## Course Description

BackMATH 511 | Real Analysis (Gerçel Analiz) | (3+0+0)3 |

Sets, countable sets, topological concepts of the set R, continuous functions, metric spaces, Lebesgue integration, Lp spaces, Riesz-Fischer theorem and Hilbert space. Normed linear spaces, Minkowski inequality, completeness theorem, L∞ space, Egoroff's theorem, Radon-Nykodym theorem, Carathéodory and Hahn Banach theorem. Lebesgue and Lebesgue-Stieltjes measure, Riesz lemma, Fubini and Tonelli theorems. | ||

MATH 512 | Complex Analysis (Karmaşık Analiz) | (3+0+0)3 |

The complex number system, metric spaces and topology of C, elementary properties and examples of analytic functions, complex integration, maximum modulus theorem, Cauchy's integral formula, properties of path integrals, conformal mapping. Schwarz-Christoffel transformation. | ||

MATH 513 | Functional Analysis (Fonksiyonel Analiz ) | (3+0+0)3 |

Linear spaces, norms, completeness. Linear mappings, continuity. Hahn-Banach theorem, normed linear spaces, Hilbert spaces. Dual spaces. Bounded linear functionals, weak convergence, uniform boundedness, weak and weak* topologies, Stone-Weierstrass theorem. Bounded linear operators: boundedness and continuity, weak and strong convergence. Open-mapping theorem. Closed graph theorem. | ||

MATH 516 | Probability (Olasılık) | (3+0+0)3 |

Monotone functions, distribution functions, absolutely continuous and singular distributions. Measure theory, classes of sets, probability measures and their distribution functions. Random variables, expected value, variance, properties of mathematical expectation. Independence, type of convergence, almost sure convergence. Borel-Cantelli lemma, vague convergence, uniform integrability, convergence of moments, law of large numbers, random series, weak law of large numbers, strong law of large numbers. Characteristic function, convolution, uniqueness and inversion, convergence theorems, central limit theorem, basic properties of conditional expectation, conditional independence and Markov property. | ||

MATH 521 | Ordinary Differential Equations (Adi Diferansiyel Denklemler) | (3+0+0)3 |

Ordinary differential equations, linear systems, nonlinear systems, existence and uniqueness theorems, continuous dependence on parameters, stability. Boundary value problems, periodic solutions. Operators on Banach spaces, contraction of mappings. | ||

MATH 522 | Partial Differential Equations I (Kısmi Diferansiyel Denklemler I) | (3+0+0)3 |

Classification of first order partial differential equations. Linear and nonlinear equations. Geometrical meaning. Monge cones and Monge equation. Characteristics. Cauchy problem. Caustics. Envelopes. Hamilton-Jacobi theory. Elements of symplectic geometry. Some equations of the second order. | ||

MATH 523 | Partial Differential Equations II (Kısmi Diferansiyel Denklemler II) | (3+0+0)3 |

Types of second order partial differential equations. Reduction to canonical form: hyperbolic, parabolic and elliptic cases. Equations of the hyperbolic type. D'Alambert formulae. Correctness. The Riemann method and Riemann function. Three dimensional wave equation. The Poisson formula. Correctness of the classical problems. Non-homogenous equations. Cylindrical waves. Point source. The Fourier method. Elliptic equations and classical problems. Harmonic functions and properties. Mean value theorem for a ball. Kelvin theorem. Uniqueness theorems for Dirichlet and Neumann problems. Dirichlet problem for a ball. Green function for Dirichlet and Neumann problem. Spherical functions and their applications. | ||

MATH 527 | Numerical Analysis (Sayısal Analiz) | (3+0+0)3 |

Polynomial approximation, Lagrange interpolation, least squares polynomial approximation, spline approximation and interpolation, the fast Fourier transform. Numerical quadrature, Richardson extrapolation, Romberg integration, Gaussian quadrature, adaptive quadrature, Monte Carlo methods for higher dimensional integrals. Direct methods of numerical linear algebra; triangular systems, Gaussian elimination and LU decomposition, pivoting, backward error analysis. Numerical solution of nonlinear systems and optimization; one-point iteration, Newton's method, unconstrained minimization, conjugate gradients. | ||

MATH 528 | Numerical Solution of Partial Differential Equations (Kısmi Diferansiyel Denklemlerin Sayısal Çözümü) | (3+0+0)3 |

Finite difference method for parabolic, elliptic and hyperbolic partial differential equations. Constructing the finite difference scheme for the model problems. Convergence, consistency and stability analysis of the numerical scheme. An introduction to the spectral methods: Fourier collocation and Fourier Galerkin methods. | ||

MATH 541 | Algebra (Cebir) | (3+0+0)3 |

Groups; isomorphism theorems, group action, simplicity of alternating groups, solvability of p-groups, Sylow theorems, Jordan-Hölder theorem, nilpotent and solvable groups. Rings; ring homomorphisms, Euclidean domains, PIDs, unique factorization, Gauss lemma, irreducibility criteria. | ||

MATH 551 | Nonlinear Continuum Mechanics I (Doğrusal Olmayan Sürekli Ortamlar Mekaniği I) | (3+0+0)3 |

Mathematical foundations of continuum mechanics, vectors and tensors, kinematics of deformation, conservation laws. | ||

MATH 552 | Nonlinear Continuum Mechanics II (Doğrusal Olmayan Sürekli Ortamlar Mekaniği II) | (3+0+0)3 |

Thermodynamics, constitutive equations of elastic, viscous, and viscoelastic materials, electromagnetic solids. | ||

MATH 554 | Perturbation Methods (Tedirgeme Yöntemleri) | (3+0+0)3 |

Matched asymptotic expansions, multiple scales, WKB and homogenization. Applications in ODEs, PDEs, difference equations, and integral equations: boundary or shock layers, nonlinear wave propagation, bifurcation and stability, and resonance. | ||

MATH 561 | Topology (Topoloji) | (3+0+0)3 |

Topological structures, open and closed sets, neighborhoods, product, order and subspace topologies, metric topology, accumulation and limit points, convergence, continuous mappings, connectedness and paths, compactness and local compactness, embeddings, separation axioms, normal spaces. Urysohn, Tychonoff and Stone-Čech theorems. | ||

MATH 564 | Differential Geometry (Diferansiyel Geometri) | (3+0+0)3 |

Differentiable manifolds, tangent and cotangent spaces, vector fields, Lie bracket, diffeomorphism, the inverse function theorem, submanifolds, hypersurfaces, standart connection of Euclidean spaces, Weingarten and Gauss maps, tensors and differential forms, Lie derivative, Riemannian connection, Riemannian manifolds, Riemannian curvature tensor. | ||

MATH 571 | Mathematical Methods in Physics and Engineering (Fizik ve Mühendislikte Matematiksel Yöntemler) | (3+0+0)3 |

Linear operators on finite dimensional vector spaces, canonical forms and functions of matrices, multilinear functions on vector spaces, tensor analysis in R3 and its applications to theory of elasticity, calculus of variations, quasi-linear partial differential equations, separation of variables, well-posed problems. Analytic functions, contour integration, conformal mapping, Banach and Hilbert spaces, expansions in orthogonal functions, classical orthogonal polynomials, integral transforms, applications to partial differential equations, Green's functions, distributions. (For students with a Bachelor of Science degree other than mathematics). | ||

MATH 581-589 | Special Topics in Mathematics I-IX (Matematikte Özel Konular I-IX) | (3+0+0)3 |

Study of special topics chosen among the recent technological or theoretical developments in mathematics. | ||

MATH 611 | Harmonic Analysis I (Harmonik Analiz I) | (3+0+0)3 |

L^p and weak-L^p spaces, convolution and approximate identities, interpolation, maximal functions. Schwartz class and the Fourier transform, classes of tempered distributions, convolution operators on L^p spaces and multipliers. Fourier coefficients, decay property of Fourier coefficients, pointwise convergence of Fourier series. The conjugate function and convergence in norm, singular integrals, the Hilbert transform and the Riesz transform, homogeneous singular integrals and the method of rotations, the Calderon–Zygmund decomposition and singular integrals, sufficient conditions for L^p boundedness. | ||

MATH 612 | Harmonic Analysis II (Harmonik Analiz II) | (3+0+0)3 |

Littlewood-Paley theory, multiplier theorems, applications of Littlewood-Paley theory, the Haar system, conditional expectation, martingales, Riesz potentials, Bessel potentials and fractional integrals, Sobolev spaces, Lipschitz spaces, Hardy spaces, singular integral on function spaces, functions of bounded mean oscilation (BMO), duality between H^1 and BMO, nontangential maximal functions, Carleson measures and the sharp maximal function. | ||

MATH 613 | Conformal Mappings (Açı Koruyan Dönüşümler) | (3+0+0)3 |

Harmonic functions, Green's formula, analytic functions, conformal mappings of simply connected domains, Riemann mapping theorem, elliptic functions, conformal mappings of multiple-connected domains, geometrical and analytical approach to conformal mappings. Quasi-conformal mappings and Teichmüller spaces. | ||

MATH 614 | Advanced Functional Analysis ( İleri Fonksiyonel Analiz) | (3+0+0)3 |

Banach algebras, elementary spectral theory in Banach spaces. Commutative Banach algebras and Gelfand theory. Integral operators. Compact operators and spectral theory. Examples of compact operators, positive compact operators. Compact symmetric operators in Hilbert spaces. Spectral theory of symmetric, normal, unitary and self-adjoint operators. | ||

MATH 615 | Functional Analysis and Applications (Fonksiyonel Analiz ve Uygulamaları) | (3+0+0)3 |

Normed spaces. Linear operators, the contraction mapping. Fixed point theorems, spectral theory. Sturm-Liouville systems. Variational methods, applications to differential equations. Linear and nonlinear elliptic partial differential equations. | ||

MATH 617 | Theory of Stochastic Processes I (Stokastik Süreçler Kuramı I) | (3+0+0)3 |

Brownian motion, its definition and basic properties, martingale, Doob's inequality, stopping times, the optional stopping theorem, convergence and regularity, some martingale applications, Markov properties of Brownian motion, Poisson process, path properties of Brownian motion, continuous semimartingales, square integrable martingales, quadratic variation, Doob-Meyer decomposition, stochastic integrals and extensions, Ito formula, Levy theorem, time change of martingales, martingale representation, Burkholder-Davis-Gundy inequalities and Stratonovich integrals. | ||

MATH 618 | Theory of Stochastic Processes II (Stokastik Süreçler Kuramı II) | (3+0+0)3 |

Ito's formula, the Girsanov theorem, Markov processes, transition properties of Markov processes, examples of Markov processes, canonical process and shift operator, extension of filtration, strong Markov property, transience and recurrence. Additive functions, continuity, harmonic functions, theory of general processes, predictable and optional processes, hitting times, processes with jumps, martingale decomposition, stochastic integrals. Ito's formula for jump processes, reduction theorem, semimartingales, the Girsanov theorem for jump processes. Stochastic differential equations, pathwise solutions and one-dimensional stochastic differential equations. | ||

MATH 619 | Advanced Differential Geometry (İleri Diferansiyel Geometri) | (3+0+0)3 |

Tensor fields, differential forms and exterior derivative. Connections. Riemannian metric, Riemannian manifold, covariant derivative, parallel translation, geodesics, exponential mapping and normal coordinates. Curvature tensors, sectional curvature, Ricci curvature and scalar curvature. Space forms. Conformal changes of Riemannian metric. Riemannian submanifolds, induced connection, second fundamental form. Equations of Gauss, Codazzi and Ricci. Cartan structure equations. | ||

MATH 653 | Nonlinear Elasticity (Doğrusal Olmayan Esneklik) | (3+0+0)3 |

Review of governing equations, linearization and stability, constitutive inequalities, large elastic deformations, exact solution of special problems, controllable deformations of incompressible materials, initial stress problems, elastic stability, nonlinear string and rod theories, membrane theory, fiber-reinforced materials, second-order elasticity, phase transformations and crystal defects. | ||

MATH 655 | Direct and Inverse Scattering of Waves (Dalgaların Doğrudan ve Ters Saçılması) | (3+0+0)3 |

Vector and scalar waves. Electromagnetic waves. Wave equation. Helmholtz equation. Method of stationary phase. Geometrical optics approximation. Elements of diffraction. Huygens-Frenel principle. Riesz-Fredholm theory for scattering. Potential theory. Weak singular integral operators. Boundary value problems for Helmholtz equation. Boundary value problems for Maxwell equations and vector Helmholtz equation. | ||

MATH 656 | Nonlinear Waves (Doğrusal Olmayan Dalgalar) | (3+0+0)3 |

Euler's equations. Dispersion, dissipation and nonlinearity. Korteweg-de Vries equation: derivation, solitary wave solutions and conserved quantities. Nonlinear Schrödinger equation (derivation, solitary wave solutions and conserved quantities). | ||

MATH 680 | Guided Research in Mathematics (Matematikte Yönlendirilmiş Araştırmalar) | (3+0+0)3 |

Guided research in a selected area of mathematics. Guidance of a doctoral student by a faculty member towards the preparation and presentation of a research proposal. | ||

MATH 681-689 | Special Studies in Mathematics I-IX (Matematikte Özel Çalışmalar I-IX) | (3+0+0)3 |

Study of current research topics in mathematics by Ph.D. students under the guidance of a faculty member and presentation of the chosen topic. | ||

MATH 690 | Ph.D. Thesis (Doktora Tezi) | Non-Credit |

Preparation of a Ph.D. thesis under the guidance of an academic advisor. |