and MATH 352.

MATH 455 Calculus of Variations (Varyasyonlar Hesabi) (3+2+0) 3
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 461 Coding Theory (Kodlar Kurami) (3+2+0) 3
Basic definitions, syndrome decoding, BCH and cyclic codes, quadratic residue codes, weight distributions, relation to design theory.
Prerequisite: MATH 322.

MATH 462 Cryptography (Sifre Kurami) (3+2+0) 3
Early crypto systems and simple systems, public key cryptography, primality and factoring, elliptic curve crypto systems.
Prerequisite: Consent of instructor.

MATH 465 Calculus of Finite Differences (Sonlu Farklar Hesabi) (3+2+0) 3
Divided differences, interpolation and integration formulas, the shift, difference and mean operators, factorials, Stirling numbers, Bernoulli and Euler polynomials, sum calculus, gamma and related functions, Euler-Maclaurin summation formula, Boole's summation formula, introduction to the theory of difference equations, applications.
Prerequisite: MATH 162.

MATH 471 Topology (Topoloji) (4+2+0) 4
Topological spaces, compactness and connectedness, continuous functions, Tychonoff's theorem, separation axioms, Urysohn and Tietze theorems, homotopy, fundamental group, covering spaces.
Prerequisite: MATH 331.

MATH 475 Differential Geometry (Diferansiyel Geometri) (3+2+0) 3
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 231 or MATH 102.

MATH 476 Differential Topology (Turevsel Topoloji) (3+2+0) 3
Smooth manifolds in Rn, transversality, Morse functions, Sard's theorem, Whitney embedding theorem, intersection theory mod 2, Jordan-Brouwer separation theorem, Borsuk-Ulam theorem, oriented intersection theory, Lefschetz fixed point theorem, Poincare-Hopf theorem, Hopf degree theorem.
Prerequisite: MATH 332.

MATH 477 Projective Geometry (Projektif Geometri) (3+2+0) 3
Projective spaces and homogeneous coordinates, subspaces, the dual space, Desargues' theorem, double ratio, collineation, projections and correlations, polarity, passage to affine and metric spaces, plane algebraic curves and their singular points, conics and cubics.
Prerequisite: MATH 201.

MATH 478 Groups and Geometries (Gruplar ve Geometriler) (3+2+0) 3
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 321.

MATH 479 Fractal Geometry (Fraktal Geometri) (3+2+0) 3
Hausdorff measure and dimension, fractal dimension, product of fractals, iterated function schemes, self-similar and self-affine sets, dimensions of geometric figures, iteration of complex functions, Julia sets, Mandelbrot set, applications to number theory, probability etc.
Prerequisite: MATH 232 or consent of instructor.

MATH 480 Seminar (Seminer) (1+0+2) 2
Student presentation on an area not covered in classes under the supervision of an instructor.

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

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

Web Site of the Department

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Contact Information Bogazici University 34342 Bebek, Istanbul Tel: 0212 359 54 00