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Matematik Doktora Programı

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Hakkında yorumlar Matematik Doktora Programı - Kurumda - Çankaya - Ankara

  • Program tanımları
    PhD. in Mathematics

    Courses for PhD. Program

    Compulsory courses to be taken for MSc. degree:
    MAT 600 (0-2-0) Ph.D. Seminar
    MAT 650 (3-0-3) Algebra II
    MAT 660 (3-0-3) Real Analysis II

    All courses for MSc. Program:
    MAT 600 (0-2-0) Ph.D Seminar
    MAT 601 (3-0-3) Fuzzy Sets
    MAT 602 (3-0-3) Fuzzy Logic
    MAT 603 (3-0-3) Symbolic Logic I
    MAT 604 (3-0-3) Symbolic Logic II
    MAT 605 (3-0-3) Set Theory
    MAT 607 (3-0-3) Category Theory I
    MAT 608 (3-0-3) Category Theory II
    MAT 631 (3-0-3) Fourier Analysis I
    MAT 632 (3-0-3) Fourier Analysis II
    MAT 633 (3-0-3) Mathematical Methods in Physic I
    MAT 634 (3-0-3) Mathematical Methods in Physic II
    MAT 635 (3-0-3) Theory of Banach Spaces
    MAT 636 (3-0-3) Optimization in Banach Spaces
    MAT 641 (3-0-3) Field Theory
    MAT 642 (3-0-3) Algebraic Numbers
    MAT 643 (3-0-3) Rings and Module Theory
    MAT 644 (3-0-3) Group Rings
    MAT 645 (3-0-3) Commutative Algebra I
    MAT 646 (3-0-3) Commutavite Algebra II
    MAT 647 (3-0-3) New Topics in Algebra
    MAT 648 (3-0-3) New Topics in Commutative Algebra
    MAT 650 (3-0-3) Algebra II
    MAT 651 (3-0-3) Algebraic Topology I
    MAT 652 (3-0-3) Algebraic Topology II
    MAT 655 (3-0-3) Bitopological Spaces I
    MAT 656 (3-0-3) Bitopological Spaces II
    MAT 657 (3-0-3) Advanced Topology I
    MAT 658 (3-0-3) Advanced Topology II
    MAT 660 (3-0-3) Real Analysis II
    MAT 661 (3-0-3) Manifold Theory I
    MAT 662 (3-0-3) Manifold Theory II
    MAT 663 (3-0-3) Semi-Riemannian Geometry I
    MAT 664 (3-0-3) Semi-Riemannian Geometry II
    MAT 665 (3-0-3) Riemannian Geometry I
    MAT 666 (3-0-3) Riemannian Geometry II
    MAT 667 (3-0-3) Lorentz Geometry I
    MAT 668 (3-0-3) Lorentz Geometry II
    MAT 669 (3-0-3) Tensor Geometry I
    MAT 670 (3-0-3) Tensor Geometry II
    MAT 672 (3-0-3) Riemann Surfaces
    MAT 673 (3-0-3) Advanced Measure Theory I
    MAT 674 (3-0-3) Advanced Measure Theory II
    MAT 675 (3-0-3) Analytic Fuphd_phd_coursescoursesnctions
    MAT 679 (3-0-3) Perturbation Theory
    MAT 680 (3-0-3) Operator Theory
    MAT 681 (3-0-3) Topological Vector Spaces I
    MAT 682 (3-0-3) Topological Vector Spaces II
    MAT 687 (3-0-3) Harmonic Analysis
    MAT 695 (3-0-3) Ordinary Differential Equations I
    MAT 696 (3-0-3) Ordinary Differential Equations II
    MAT 697 (3-0-3) Functional Analytic Methods in PDE's
    MAT 698 (3-0-3) Long-time behavior of Solutions to Nonlinear Evolutionary PDE's
    MAT 701 (3-0-3) New Topics in Geometry
    MAT 702 (3-0-3) New Topics in Differential Geometry
    MAT 705 (3-0-3) Advanced Set Theory
    MAT 706 (3-0-3) Continuous Lattices
    MAT 711 (3-0-3) Attractors of the Dissipative Evolutionary Equations
    MAT 731 (3-0-3) Topological Groups
    MAT 732 (3-0-3) Banach Algebras
    MAT 741 (3-0-3) Semigoup Theory I
    MAT 742 (3-0-3) p-adic Analysis
    MAT 745 (3-0-3) Homological Algebra I
    MAT 746 (3-0-3) Homological Algebra II
    MAT 747 (3-0-3) Localization I
    MAT 748 (3-0-3) Localization II
    MAT 755 (3-0-3) Algebraic Geometry I
    MAT 756 (3-0-3) Algebraic Geometry II
    MAT 757 (3-0-3) New Topics in Bitopology I
    MAT 758 (3-0-3) New Topics in Bitopology II
    MAT 761 (3-0-3) Lie Groups I I
    MAT 762 (3-0-3) Lie Groups II
    MAT 763 (3-0-3) Minimal Surfaces I
    MAT 764 (3-0-3) Minimal Surfaces II
    MAT 765 (3-0-3) Kinematics I
    MAT 766 (3-0-3) Kinematics II
    MAT 767 (3-0-3) Minkowski Geometry I
    MAT 768 (3-0-3) Minkowski Geometry II
    MAT 769 (3-0-3) C *-Algebras I
    MAT 770 (3-0-3) C *-Algebras II
    MAT 777 (3-0-3) New Topics in Real Analysis
    MAT 778 (3-0-3) New Topics in Functional Analysis
    MAT 781 (3-0-3) Convex Analysis I
    MAT 782 (3-0-3) Convex Analysis II
    MAT 786 (3-0-3) New Topics in the Theory of Banach Spaces
    MAT 791 (3-0-3) Differential Operators II
    MAT 792 (3-0-3) Differential Operators II
    MAT 797 (3-0-3) New Topics in Ordinary Differential Equations
    MAT 798 (3-0-3) New Topics in Partial Differential Equations


    Course Descriptions
     
    MAT 600 Ph.D Seminar (0 2 0)
    Each Ph.D. student checks the literature, works on a subject and presents this work before all academic staff.
     
    MAT 601 Fuzzy Sets (303)
    Standart Theory of Fuzzy Sets, Special Topics and Applications to Computing and other Branches of Science.
     
    MAT 602 Fuzzy Logic (303)
    Representation of Inexact or Incomplete knowledge using Fuzzy Sets and its Manipulation using Multivalued or Fuzzy Logic Systems; Applications to Control Systems; Expert Systems etc., Computer Implementations.

    MAT 603 Symbolic Logic I (303)
    Propositional Calculus-wff, Normal Form, Completeness, Consequences of a Set of wff; First Order Predicate Calculus - Structures.swff and Models, Proofs, Duality. Gödel Theorems.

    MAT 604 Symbolic Logic II (303)
    Extended Completeness Theorem, Algebraic Structures, High Order Predicate Calculus.

    MAT 605 Set Theory (303)
    Zermelo - Fraenkel Axioms, Development of the Theory of Sets, Ordinal Numbers, Cardinal Numbers, Axiom of Choice, Zorn's Lemma, Well Ordering Principle.

    MAT 607 Category Theory I (303)
    Sets, Classes and Conglmerates, Concrete Categories, Abstract Categories, Formation of Categories, Special Morphisma and Special Objects, Functors, Natural Transformations, Limits in Categories, Complete Categories.

    MAT 608 Category Theory II (303)
    Universal Maps, Adjoint Functors, HOM-Functors, Representable Functors, Free Objects, Algebraic Categories and Functors, Subobjects, Quotient Objects and Factorizations, Reflective Subcategories, Pointed categories.

    MAT 631 Fourier Analysis I (303)
    Fourier Series, Fourier Integrals.

    MAT 632 Fourier Analysis II (303)
    Convergence of Fourier Series and Summability Theorems, Theorems Relating to the Coefficients, Uniqueness Properties.

    MAT 633 Mathematical Methods in Physics I (303)
    Existence Theorems, Linear and Non-linear Differential Equations, Regular and Singular Boundary Vaule Problems, Characteristic, Liapunov Method, Fredholm Theory, Hilbert-Schmitd Theorem, Singular Integral Eguations, Wiener-Hopf Equation, Potential Theory.

    MAT 634 Mathematical Methods in Physics II (303)
    Cauchy Problems for Partial Differential Equations, Classification of Second Order Linear Partial Differential Equations, Solution of Elipric, Parabolic and Hyperbolic Equations, Existence, Fourier and Laplace Transformations, Potential, Green's Functions, Sobolev Spaces, Schwarts Distribution.

    MAT 635 Theory of Banach Spaces (303)
    Linear topological spaces. Conjugate spaces. Cones. Wedges. Order relations. Krein-Smulyan Theorem, ew*-Topology. Completeness. Weakly sequentially / countably compactness. Quasi-completeness. Completely continuous linear operators. Reflexivity. Weakly compactly generated spaces. Unconditional convergence. Tensor products of LCS. Schauder/unconditional bases. Extreme points. Fixed point theorems. Banach Stone Theorem. Jerinson Theorem. Normed lattices. Stone-Weierstrass Theorem. Monotone projections. Dunford-Pettis Theorem. Isometry. Fenchel duality of convex functions. Characterizations of inner- product spaces.

    MAT 636 Optimization in Banach Spaces (303)
    Convexity in topological vector spaces. Duality in normed spaces. Bochner integral, Vector measures, and distributions on real intervals. Maximal monotone operators and evolution systems in Banach spaces. Regularization of convex functions. Perturbation of cyclically monotone operators and subdifferential calculus. Minimax theorem of concave-convex functions. The maximum principle and optimality theorem.

    MAT 641 Field Theory (303)
    Field Extensions; p-adic Field, p-adic Completions; Transcendence Bases, Separability.

    MAT 642 Algebraic Numbers (303)
    Basic and Discriminants of the Number Fields; Quadratic, Cubic and Algebraic Extensions; Prime Factorization and Ramification; Units, Dirichlet Theorem, The Geometry of Numbers; Class Number, Zeta Function and Density Theorems.
     
    MAT 643 Rings and Module Theory (303)
    Over Noncommutative Rings: Category Theory; Direct Sums of Modules; Chains; Classic Ring Theory; Functors; Duality; Injective and Projective Modules; Perfect Rings and Semi-perfect Rings.

    MAT 644 Group Rings (303)
    Idempotents in Group Rings; Super Solvable Groups; Units in Group Rings; Automorphisms; Lie Conditions.
     
    MAT 645 Commutative Algebra I (303)
    Over Commutative Rings: Chain Conditions, Spectrum, Hilbert Nullstellensatz; Artin-Ress Lemma; Dimensions; Koszul Complex; Cohen-Macaulay Rings; Regular Rings.

    MAT 646 Commutative Algebra II (303)
    Over Commutative Rings: Ideals, Factorizations; Jacobson's Radical, Direct Factorization; Modules, Projective, Injective and Flat Modules; Chain Conditions, Tensor Products; Morita Context.

    MAT 647 New Topics in Algebra (303)
    New topics in the area of Algebra.

    MAT 648 New Topics in Commutative Algebra (303)
    New topics in the area of commutative algebra

    MAT 650 Algebra II (303)
    Artinian and Noetherian Modules and Rings, Decompositions of Modules and Rings, Radicals, Wedderburn-Artinian Theorem; Free, Projective and Injective Modules, Tensor Products, Projective and Injective dimensions, Functors, Hom-Ext Functors and applications to groups.

    MAT 651 Algebraic Topology I (303)
    Homology Classes; Fundamental Group; Singular Homology Theory; Singular and Simplecial Homology.

    MAT 655 Bitopological Spaces I (303)
    Seperation Properties, Dual Covers; Binormality; Local Finiteness Properties p-q Metrizability and Sequential Normality.

    MAT 656 Bitopological Spaces II (303)
    Compertmental Dual Families; Quasi-uniform Bitoplogical Spaces and Extensions; Para-quasi Uniformities.

    MAT 657 Advanced Topolgy I (303)
    Filters and Nets; Convergence; Compactness Real Compactness, Cardinality Functions.

    MAT 658 Advanced Topolgy II (303)
    Paracompactness and Generalized Metric Spaces; Uniform Structure and Proximities; Metrizability Theorems.

    MAT 660 Real Analysis II (303)
    Integration on locally compact spaces. Outer measures induced by positive functionals. Riesz representation theorem for bounded functionals on C0(X). Bounded fuunctionals on Lp , 1 £ p£ ¥ . Abstract Hilbert spaces. Complete orthonormal sets, Parceval's identity. A complete orthonormal set in L ([-p, p],l). Fourier transform and the Riemann-Lebesque lemma. Riesz-Fischer theorem. Product measures, Fubini's theorm. Convolution, and the group algebra L1(R). Fourier inversion theorem, Plancheral's theorem. Introduction to Banach spaces. Baire's theorem, Hahn-Banach extension theorem. Open mapping theorem and closed graph theorem. The principle of uniform boundedness.Weak and weak*topologies,Alaoðlu's theorem.

    MAT 661 Manifold Theory I (303)
    Smooth Manifolds, Smooth Mapings, Tangent Vectors, Differential Maps, Curves, Vector Fields, One Forms, Submanifolds, Some Special Manifolds, Integral Curves.
     
    MAT 662 Manifold Theory II (303)
    Jacobi fields, Tidal faces, Locally symmetric manifolds, Isometries of normal neighborhoods, Symmetric spaces, Simply connected space forms, Transvections.

    MAT 663 Semi-Riemannian Geometry I (303)
    Isometries, The Levi-Civita connection, Parallel Translation, Geodesics, The Exponential map, Curvature, Sectional curvature, Semi-Riemannian surfaces, Type changing and metric contraction, Frame fields, Some differential operator Ricci and scalar curvature, Semi-Riemannian product manifolds, Local isometries, Levels of structure.

    MAT 664 Semi-Riemannian Geometry I (303)
    Tangent and normals, The indcuced connection, Geodesics in submanifolds, Totally geodesic submanifolds, Semi-Riemannian Hypersurfaces, Hyperquadrics, The Codazzi equation, Totally umbilic hypersurfaces, the normal connection, A Conguruence theorem, Isometric immersions, Two parameter maps.

    MAT 665 Riemannian Geometry I (303)
    Complete manifolds, Hopf-Rinow theorem, the theorem of Hadamard, theorem of Cartan on the determination of the metric by means of the curvature, Hyperbolic space, Space forms, Ýsometries of the hyperbolic space, Theorem of Liouvile, Formulas for the first and second variations of energy, the theorems of Bonnet-Myers and of Synge-Weinstein.

    MAT 666 Riemannian Geometry II (303)
    The theorem of Rauch, Applications of the index lemma to imersions, Focal points and an extension of Rauchs theorem, The index theorem, Existence of closed geodesics, The cut locus, the estimate of the injectivity radius, The Sphore theorem, Some further developments.

    MAT 667 Lorentz Geometry I (303)
    The Gauss Lemma, Convex open sets, Arc length, Riemannian distance, Riemannian completeness, Lorentz Casual character, Time cones, Local Lorentz Geometry , Geodesics in hyperquadrics, Geodesics in surfaces, Completenes and extendibility.

    MAT 668 Lorentz Geometry II (303)
    Deck transformations, Orbit manifolds, Orientability, Semi-Riemannian coverings, Lorentz time orientability, Volume elements, Vector bundles, Local isometries, Matched coverings, Warped products, Warped product geodesics, Curvature of warped products, Semi-Riemannian submersions.

    MAT 669 Tensor Geometry I (303)
    Basic Algebra, Tensor fields, Interpretations, Tensors at a point, Tensor components, Contraction, Covariant tensors, Tensor derivations.

    MAT 670 Tensor Geometry II (303)
    First variation, Second variation, The index form, Conjugate points, Local minima and maxima, some global consequences, The end manifold case, Focal points along null geodesics, A casuality theorem.

    MAT 672 Riemann Surfaces (303)
    Manifolds, Riemann Surfaces of Analytic Functions; Covering Manifolds, Combinatoric Topology, Differentials and Integrals, Hilbert Space of Differentials, Uniformization, Compact Riemann Surfaces Riemann-Roch Theorem, Abel Theorem, Jacobi Inverse problem.

    MAT 673 Advanced Measure Theory I (303)
    Measure Spaces and their Productts; Theorems on Extension, Completion and Approximation; Inner and Outer Measures; The Lebesgue Measure;Theory of the Lebesgue Integral; The Lp-spaces (1£ p £ oo); Modes of Convergence.

    MAT 674 Advanced Measure Theory II (303)
    Decompositions of Measures; Signed and Complex Measures; Absolute Continuity; The Rodon-Nikodym Theorem, Differentation, The Dini Derivatives; Measures on Locally Compact Spaces; The Riesz Representation Theorem; Daniell and Radon Measures; Analytic Sets; Haar Measure.

    MAT 675 Analytic Functions (303)
    The Space of Analytic Functions, Weierstrass Factorization Theorem, Gamma Function, Riemann Zeta Function, Mittag-Leffler Theorem, Analytic Continuation; Monodromy Theorem, Seed and Sheaaf of Analytic Functions, Analytic Manifolds, Harmonic Functions, Entire Functions, Picards Theorem.

    MAT 679 Perturbation Methods (303)
    Asymptotic expansions: Taylor expansion. Asymptotic expansions for definite integrals. Uniform and non-uniform asymptotic expansions. Regular asymptotic expansions. Applications to ordinary and partial differential equations. Examples from Fluid Dynamics. Thin aerofoil theory for subsonic flows. Singular perturbations: Matched asymptotic expansion theory. Concept of the boundary layer. Inner and outer solutions. Overlap region. Matching of the asymptotic expansions. Method of multiple scales: Uniformly valid solutions, Amplitude equations, WKE Method.Turningpointproblems.

    MAT 680 Operator Theory (303)
    Operators on Banach spaces, adjoint of an operator, spectral properties of operators, the numerical range of a bounded linear operator, the numerical range and spectrum, the closure of the numerical range, the maximal range of operators on Hilbert spaces, extreme points of the numerical range, the numerical radius, the spectral radius and the norm of a bounded linear operator, Hermitian and normal operators on Banach spaces, classes of elements in Banach algebras with unit, Vidav-Palmer theorem, properties of Hermitian and normal elements in a Banach algebra, the numerical range of elements of locally m-convex algebras, spectral sets and dilations of operators with property s (T)= s (RET), conditions implying normality (Hermitianity, unitarity) of an operator, invariant subspaces, reducing invariant subspaces, some structure theorems analytic functions of operators, semigroups of operators.

    MAT 681 Topological Vector Spaces I (303)
    Bounded sets in TVS. Metrizability. Frechet space. Open and closed grah theorems. FH space. Completeness and compactness in TVS. Local convexity and duality in TVS. Polar topologies. Equicontinuous sets. Mackey-Arens theorem. Barrelled spaces. Separable spaces. Natural embedding. Semireflexivity. Banach-Mackey Theorem. Dual operators. The Hellinger-Toeplitz tehorem. Weakly compact operators in Banach spaces. Precompact convergence. Almost weak* closed sets. Strict hypercompleteness. Full completeness. Ptaks and N.J. Kaltons closed theorems. Inductive limits. Weak and convex compactness. Extreme points. Phillips lemma. L,M(H), G(B) and G spaces, Barrelled subspaces. Separable quotient problem. Inclusion theorem.

    MAT 682 Topological Vector Spaces II (303)
    Baralled spaces, The Banach-Steinhaus theorem, Inductive and Projective limits, Quotient spaces, Product spaces, Closed graph Theorem, Riezs Theorem.
     
    MAT 687 Harmonic Analysis (303)
    Fourier transforms on the line, Fourier-Stieltjes transforms, Fourier transforms on L(R), almost periodic functions on the line, the Paley_Wiener theorems, Fourier analysis on locally compact groups, the Heaar measure, the Haar integral, Fourier transforms on locally compact groups, the character group of a locally compact group, the duality theory.

    MAT 695 Numerical Solutions Differential Equations I (303)
    Introduction, 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 (Floquet theory), Asymtotic behaviour, Second order differential equations, Boundedness of solutions, Oscillatiory equations, Application to some classical equations,

    MTK 696 Numerical Solutions of Differential Equations II (303)
    Introduction, The concept of equilibrium point, Definition of the stability, Stability of linear systems, Stability of weakly nonlinear systems, Two-dimensional systems, Stability by Lyapunovs second method, Autonomous systems, Nonautonomous systems, Perturbation theorems, Periodic solutions, Poincare-Bendixson theory, Hopf- bifurcation.

    MAT 697 Functional Analytic Methods in PDE's (303)
    Sobolev spaces, Elements of nonlinear analysis : elliptic boundary value problems 's, Nonstationary problems: existence , uniqueness of weak solutions to initial boundary value problems for semilinear parabolic and hyperbolic equations, regularity of solutions.

    MAT 698 Long-time behavior of Solutions to Nonlinear  Evolutionary PDE's. (303)
    Existence and uniqueness of a weak solution to the initial boundary value problems for semilinear hyperbolic, parabolic equations and nonlinear evolutionary equations of viscoelasticity. Regularity of solutions. Global asymptotic stability of the zero solution to the initial boundary value problem for the semilinear wave equations with the linear dissipative terms and nonlinear reaction diffusion equations. Asymptotic behavior of solutions to nonlinear wave equation with nonlinear disssipative term. Blow up of solutions initial boundary value problems for the nonlinear wave equations , nonlinear parabolic equations and nonlinear Schrödinger equation. Stable and unstable sets for nonlinear parabolic and hyperbolic equations.

    MAT 701 New Topics in Geometry (303)
    New developments in the area of Geometry

    MAT 702 New Topics in Differential Geometry (303)
    New developments in the area of Differential Geometry

    MAT 705 Advanced Set Theory (303)
    Infinitary Combinatorics: Almost disjoint and Quasi-disjoint sets, Martins Axiom, Equivalent axioms, Suslin Problem, Trees, C.U.B filter, Diamond Axioms.

    MAT 706 Continuous Lattices (303)
    Complete Lattices, Lattice Theory of Continuous Lattices, Scott Topology, Scott Continuous Functions, Lawson Topology, Meet-Continuous Lattices, Morphisms and Functors, Spectral Theory of Continuous Lattices.

    MAT 711 Attractors of the Dissipative Evolutionary Equations (303)
    Point dissipative and bounded dissipative semigroups of operators in metric spaces. Attractors of compact and assymptotically compact semigrous. Invariant sets and estimates of Hausdorf and fractal dimensions of invariant sets. Exponential attrtactors. Existence and estimates of dimensions of attractors of semigroups , generated by evolutionary.

    MAT 731 Topological Groups (303)
    Semitopological and topological groups, Locally compact semitopological groups, Translations in topological groups, Neighborhood systems of identity, Separation axioms in topological groups, Uniform structure on a topological group, Subgroups, Quotient groups, Products and inverse limits of groups, Locally compact groups, Classical linear groups, Continuous and open homorphisms, The open homomorphism and closed graph theorems Schurs lemma Orthogonality relations Orthonormal family of functions on metrizable compact groups, Structure of metrizable compact groups, Integral equations on compact groups, The Peter-Weyl theorem, Topologies of dual groups, Dual groups of locally compact abelian groups, Dual groups of compact and discrete groups, some applications of duality theory.

    MAT 732 Banach Algebras (303)
    Basic definitions, examples of Banach algebras, ideals and subalgebras, regular and singular elements in a Banach algebra, quasi-regular and quasi-singular elements in a A Banach algebra, division algebras, Gelfand-Manzur theorem, topological zero divisors, the spectrum (spectral radius) of an element in a Banach algebra, spectral mapping theorem, maximal regular ideals, Gelfands topology, maximal ideal space, Gelfands representation theorem, semi-simple Banach algebras, the radical of an algebra, homomorphisms and isomorphisms of commmutative Banach algebras, the hull-kernel topology, regular commutative Banach algebras.

    MAT 741 Semigroup Theory I (303)
    Uniformly Continuous Semigroups, Strongly Continuous Semigroups, The Hille-Yosida Theorem, C0-Semigroups, Dual Semigroups, Spectral Properties, Compact Semigroups, Analytic Semigroups, Differentiable Semigroups, Lattice Semigroups, Bounded Perturbation of General C0-Semigroups, The Trotter Approximation Theorem, Abstract Cauchy Problem, Evolution Equations,

    MAT 742 p-Adic Analysis (303)
    Valuations of the Fields, p-adic Valuations of Q; Ostrowski Theory, Equivalent Valuations; Valuation Rings, p-adic Number Fields; Completion of a Valued Field; Qp Field, Analysis on Qp, Newtons Medhod; Roots of Unity in Qp.
     
    MAT 745 Homological Algebra I (303)
    Cohomology of Groups, Ext and Tor Functors, Extensions, Tates Cohomologies.

    MAT 746 Homological Algebra II (303)
    Equences of Projective and Injective Modules; Exact Sequences; Global and Homological Dimension; Dimensions of Certain Rings.

    MAT 747 Localization I (303)
    Ore Process; Divisions and Rings and their Modules; Localization on Semi-prime Ideals;Noether Bimodules; First Layer Condition.

    MAT 748 Localization II (303)
    Classical Localization; Cliques, Second Layer Condition; Noether Modules and Localization; Indecomposable Injective Modules and Layer Relations.
     
    MAT 755 Algebraic Geometry I (303)
    Group and Field Theory; Affine and Projective Geometry; Resargues and Pappus Theorms, Harmonic Points; Projective Plane.

    MAT 756 Algebraic Geometry II (303)
    Symplectic and Orthogonal Geometry; Special Cases on Symplectic and Orthogonal Geometry; GLN(K) Group, Symplectic Group Structure; Orthogonal Group, Elliptic Spaces, Clifford Algebra.

    MAT 757 New Topics in Topology (303)
    New developments in the area of Topology

    MAT 758 New Topics in Bitopology (303)
    New developments in the area of Bitopology

    MAT 761 Lie Groups I (303)
    Semi orthogonal groups, Some isometry groups, Time arientability and space, Orientability, Linear Algebra, Space forms, Killing vector fields, The Lie algebra i(M), I(M) as a Lie Group, Homogeneous spaces.
     
    MAT 762 Lie Groups II (303)
    More about Lie groups, Bi-invariant metrics, Coset manifolds, Reductive homogeneous spaces, Symetric spaces, Duality, Some complex geometry.

    MAT 763 Minimal Surfaces I (303)
    Parametric surfaces:Local Theory, Non-parametric surfaces, Surfaces that minimize area, Isothermal parameters, Bernsteins Theorem, Parametric Surfaces:Global Theory
     
    MAT 764 Minimal Surfaces II (303)
    Minimal surfaces with Boundary, Parametric surfaces in E3.Gauss map, Surfaces in E3 .Gauss Curvature and total curvature, Non-parametric surfaces in E3, Application of parametric methods to non-parametric problems, Parametric surfaces in En: Generalized Gauss map.
     
    MAT 765 Space Kinematics I (303)
    Selection of basic concepts from algebra Matrices and determinants, Vectors in three dimensional Euclidean space, Vector spaces, Mapings of sets and Vector spaces, Basic concepts from group theory, Basic Topological concepts, differential geometry of curves and ruled surfaces in E3, Differentiable manifolds, Lie groups.

    MAT 766 Space Kinematics II (303)
    Lie group of congruences of E3 and its Lie Algebra, Klein Quadrics, Representation of space mation, Associated spherical mation, Directing cones of spaces motion, Axoids, Elementary motions, Invariants of motion, Invariant Axoids and Relations between the invariants of motion and the invariants of Axoids, Trajectory of a Point.

    MAT 767 Minkowski Geometry I (303)
    Newtonian space and time, Newtonian spacetime, Minkowski spacetime, Minkowski geometry , Particles observed, Some relativistic effects, Lorent-Fitzgerald contraction, Energy, Momentum, Collisions, An accerelerating observer.

    MAT 768 Minkowski Geometry II (303)
    Foundation, The Einstein equation, Perfect fluids, Robertson, Walker spacetimes, the Robertson-Walker flow, Robertson-Walker cosmaology, Friedmann models, Geodesics and Redshift, Observer fields, Static spacetimes.

    MAT 769 C*-Algebras I (303)
    Involutive algebras, normed involutive algebras, C*-algebras, commutative C*-algeras, a functional calculus in C*-algebras, approximate identities in C*-algebras, quotient C*-algebras, positive forms, representations, irreducible representations, pure aforms, the enveloping C*-algebras, ideals in C*-algebras,

    MAT 770 C*-Algebras II (303)
    The seconud dual of a C*-algebra, polar devomposition of a linear form, positive part of an ideal in a C*-algebra, tensor product of C*-algebras derivations of C*-algebra, automorphisms of C*-algebras, inner automorphisms, derivable automorphisms.

    MAT 777 New Topics in Real Analysis (303)
    New developments in the area of Real Analysis


    MAT 778 New Topics in Functional Analysis (303)
    New developments in the area of Functional Analysis

    MAT 781 Convex Analysis I (303)
    The Concept of Derivative in Normed Spaces; Convex Functions Defined on R , their Characterization and Differentiability; Conjugate Convex Functions; Solution of Certain Inequalities using Convex Functions.

    MAT 782 Convex Analysis II (303)
    Convex Functions Defined on Normed Spaces; Notions of Continuity and Differentiability of such Functions; Application of Convex Functions to Optimization Problems, Dual Problems, Kuhu-Tuckers Problem.

    MAT 786 New Topics in the Theory of Banach Spaces (303)
    Selected topis from the following: Lpspaces; Type and cotype. Projection constants. L1 ( m )-spaces. C(k)-spaces. The Disc Algebra. Absolutely summing and related topics. Schatten Neumann classes. Factorization theorems. Absolutely summing operators disk algebra.

    MTK 797 New Topics in Ordinary Differential Equations (303)
    New developments in the area of Ordinary Differential Equations

    MTK 798 New Topics in Partial Differential Equations (303)
    New developments in the area of Partial Differential Equations

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