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.
Please check Graduate Catalogue
Graduate Courses Offered by Mathematics Department

MATH 521 
Algebra I 
(4+0+0) 4 
Free groups, group actions, group with operators, Sylow theorems, JordanHölder theorem, nilpotent and solvable groups. Polynomial and power series rings, Gauss's lemma, PID and UFD, localization and local rings,chain conditions, Jacobson radical. 
MATH 522 
Algebra II 
(4+0+0) 4 
Galois theory, solvability of equations by radicals, separable extensions, normal basis theorem, norm and trace, cyclic and cyclotomic extensions, Kummer extensions. Modules, direct sums, free modules, sums and products, exact sequences, morphisms, Hom and tensor functors, duality, projective, injective and flat modules, simplicity and semisimplicity, density theorem, WedderburnArtin theorem, finitely generated modules over a principal ideal domain, basis theorem for finite abelian groups. 
MATH 525 
Algebraic Number Theory 
(4+0+0) 4 
Valuations of a field, local fields, ramification index and degree, places of global fields, theory of divisors, ideal theory, adeles and ideles, Minkowski's theory, extensions of global fields, the Artin symbol. 
MATH 527 
Number Theory 
(4+0+0) 4 
Method of descent, unique factorization, basic algebraic number theory, diophantine equations, elliptic equations, padic numbers, Riemann zeta function, elliptic curves, modular forms, zeta and Lfunctions, ABCconjecture, heights, class numbers for quadratic fields, a sketch of Wiles' proof. 
MATH 528 
Analytic Number Theory 
(4+0+0) 4 
Primes in arithmetic progressions, Gauss' sum, primitive characters, class number formula, distribution of primes, properties of the Riemann zeta function and Dirichlet Lfunctions, the prime number theorem, Polya Vinogradov inequality, the large sieve, average results on the distribution of primes. 
MATH 529 
Analytic Number Theory II 
(3+0+0) 3 
The prime number theorem for arithmetic progressions. Sums over primes, exponential sums. The large sieve, BombieriVinogradov theorem,Selberg’s sieve. Results on the distribution of primes. 
MATH 531 
Real Analysis I 
(4+0+0) 4 
Lebesgue measure and Lebesgue integration on Rn, general measure and integration, decomposition of measures, RadonNikodym theorem, extension of measures, Fubini's theorem. 
MATH 532 
Real Analysis II 
(4+0+0) 4 
Normed and Banach spaces, Lpspaces and duals, HahnBanach theorem, category and uniform boundedness theorem, strong, weak and weak*convergence, open mapping theorem, closed graph theorem. 
MATH 533 
Complex Analysis I 
(4+0+0) 4 
Review of the complex number system and the topology of C, elementary properties and examples of analytic functions, complex integration, singularities, maximum modulus theorem, compactness and convergence in the space of analytic functions. 
MATH 534 
Complex Analysis II 
(4+0+0) 4 
Runge's theorem, analytic continuation, Riemann surfaces, harmonic functions, entire functions, the range of an analytic function. 
MATH 535 
Functional Analysis 
(4+0+0) 4 
Topological vector spaces, locally convex spaces, weak and weak* topologies, duality, Alaoglu's theorem, KreinMilman theorem and applications, Schauder fixed point theorem, KreinSmulian theorem, EberleinSmulian theorem, linear operators on Banach spaces. 
Prerequisite: 
MATH 531 and MATH 532 
MATH 541 
Probability Theory 
(4+0+0) 4 
An introduction to measure theory, Kolmogorov axioms, independence, random variables, expectation, modes of convergence for sequences of random variables, moments of a random variable, generating functions, characteristic functions, product measures and joint probability, distribution laws, conditional expectations, strong and weak law of large numbers, convergence theorems for probability measures, central limit theorems. 
MATH 544 
Stochastic Processes and Martingales 
(4+0+0) 4 
Stochastic processes, stopping times, DoobMeyer decomposition, Doob's martingale convergence theorem, characterization of square integrable martingales, RadonNikodym theorem, Brownian motion, reflection principle, law of iterated logarithms. 
MATH 545 
Mathematics of Finance 
(4+0+0) 4 
From random walk to Brownian motion, quadratic variation and volatility, stochastic integrals, martingale property, Ito formula, geometric Brownian motion, solution of BlackScholes equation, stochastic differential equations, FeynmanKac theorem, CoxIngersollRoss and Vasicek term structure models, Girsanov's theorem and risk neutral measures, HeathJarrowMorton term structure model, exchangerate instruments. 
MATH 551 
Partial Differential Equations I 
(4+0+0) 4 
Existence and uniqueness theorems for ordinary differential equations, continuous dependence on data. Basic linear partial differential equations : transport equation, Laplace's equation, diffusion equation, wave equation. Method of characteristics for nonlinear firstorder PDE's, conservation laws, special solutions of PDE's, CauchyKowalevskaya theorem. 
MATH 552 
Partial Differential Equations II 
(4+0+0) 4 
Hölder spaces, Sobolev spaces, Sobolev embedding theorems, existence and regularity for secondorder elliptic equations, maximum principles, secondorder linear parabolic and hyperbolic equations, methods for nonlinear PDE's, variational methods, fixed point theorems of Banach and Schauder. 
MATH 571 
Topology 
(4+0+0) 4 
Fundamental concepts, subbasis, neighborhoods, continuous functions, subspaces, product spaces and quotient spaces, weak topologies and embedding theorem, convergence by nets and filters, separation and countability, compactness, local compactness and compactifications, paracompactness, metrization, complete metric spaces and Baire category theorem, connectedness. 
MATH 572 
Algebraic Topology 
(4+0+0) 4 
Basic notions on categories and functors, the fundamental group, homotopy, covering spaces, the universal covering space, covering transformations, simplicial complexes and their homology. 
MATH 575 
Differentiable Manifolds 
(3+0+0) 3 
Differentiable manifolds, smooth maps, submanifolds, vectors and vector fields, Lie brackets, Lie Groups, Lie group actions, integral curves and flows, Lie algebras, Lie derivative, Killing fields, differential forms, Integration. 
MATH 576 
Riemannian Geometry 
(3+0+0) 3 
Differentiable manifolds, vectors and tensors, riemannian metrics, connections, geodesics, curvature, jacobi fields, riemannian submanifolds, spaces of constant curvature. 
MATH 577 
Complex Manifolds 
(3+0+0) 3 ECTS 7 
Complex Manifolds, Kahler and CalabiYau
Manifolds, Homology and Cohomology, Fiber Bundles, Connections on
Fiber Bundles, Characteristic Classes, Index Theorems. 
Prerequisite: 
MATH 576 or consent of the instructor 
MATH 579 
Graduate Seminar 
(0+1+0) Noncredit 
Presentation of topics of interest in mathematics through seminars offered by faculty, guest speakers and graduate students. 
MATH 581 
Selected Topics in Analysis I 
(3+0+0) 3 
MATH 582 
Selected Topics in Analysis II 
(3+0+0) 3 
MATH 583 
Selected Topics in Foundations of Mathematics 
(3+0+0) 3 
MATH 584 
Selected Topics in Algebra and Topology 
(3+0+0) 3 
MATH 585 
Selected Topics in Probability and Statistics 
(3+0+0) 3 
MATH 586 
Selected Topics in Differential Geometry 
(3+0+0) 3 
MATH 587 
Selected Topics in Differential Equations 
(3+0+0) 3 
MATH 588 
Selected Topics in Applied Mathematics I 
(3+0+0) 3 
MATH 589 
Selected Topics in Combinatorics 
(3+0+0) 3 
MATH 590 
Readings in Mathematics 
(0+0+2) 1 
Literature survey and presentation on a subject to be determined by the instructor. 
MATH 601 
Measure Theory 
(4+0+0) 4 
Fundamentals of measure and integration theory, RadonNikodym Theorem, L_{p} spaces, modes of convergence, product measures and integration over locally compact topological spaces. 
MATH 611 
Differential Geometry I 
(4+0+0) 4 
Survey of differentiable manifolds, Lie groups and fibre bundles, theory of connections, holonomy groups, extension and reduction theorems, applications to linear and affine connections, curvature, torsion, geodesics, applications to Riemannian connections, metric normal coordinates, completeness, De Rham decomposition theorem, sectional curvature, spaces of constant curvature, equivalence problem for affine and Riemannian connection. 
MATH 612 
Differential Geometry II 
(4+0+0) 4 
Submanifolds, fundamental theorem for hypersurfaces, variations of the length integral, Jacobi fields, comparison theorem, Morse index theorem, almost complex and complex manifolds, Hermitian and Kaehlerian metrics, homogeneous spaces, symmetric spaces and symmetric Lie algebra, characteristic classes. 
MATH 623 
Integral Transforms 
(4+0+0) 4 
Fourier transforms, exponential, cosine and sine, Fourier transform in many variables, application of Fourier transform to solve boundary value problems, Laplace transform, use of residue theorem and contour integration for the inverse of Laplace transform, application of Laplace transform to solve differential and integral equations, FourierBessel and Hankel transforms for circular regions, Abel transform for dual integral equations. 
MATH 624 
Numerical Solutions of Partial Differential and Integral Equations

(4+0+0) 4 
Parabolic differential equations, explicit and implicit formulas, elliptic equations, hyperbolic systems, finite elements characteristics, Volterra and Fredholm integral equations. 
MATH 627 
Optimization Theory I 
(4+0+0) 4 
Fundamentals of linear and nonlinear optimization theory. Unconstrained optimization, constrained optimization, saddlepoint conditions, KuhnTucker conditions, postoptimality, duality, convexity, quadratic programming, multistage optimization. 
MATH 628 
Optimization Theory II 
(4+0+0) 4 
Design and analysis of algorithms for linear and nonlinear optimization. The revised simplex method, algorithms for network problems, dynamic programming, quadratic programming techniques, methods for constrained nonlinear problems. 
MATH 631 
Algebraic Topology I 
(4+0+0) 4 
Basic notions on categories and functions, the fundamental groups, homotopy, covering spaces, the universal covering space, covering transformations, simplicial complexes and homology of simplicial complexes. 
MATH 632 
Algebraic Topology II 
(4+0+0) 4 
Singular homology, exact sequences, the MayerVietoris exact sequence, the Lefschetz fixedpoint theorem, cohomology, cup and cap products, duality theorems, the Hurewicz theorem, higher homotopy groups. 
MATH 635 
An Introduction to Nonlinear Analysis 
(3+0+0) 3 
Calculus in Banach spaces. Implicit function theorems. Degree theories. Fixed Point Theorems. Bifurcation theory. Morse Lemma. Variational methods. Critical points of functionals. PalaisSmale condition. Mountain Pass Theorem. 
Prerequisite: 
MATH 535 or equivalent 
MATH 643 
Stochastic Processes I 
(4+0+0) 4 
Survey of measure and integration theory, measurable functions and random variables, expectation of random variables, convergence concepts, conditional expectation, stochastic processes with emphasis on Wiener processes, Markov processes and martingales, spectral representation of secondorder processes, linear prediction and filtering, Ito and Saratonovich integrals, Ito calculus, stochastic differential equations, diffusion processes, Gaussian measures, recursive estimation. 
Prerequisite: 
MATH 552 or consent of instructor. 
MATH 644 
Stochastic Processes II 
(4+0+0) 4 
Tightness, Prohorov's theorem, existence of Brownian motion, Martingale characterization of Brownian motion, Girsanov's theorem, FeynmannKac formulas, Martingale problem of Stroock and Varadhan, applications to mathematics of finance. 
MATH 645 
Mathematical Statistics 
(4+0+0) 4 
Review of essentials of probability theory, subjective probability and utility theory, statistical decision problems, a comparison game theory and decision theory, main theorems of decision theory with emphasis on Bayes and minimax decision rules, distribution and sufficient statistics, invariant statistical decision problem, testing hypotheses, the NeymanPearson lemma, sequential decision problem. 
Prerequisite: 
MATH 552 or consent of instructor. 
MATH 660 
Advanced Number Theory 
(4+0+0) 4 
Basic algebraic number theory; number fields, ramification theory, class groups, Dirichlet unit theorem; zeta and Lfunctions; Riemann, Dedekind zeta functions, Dirichlet, Hecke Lfunctions, primes in arithmetic progressions, prime number theorem; cyclotomic fields, reciprocity laws, class field theory, ideles and adeles, modular functions and modular forms. 
MATH 680 
Seminar in Pure Mathematics I 
(4+0+0) 4 
Recent developments in pure mathematics. 
MATH 681 
Seminar in Pure Mathematics II 
(4+0+0) 4 
Recent developments in pure mathematics. 
MATH 682 
Seminar in Applied Mathematics I 
(4+0+0) 4 
Recent developments in applied mathematics. 
MATH 683 
Seminar in Applied Mathematics II 
(4+0+0) 4 
Recent developments in applied mathematics. 
MATH 699 
Guided Research 
(2+4+0) 4 
Research in the field of Mathematics, by arrangement with members of the faculty; guidance of doctoral students towards the preparation and presentation of a research proposal. 
