Program tanımları
Matematik Yüksek Lisans Programı
Compulsory courses to be taken for MSc. degree:
MAT 500 (0-2-0) M.Sc. Seminar
MAT 550 (3-0-3) Algebra I
MAT 551 (3-0-3) Topology I
MAT 560 (3-0-3) Real Analysis I
All courses for MSc. Program:
MAT 500 (0-2-0) M.Sc. Seminar I
MAT 535 (3-0-3) Nonlinear Functional Analysis and Applications I II
MAT 536 (3-0-3) Nonlinear Functional Analysis and Applications III
MAT 550 (3-0-3) Algebra I
MAT 551 (3-0-3) Topology I
MAT 552 (3-0-3) Topology II
MAT 560 (3-0-3) Real Analysis I
MAT 561 (3-0-3) Geometry I
MAT 562 (3-0-3) Geometry II
MAT 563 (3-0-3) Curves and Surfaces I
MAT 564 (3-0-3) Curves and Surfaces II
MAT 571 (3-0-3) Measure and Integration Theory I
MAT 572 (3-0-3) Measure and Integration Theory II
MAT 575 (3-0-3) Complex Analysis I
MAT 576 (3-0-3) Complex Analysis II
MAT 581 (3-0-3) Functional Analysis I
MAT 582 (3-0-3) Functional Analysis II
MAT 584 (3-0-3) Applied Functional Analysis
MAT 591 (3-0-3) Ordinary Differential Equations
MAT 592 (3-0-3) Partial Differential Equations
MAT 593 (3-0-3) Second Order Partial Differential Equations
MAT 595 (3-0-3) Computational Geometry I
MAT 596 (3-0-3) Computational Geometry II
Course Descriptions
MAT 500 M.Sc. Seminar
A one hour seminar in which student presents his M.Sc thesis and answers questions on the topic of the thesis.
MAT 535 Nonlinear Functional Analysis and Applications I
Fixed points methods. Nonexpansive mappings. Differential and integral calculus in Banach spaces. Implicit and inverse function theorems. Potential operators and variational methods for linear and nonlinear operator equations. Extrema of functionals. Monotone operators and monotonicity methods for nonlinear operator equations. Applications to differential and integral equations and physical problems.
MAT 536 Nonlinear Functional Analysis and Applications II
Topological methods in nonlinear analysis. Leray-Schauder degree. Topological degree theory in Banach space. Critical points and introduction to Morse theory. a –set contractions. Maximal monotone relations and applications to nonlinear operator equations. Variational inequalities. Biforcation theory. Applications to partial differential equations, nonlinear intgral equations and Hamiltonian systems.
MAT 550 Algebra I
Free groups, Principal Ideal Domains, Unique Factorization Rings, Euclidean Rings, Basics about Modules, Modules over Principal Ideal Domains and Dedekind domains, Jacobson radical of a ring, Classical structures of Rings, Simple and Semi-simple rings, Prime and Semiprime rings, Subdirectly irreducible rings.
MAT 551 Topology I
Fundamental Properties; Subspaces; Product Spaces; Quotient Spaces, Special Results for Metric Spaces, Separation Axioms, Countability Properties, convergence of filters and nets.
MAT 552 Topology II
Connectedness and Arcwise Connectedness, Compactness compactification, Local Compactness, Metrizability, Function Spaces, Topologies on Function spaces.
MAT 560 Real Analysis I
s -algebras, measures, measurable functions. Convergence in measure. Riesz and Egorof Theorems. The abstract Lebesgue integral. Monotone and dominated convergence theorems. Hölder and Minkowski inequalities. The Banach space Lp. Outer measures and Caratheodory’s definition of measurability. Hahn extension theorem. Construction of the Lebesgue measure on the real line. Signed and complex measures, Hahn decomposition and total variation. Absolute continuity and the Radon-Nikodym theorem. Singular measures. Lebesgue decomposition. Differentiation. Jordan decomposition of functions of bounded variation. Absolutely continuous functions. The relationship between absolute continuity for functions and for measures on R.
MAT 561 Geometry I
Differentiable manifolds, Tangent space, imersions and embeddings, Other examles of manifolds, Orientation, Riemann metrics, Affine connections, Riemannian connections, The geosdesic flow, Minimizing properties of geodesics, Convex neighborhoods.
MAT 562 Geometry II
Curvature, Sectional curvature, Ricci curvature and skalar curvature, Tensors on Riemannian manifolds, The Jacobi equation, Conjugate points, The second fundamental form, The fundamental equations.
MAT 563 Curves and Surfaces I
Polygon Meshes, Parametric Curves, Bernstein Bases and Bezier Curves, Splines, B-Splines, Rational Splines.
MAT 564 Curves and Surfaces II
Parametric Surfaces, Tensorproduct Surfaces, Splines-Surfaces Interpolation, Spline-Surfaces Approximation, Bezier-Surfaces, Rational Splines-Surfaces, NURBS, Implicit Surfaces, Solid Modelling, Algebraic Surfaces, Deformations, Patches
MAT 571 Measure and Integration Theory I
Measurable Spaces; Measurable Functions; Concept of the Lebesque Measure on Rn and their Fundamental Properties; The Lebesgue Integral; The Stieltjes Integral.
MAT 572 Measure and Integration Theory II
Classes; Set Functions; Abstract Measure Spaces and their Products; Integration on the Product Spaces; The Convergence Theorems.
MAT 575 Complex Analysis I
Complex Number system,Topology of Complex plane,Pover series, Analytic functions, Möbiues transformations,Complex integration,Cauchy's theorems, Cauchy integral formula,Local properties of analytic functions,Zeros and singularities, Open mapping Theorem, Maximum modules principal, Schwarz Lemma, Residue theorem,Argument principal,Rouchet theorem, Normal families, Arzela-Ascoli theorem, Montel's theorem, Riemann mapping theorem.
MAT 576 Complex Analysis II
Weierstrass factorization theorem, Runge’s theorem, Mittag-Leffler’s theorem, Schwarz’s reflection principle, Analytic continuation, Monodromy theorem, Sheaf of germs of analytic functions, Covering spaces, Harmonic functions, Subharmonic and superharmonic functions, The Dirichlet problem, Green’s function, Jensen’s formula, Hadamard factorization theorem, Litte Picard theorem, Schottkyt’s theorem, Great Picard theorem.
MAT 581 Functional Analysis I
Normed linear spaces. Bounded and unbounded linear operators. Convex sets. Linear functionals. Duality. Hilbert spaces. Open mapping theorem.Closed operators. Closed graph theorem. Uniform boundedness principle. Bilinear forms and the Lax-Milgram lemma. Adjoints and closed range theorem. Sectrum, spectral representation and operator calculus for bounded operators. The spectral radius formula. Compact operators. Self-adjoint operators in Hilbert saces.
MAT 582 Functional Analysis II
Normed algebras. Algebra of bounded operators. Banach algebras. Invariant sub-spaces, Gelfand-Mazur theorem. Positive operators. Polar decomositions. Spectral theory for normal operators. Fredholm and semi-Fredholm operators. Separation theorems. The Krein-Milman theorem. Reflexive spaces. Uniformly convex sapaces. Weak topology and weak convergence, Alaoðlu-Mazur theorem. Sobolev spaces and the Sobolev Imbedding Theorems.
MAT 584 Applied Functional Analysis
Bases in in Hilbert spaces, Fourier series, the contraction mapping principle ,the space of linear bounded operators, Banch - Steinhaus theorem, theorems on existence of inverse operators, Hahn - Banach theorem, Riesz's representation theorem, selfadjoint operators, spectr of compact operators. Riesz - Schauder theory, weak convergence and weak compactness, properties of weak convergent sequences. Poincare and Friedrichs inequalities, Rellich's embedding theorem, weak solution of Dirichlet's problem for Poisson equation, the Cauchy problem for differential - operator equations in Hilbert space and applications
MAT 591 Ordinary Differential Equations
Existence and uniqueness theorems, Continuity of solutions with respect to parameters, Basic inequalities and comparison theory, Differential and integral inequalities, Fixed-point methods, Properties of linear homogeneous systems, Periodic coefficients, Asymtotic behaviour, Second order differential equations, Boundedness of solutions, Oscillatiory equations,
MAT 592 Partial Differential Equations
Definition of the stability, Stability of linear systems, Stability of weakly nonlinear systems, Two-dimensional systems, Stability by Lyapunov’s second method, Autonomous systems, Nonautonomous systems, Perturbation theorems, Periodic solutions, Poincare-Bendixson theory.
MAT 593 Second order Partial Differential Equations
Two - point boundary value problems for one dimensional, Poisson equation, The heat equation The wave equation, Poisson equation,
MAT 595 Computational Geometry I
Analysis of Algorithms, Data Structures, Geometric Data Structures, Line Segment Intersection, Polygon Triangulation, Linear Programming, Othogonal Range Searching, Point Location, Voronoi Diagrams,
MAT 596 Computational Geometry II
Arrangements and Duality, Delaunay Triangulations, Convex Hulls, Painter’s Algorithm, Motion Planning, Quadtrees , Visibility Graphs, Simplex Range Searching
