Program tanımları
MATEMATİK ANABİLİM DALI LİSANSÜSTÜ PROGRAMLARI
Matematik kabul edilebilir aksiyomlar üzerine kurulu, mantık kurallarıyla çıkarımlar yapılan, mutlak denilebilecek gerçekliğe sahip bir çalışma alanıdır. Aksiyom ve sonuçların fiziksel gerçeklikle örtüştüğü durumlarda hemen hemen bütün bilim ve mühendislik dallarının konu edilebileceği uygulamalı matematik söz konusudur. Bilginin ve bilimin hızla değer kazanarak teknoloji, sanayi ve finans dünyasını şekillendirdiği günümüzde; matematik ve matematiksel düşünce hem değişik akademik alanlarda araştırma yapmak isteyenler hem de profesyonel iş yaşamını hedefleyenler için bir gerekliliktir.
Matematikte lisans ve/veya yüksek lisans derecesi ile iş bulabilmek çoğu zaman uygulamalı matematiğin bazı alanlarında ve bilgisayar bilimlerinde eğitimi gerektirmektedir.
Geleneksel uygulamalı matematik, fiziksel bilimlere (fizik, mühendislik, endüstri vb) yönelimli matematik anlamındadır ve bu alanlarda son derece geniş bir yelpazede çalışma olanağı vardır. Finas kuramcıları, ekonomistler, mühendisler, fizikçiler, bir kısım biyolog, kimyacı vb meslek sahiplerinin, kendilerine matematikçi denmese bile, iyi ve yeterli matematik eğitimi almış olmaları gerekmektedir.
Öte yandan, karar verme bilimleri denilebilecek operasyon analizi, sistem analizi vb için, matematik ve sistem mühendisliğinin birinde lisans diğerinde yüksek lisans derecesi almak oldukça uygun bir eğitim olabilir. Aktuarya (sigortacılık) ve işletme için lineer cebir ve istatistik, bilgisayar bilimlerinde özellikle yazılım ve ağ mühendisliği konularında mantık, kombinatorik, sayılar kuramı ve cebir gerekli matematik olarak görülmektedir.
Ortaöğrenim ve lise matematik öğretmenlerinin geniş bir matematik kültürüne sahip olmaları öğrettikleri parçaların bütün içindeki yerini gösterebilmeleri ve öğrenciler üzerinde matematik hakkında iyi bir izlenim bırakabilmeleri bakımından gereklidir.
Bu veriler ışığında hazırlanan Matematik Yüksek Lisans ve Doktora programları Fen ve Mühendislik Fakülteleri (Matematik, Fizik, Kimya, Biyoloji bölümleri ile Makine, Elektrik ve Elektronik, Bilgisayar, Sistem, Kimya Mühendislikleri bölümleri) ile Eğitim Fakültelerinin temel bilimler bölümlerinden Lisans ve Yüksek Lisans dereceleri alan öğrencilerin tümüne açık bir programdır.
Yetenekli ve matematiği seven öğrencilerin doktora derecesi alarak üniversitede eğitim ve araştırma çalışmalarını hedeflemeleri idealdir. Bu alandaki iş fırsatları ülkemizde (özellikle batılı ülkelere göre) daha fazladır. Doktora mezunları araştırma grupları bulunan büyük şirketlerde ve devlet kurumlarında da iş bulabilirler.
Başka bir Enstitüde bir yarıyılını başarı ile tamamlayan öğrenciler de programlara başvurabilirler.
Doktora programına, Yüksek Lisans derecesi ile kabul edilen öğrenciler yirmibir kredilik yedi ders almalı ve tez çalışması yapmalıdır. Lisans derecesi ile kabul edilen öğrenci en az kırkiki kredilik ondört ders almalı ve tez çalışması yapmalıdır. Derslerden altı tanesi zorunludur. Yedi dersini tamamlayan öğrenci dilerse Yüksek Lisans programına geçebilir.
Matematik Bölümü araştırma görevlilerini lisanüstü öğrencileri arasından seçmeyi benimsemektedir.
Doğrusal olmayan türevli denklemler, dinamik sistemler, türevsel geometri, fonksiyonel analiz, yaklaşım kuramı, tam çözülebilir sistemler, Hamilton yapıları, simplektik geometri, geometrik mekanik, Lie grupları ve Lie cebirleri bölümün temel araştırma alanlarına örnek olarak sayılabilir. Lisanüstü derslerin yanısıra düzenli seminerler öğrencileri çalışma alanlarında daha ileri düzeye taşımayı hedeflemektedir. Seminerlerin konusu bölüm elemanlarının ve ziyaretçilerin araştırma konularına göre zamanla değişmektedir.
DERS PROGRAMI
Ph.D. DEGREE IN MATHEMATICS
The aim of this program is to make research in various areas of mathematics such as geometry, functional analysis, nonlinear differential equations as well as in applications of mathematics to basic problems from physics and engineering.
Program:
Students accepted with M.S. degree must complete seven courses of at least twenty-one credits and a thesis work.
Students accepted with B.S. degree must complete fourteen courses of at least fourty-two credits and a thesis work. Six of the courses are compulsory. In this program, students completing seven courses may transfer to M.S. program.
Basic research areas of the department are nonlinear differential equations, dynamical systems, differential geometry, gravitation and field theories, functional analysis, approximation theory, completely integrable systems and geometric mechanics.
COURSES
Compulsory (for students accepted with B.S. degree)
MATH 511 Differential Geometry
MATH 521 Algebra I
MATH 531 Topology
MATH 541 Ordinary Differential Equations
MATH 551 Functional Analysis I
MATH 553 Complex Analysis
LIST OF COURSES:
MATH 510 CONCEPTS OF GEOM. FOR MATH. TEACHERS
MATH 511 DIFFERENTIAL GEOMETRY
MATH 512 RIEMANNIAN GEOMETRY
MATH 513 TOPICS IN GEOMETRY
MATH 521 ALGEBRA I
MATH 522 ALGEBRA II
MATH 523 LIE GROUPS AND LIE ALGEBRAS
MATH 524 GEOMETRIC CONTROL THEORY
MATH 531 TOPOLOGY
MATH 532 ALGEBRAIC TOPOLOGY
MATH 533 INTRODUCTION TO DIFFERENTIAL TOPOLOGY
MATH 541 ORDINARY DIFFERENTIAL EQUATIONS
MATH 542 PARTIAL DIFFERENTIAL EQUATIONS
MATH 544 TOPICS IN APPLIED MATHEMATICS
MATH 550 CONCEPTS OF ANALYSIS FOR MATH. TEACHERS
MATH 551 FUNCTIONAL ANALYSIS I
MATH 552 FUNCTIONAL ANALYSIS II
MATH 554 APPROXIMATION THEORY
MATH 555 MEASURE AND INTEGRATION THEORY
MATH 556 TOPICS IN ANALYSIS
MATH 590 SEMINAR
MATH 600 M.S. THESIS
MATH 611 INTRODUCTION TO SYMPLECTIC GEOMETRY
MATH 612 GEOMETRY OF DIFFEOMORPHISM GROUPS
MATH 621 APPL. OF LIE GROUPS TO DIFFERENTIAL EQUATIONS
MATH 641 EXTERIOR DIFFERENTIAL FORMS I
MATH 642 EXTERIOR DIFFERENTIAL FORMS II
MATH 643 PARTIAL DIFFERENTIAL EQUATIONS II
MATH 644 GEOMETRIC THEORY OF DYNAMICAL SYSTEMS
MATH 645 SOBOLEV SPACES AND ELLIPTIC OPERATORS
MATH 690 SEMINAR
MATH 700 PH.D. THESIS
DERS İÇERİKLERİ
DESCRIPTIONS OF COURSES:
MATH 510 CONCEPTS OF GEOMETRY FOR MATHEMATICS TEACHERS
Euclidean and non-Euclidean geometries. Calculus on and submanifolds of . Differential geometry of curves and surfaces. Vectors, tensors, multivectors and differential forms. Flows, symmetries and geometries. Structure equations. Lie groups and Lie algebras. Constructions in modern differential geometry.
MATH 511 DIFFERENTIAL GEOMETRY
Smooth mappings. Implicit function theorem. Submanifolds of Euclidean space. , analytic and smooth manifolds. Examples: projective spaces, Grassmann manifolds, Riemann surfaces. Manifolds with boundary. Partition of unity. Mappings of manifolds, regular and singular points, immersions, submersions and embeddings. Sard’s theorem. Tangent bundle. Existence of Riemannian metric. Vector fields, flows and differentials. Algebra of vector fields. Cotangent bundle. Tensor fields, multi-vectors, exterior forms and their algebras. Applications to mechanics and Lie groups.
MATH 512 RIEMANNIAN GEOMETRY
Riemannian manifolds. Absolute differentiation and connection. Riemann curvature, Bianchi identities. Geometry of hypersurfaces, Riemannian immersions and submersions. Completeness. Isometries and Killing vectors. Homogeneous and symmetric spaces. Properties of curvature tensors. Variational principles.
Prerequisite: Math 511.
MATH 521 ALGEBRA I
Groups; Sylow theorems. Direct sums and direct products. Free groups. Action of a group on a set. Rings; homomorphisms, commutative rings. Principal ideal domains, unique factorization domains. Noetherian rings. Rings of quotients. Localization.
MATH 522 ALGEBRA II
Galois theory. Categories and functors. Module categories. Tensor products. Projective and injective modules. Primitive rings. Jacobson radical. Semi-simple rings. Decomposition theorems.
MATH 523 LIE GROUPS AND LIE ALGEBRAS
Manifolds, Lie groups and Lie algebras. Exponential map, Baker-Campbell-Hausdorff formula. Lie’s fundamental theorems. Nilpotent and solvable Lie algebras. Cartan’s criterion. Semisimple Lie algebras. Casimirs. The theorem of Weyl. Levi decomposition. Global results.
Prerequisite: Math 511.
MATH 524 GEOMETRIC CONTROL THEORY
General control systems, orbits, transitivity, reachability, controllability, observability, minimal realization. Linear control systems on , rank conditions. Polynomial control systems, feedback, bounded controls, bang-bang principle. Systems on Lie groups and homogeneous spaces, controllability of affine systems, characterization of observability, normalizer, drift vector field. Applications.
Prerequisite: Math 523.
MATH 531 TOPOLOGY
Topological spaces and continuous mappings. Metric topology. Topology of and . Factor spaces and quotient topology. Classification of surfaces. Orbit spaces, projective and lens spaces. Operations on sets, completeness. Connectedness, countability and separation axioms. Normal spaces. Compactness and compactifications. Metrization.
MATH 532 ALGEBRAIC TOPOLOGY
Topology of space of continuous mappings. Homotopy. Extension. Retraction and deformation. Algebraization of topological problems. Homotopy groups. Fundamental group. Computations of the fundamental and homotopy groups of closed surfaces, topological invariance of the Euler characteristics. Homology groups of simplicial complexes and polyhedra. Barycentric subdivision, simplicial mappings. Singular homology. Homology groups of spheres, cell complexes and projective spaces. Degree of a mapping. Lefschetz number of simplicial and continuous mappings.
Prerequisite: Math 531.
MATH 533 INTRODUCTION TO DIFFERENTIAL TOPOLOGY
Locally trivial fiber spaces. Lifts of mappings and covering homotopy. Vector bundles and morphisms. Homotopy groups, universal covering. Monodromy. Cellular structure. Index of critical points. Morse lemma. Gradient fields. Homotopy type and change. de Rham cohomology, homotopy operator and Poincaré lemma. Stokes’ theorem. de Rham’s isomorphism theorem. Applications.
Prerequsite: Math 531, Math 511.
MATH 541 ORDINARY DIFFERENTIAL EQUATIONS
First order equation. Cauchy-Euler method. Continuation of solution. Systems of equations. Lipschitz conditions. Linear systems. Green’s function. Singularities of linear autonomous systems. Nonlinear equations. Poincare-Bendixson theorem. Poincare index. Limit cycles.
MATH 542 PARTIAL DIFFERENTIAL EQUATIONS I
Well-posed problems. Classical solutions. Weak solutions and regularity. Transport, Laplace, Heat, Wave equations. Non-linear first order PDE’s, Hamilton-Jacobi theory. Separation of variables, similarity solutions, transform methods. Converting non-linear into linear PDE. Asymptotics, power series method.
MATH 550 CONCEPTS OF ANALYSIS FOR MATHEMATICS TEACHERS
Cartesian tradition. Calculus. Algebraic conceptions of Euler and Lagrange. Functions. Analytical mechanics. Potential theory. Gauss, Green and Stokes theorems. Cauchy and Weierstrass. Complex functions. Riemann and Lebesgue integrals. Modern foundation of analysis. Topics in the theory of differential equations, variational calculus and functional analysis.
MATH 551 FUNCTIONAL ANALYSIS I
Linear vector spaces, subspaces, direct sum. Linearly independent sets, Hamel bases. Linear transformations, linear functionals. Eigenvalues and eigenvectors of linear operators. Introduction to topology. Numerical functions. Measures of sets, integration of numerical functions. Metric spaces, completeness. Contraction mappings. Compact metric spaces. Approximations. Normed linear spaces, norm and semi-norm topologies. Bounded linear operators.
MATH 552 FUNCTIONAL ANALYSIS II
Normed linear spaces, topological dual, weak and strong topologies. Compact and closed operators. Inner product spaces, orthogonal subspaces. Orthonormal sets and Fourier series. Duals of Hilbert spaces. Linear operators on Hilbert spaces and their adjoints. Spectral theory of linear operators, resolvent set and spectrum. Spectrum of bounded linear operators. Spectral analysis on Hilbert spaces. Introduction to non-linear functional analysis, Gâteaux and Fréchet derivatives of non-linear operators. Integration of operators.
MATH 553 COMPLEX ANALYSIS
Analytic functions. The argument principle. Conformal mappings. The Riemann mapping theorem. Infinite products. The Weierstrass factorization theorem. The Mittag–Lefler theorem. Analytic continuation. The Picard theorem.
MATH 554 APPROXIMATION THEORY
Preliminaries, polynomials of best approximation. Existence, characterizations and uniqueness of the best polynomial approximation. Best trigonometric approximation. Degree of approximation by trigonometric functions. Inverse theorems for periodic functions. Rational approximation.
MATH 555 MEASURE AND INTEGRATION THEORY
Measures, outer measures. Extension of measures. Measurable functions. Integrable functions. Sequences of integrable functions. Properties of integrals. Signed measures. Hahn and Jordan decompositions. The Radon-Nikodym theorem. Product Spaces.
MATH 645 SOBOLEV SPACES AND ELLIPTIC OPERATORS
Review of Banach and Hilbert spaces. Hölder continuity. Sobolev spaces. Sobolev inequality and embedding theorem. Differential operators. Adjoints. Principle symbols. Elliptic operators. Linearization of nonlinear differential operators. Fredholm alternative. Regularity of solutions for elliptic equations. Survey of methods for the existence problems.
MATH 611 INTRODUCTION TO SYMPLECTIC GEOMETRY
Poisson brackets and Poisson manifolds. Hamiltonian vector fields. Symplectic manifolds. Darboux theorem. Lagrangian submanifolds. Special symplectic structures. Legendre transforms. Hamiltonian symmetries. Symplectic reduction. Applications.
MATH 612 GEOMETRY OF DIFFEOMORPHISM GROUPS
Mappings and flows. Manifolds of maps. Group of diffeomorphisms and its subgroups. Diffeomorphisms of circle. Lie groups of volume preserving and symplectic diffeomorphisms. Lie algebras of divergence free and symplectic vector fields. Lie-Poisson structures on duals of Lie algebras. Poisson maps. Generalizations to semi-direct products. Variational problems. Applications to integrable non-linear partial differential equations, symplectic topology, fluids and plasmas.
MATH 621 APPLICATIONS OF LIE GROUPS TO DIFFERENTIAL EQUATIONS
Symmetries and generalized symmetries of differential equations, Lagrangians and Hamiltonian equations in finite and infinite dimensions. Group invariant solutions. Formal variational calculus. Bi-Hamiltonian systems. Integrability.
MATH 641 EXTERIOR DIFFERENTIAL FORMS I
Alternating multilinear functionals, exterior forms and exterior algebra. Differentiable manifolds, vector fields, tangent spaces. Lie derivatives, Lie algebras. Mapping between manifolds. Cotangent spaces, exterior differential forms and exterior algebra. Interior product, dual forms. Ideals of exterior algebra, exterior derivative, closed ideals. Lie derivatives of exterior forms, Frobenius and Cartan theorems. Isovector fields.
MATH 642 EXTERIOR DIFFERENTIAL FORMS II
Integration of differential forms, Stokes’ theorem. Conservation laws. Homotopy operator. Decomposition of a form into exact and antiexact parts. Canonical forms of 1-forms and closed 2-forms. Caratheodory theorem. Solutions of exterior differential equations. Affine connections on manifolds. Covariant derivative, curvature and torsion forms. Exterior forms equivalent to system of partial differential equations. Symmetry groups. Variational Calculus. Various applications.
MATH 644 GEOMETRICAL THEORY OF DYNAMICAL SYSTEMS
Review of differentiable manifolds and vector fields. Transversality. Structural stability. Tubular flows. Local stability. Invariant manifolds. Kupka-Smale theorem. Poincare map. Morse-Smale vector fields.
