# Department of Mathematics

Departmental Courses

All courses and course definitions of the Mathematics department may be seen under the undergraduate catalogue and graduate catalogue pages of Boğaziçi University. The links indicated below provide the most recent information about new courses or changes in course titles, definitions or credits as accepted by the university senate.

 Courses Offered by Mathematics Department
 Math 101 Calculus 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.
 Prerequisite: None

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

 Math 105 Introduction to Finite Mathematics (4+2+0) 4
 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.
 Prerequisite: None

 Math 106 Introduction to Calculus for Social Sciences (4+2+0) 4
 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.
 Prerequisite: None

 Math 111 Introduction to Mathematical Structures (4+2+0) 4
 Propositional logic, quantification,  methods of proof, sets, relations, functions and operations, equivalence relations, cardinality, introduction to algebraic structures.
 Prerequisite: None

 Math 131 Calculus for Mathematics Students I (4+2+0) 4
 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.
 Prerequisite: None

 Math 132 Calculus for Mathematics Students II (4+2+0) 4
 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 (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.
 Prerequisite: None

 Math 201 Matrix Theory (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.
 Prerequisite: None

 Math 202 Differential Equations (4+2+0) 4
 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 132 and MATH 201) or (MATH 101 and MATH 201)

 Math 224 Linear Algebra I (3+2+0) 3
 Vector spaces, linear transformations, rank and nullity, change of basis, canonical forms, Euclidean spaces, Gram-Schmidt orthogonalization process.
 Prerequisite: Math 111 and MATH 201

 Math 231 Calculus for Mathematics Students III (4+2+0) 4
 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 (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.
 Prerequisite: ( MATH 131 and MATH 111) or (MATH 101 and 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.
 Prerequisite: MATH 111 and MATH 201

 Math 322 Algebra 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 the instructor

 Math 327 Number Theory (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 (4+2+0) 4
 Metric spaces, convergence, completeness, continuity, compactness, connectedness, contraction mapping principle.
 Prerequisite: MATH 232

 Math 332 Real Analysis II (4+2+0) 4
 Sequences and series of functions, Arzela-Ascoli theorem, Stone-Weierstrass theorem, Fourier series, inverse and implicit function theorems, integration.
 Prerequisite: (MATH 231 and MATH 331) or consent of the instructor

 Math 333 Fourier Series (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,  Fejer and Poisson kernels, applications.
 Prerequisite: MATH 232

 Math 336 Numerical Analysis (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
 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 dividents.
 Prerequisite: None

 Math 343 Probability (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 (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 tests, goodness of fit tests, linear regression, analysis of variance, nonparametric tests.
 Prerequisite: None

 Math 351 Qualitative Theory of Ordinary Differential Equations (3+2+0) 3
 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 (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 functions.
 Prerequisite: (MATH 202 and MATH 231) or (MATH 102 and MATH 202)

 Math 363 Graph Theory (3+2+0) 3
 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 162 or consent of instructor

 Math 401 History of Mathematics (3+2+0) 3
 Selected topics in the history of mathematics and related fields.
 Prerequisite: Consent of instructor

 Math 404 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.
 Prerequisite: Two MATH courses and consent of instructor
 Math 411 Mathematical Logic (3+2+0) 3
 Propositional 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 (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.
 Prerequisite: Math 111 and consent of instructor

 Math 421 Algebra 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 (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.
 Prerequisite: Math 224 and MATH 322

 Math 431 Complex Analysis 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 231 and Math 232

 Math 432 Complex Analysis 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 (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 (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 (3+2+0) 3
 Definition of risk, risk treatment, measures of risk and risk aversion, modelling of loss distributions, basic ratemaking and reserving techniques, reinsurance, individual and collective risk theory, credibility theory and ruin theory.
 Prerequisite: Consent of the instructor

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

 Math 444 Ratemaking Models (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.
 Prerequisite: Math 442 and consent of the instructor

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

 Math 451 Numerical Solutions of Differential Equations (3+2+0) 3
 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 problems.
 Prerequisite: Math 336 and MATH 352

 Math 455 Calculus of Variations (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, the principle of least action, conservation laws, Hamilton-Jacobi equation.
 Prerequisite: Math 202

 Math 461 Coding Theory (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 (3+2+0) 3
 Early crypto systems and simple systems, public key cryptography, primality and factoring, elliptic curve crypto systems.
 Prerequisite: Consent of the instructor

 Math 465 Calculus of Finite Differences (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 (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 (3+2+0) 3
 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 231 or MATH 102

 Math 476 Differential Topology (3+2+0) 3
 Smooth manifolds in R, 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 (3+2+0) 3
 Projective spaces and homogenous 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 (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 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 321

 Math 479 Fractal Geometry (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 the instructor

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

 Math 481-489 Selected Topics in Mathematics (3+0+0) 3
 Selected topics in pure and applied mathematics.
 Prerequisite: Consent of the instructor.

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

 Math 491-499 Selected Topics in Mathematics (3+0+0) 3
 Selected topics in pure and applied mathematics.
 Prerequisite: Consent of the instructor.

### Current Semester (Spring '13)

 Offered Courses Course Webpages