Lisansüstü Ders Kataloğu


Math 521

Algebra I (Cebir I)

(4+0+0) 4

 

Free groups, group actions, group with operators, Sylow theorems, Jordan-Hölder theorem, nilpotent and solvable groups. Polynomial and power series rings, Gauss's lemma, PID and UFD, localization and local rings,chain conditions, Jacobson radical.

 

Önkoşul:

Yok

 

Math 522

Algebra II (Cebir 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, Wedderburn-Artin theorem, finitely generated modules over a principal ideal domain, basis theorem for finite abelian groups.

 

Önkoşul:

Yok

 

Math 525

Algebraic Number Theory (Cebirsel Sayı Kuramı)

(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.

 

Önkoşul:

Yok

 

Math 527

Number Theory (Sayılar Kuramı)

(4+0+0) 4

 

Method of descent, unique factorization, basic algebraic number theory, diophantine equations, elliptic equations, p-adic numbers, Riemann zeta function, elliptic curves, modular forms, zeta and L-functions, ABC-conjecture, heights, class numbers for quadratic fields, a sketch of Wiles' proof.

 

Önkoşul:

Yok

 

Math 528

Analytic Number Theory (Çözümsel Sayılar Kuramı)

(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 L-functions, the prime number theorem, Polya- Vinogradov inequality, the large sieve, average results on the distribution of primes.

 

Önkoşul:

Math 533

 

Math 529

Analytic Number Theory II (Çözümsel Sayılar Kuramı II)

(3+0+0) 3

 

The prime number theorem for arithmetic progressions. Sums over primes, exponential sums. The large sieve, Bombieri-Vinogradov theorem,Selberg’s sieve. Results on the distribution of primes.

 

Önkoşul:

Math 528

 

Math 531

Real Analysis I (Reel Analiz I)

(4+0+0) 4

 

Lebesgue measure and Lebesgue integration on Rn, general measure and integration, decomposition of measures, Radon-Nikodym theorem, extension of measures, Fubini's theorem.

 

Önkoşul:

Yok

 

Math 532

Real Analysis II (Reel Analiz II)

(4+0+0) 4

 

Normed and Banach spaces, Lp-spaces and duals, Hahn-Banach theorem, category and uniform boundedness theorem, strong, weak and weak*-convergence, open mapping theorem, closed graph theorem.

 

Önkoşul:

Math 531

 

Math 533

Complex Analysis I (Karmaşık Analiz 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.

 

Önkoşul:

Yok

 

Math 534

Complex Analysis II (Karmaşık Analiz II)

(4+0+0) 4

 

Runge's theorem, analytic continuation, Riemann surfaces, harmonic functions, entire functions, the range of an analytic function.

 

Önkoşul:

Math 533

 

Math 535

Functional Analysis (Fonksiyonel Analiz)

(4+0+0) 4

 

Topological vector spaces, locally convex spaces, weak and weak* topologies, duality, Alaoglu's theorem, Krein-Milman theorem and applications, Schauder fixed point theorem, Krein-Smulian theorem, Eberlein-Smulian theorem, linear operators on Banach spaces.

 

Önkoşul:

Math 531 ve Math 532

 

Math 541

Probability Theory (Olasılık Kuramı)

(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.

 

Önkoşul:

Yok

 

Math 544

Stochastic Processes and Martingales (Rassal Süreçler ve Martingeller)

(4+0+0) 4

 

Stochastic processes, stopping times, Doob-Meyer decomposition, Doob's martingale convergence theorem, characterization of square integrable martingales, Radon-Nikodym theorem, Brownian motion, reflection principle, law of iterated logarithms.

 

Önkoşul:

Math 541

 

Math 545

Mathematics of Finance (Finans Matematiği)

(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 Black-Scholes equation, stochastic differential equations, Feynman-Kac theorem, Cox-Ingersoll-Ross and Vasicek term structure models, Girsanov's theorem and risk neutral measures, Heath-Jarrow-Morton term structure model, exchange-rate instruments.

 

Önkoşul:

Yok

 

Math 551

Partial Differential Equations I (Kısmi Diferansiyel Denklemler 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 non-linear first-order PDE's, conservation laws, special solutions of PDE's, Cauchy-Kowalevskaya theorem.

 

Önkoşul:

Yok

 

Math 552

Partial Differential Equations II (Kısmi Diferansiyel Denklemler II)

(4+0+0) 4

 

Hölder spaces, Sobolev spaces, Sobolev embedding theorems, existence and regularity for second-order elliptic equations, maximum principles, second-order linear parabolic and hyperbolic equations, methods for non-linear PDE's, variational methods, fixed point theorems of Banach and Schauder.

 

Önkoşul:

Math 551

 

Math 571

Topology (Topoloji)

(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.

 

Önkoşul:

Yok

 

Math 572

Algebraic Topology (Cebirsel Topoloji)

(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.

 

Önkoşul:

Math 571

 

Math 575

Differentiable Manifolds (Türevlenebilir Çok Katmanlılar)

(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.

 

Önkoşul:

Yok

 

Math 576

Riemannian Geometry (Riemann Geometrisi)

(3+0+0) 3

 

Differentiable manifolds, vectors and tensors, riemannian metrics, connections, geodesics, curvature, jacobi fields, riemannian submanifolds, spaces of constant curvature.

 

Önkoşul:

Yok

 

Math 577

Complex Manifolds (Karmaşık Çok Katmanlılar)

(3+0+0) 3 ECTS 7

 

Complex Manifolds, Kahler and Calabi-Yau Manifolds, Homology and Cohomology,
Fiber Bundles, Connections on Fiber Bundles, Characteristic Classes, Index Theorems.

 

Önkoşul:

Math 576 veya dersi veren öğretim üyesinin izni ile

 

Math 579

Graduate Seminar (Lisansüstü Semineri)

(0+1+0) Non-credit

 

Presentation of topics of interest in Mathematics through seminars offered by faculty, guest speakers and graduate students.

 

Önkoşul:

Yok

 

Math 581

Selected Topics in Analysis I (Analizden Seçme Konular I)

(3+0+0) 3

 

Math 582

Selected Topics in Analysis II (Analizden Seçme Konular II)

(3+0+0) 3

 

Math 583

Selected Topics in Foundations of Mathematics (Matematiğin Temellerinden Seçme Konular)

(3+0+0) 3

 

Math 584

Selected Topics in Algebra and Topology (Cebir ve Topolojiden Seçme Konular)

(3+0+0) 3

 

Math 585

Selected Topics in Probability and Statistics (Olasılık ve İstatistikten Seçme Konular)

(3+0+0) 3

 

Math 586

Selected Topics in Differential Geometry (Diferansiyel Geometriden Seçme Konular)

(3+0+0) 3

 

Math 587

Selected Topics in Differential Equations (Diferansiyel Denklemlerden Seçme Konular)

(3+0+0) 3

 

Math 588

Selected Topics in Applied Mathematics (Uygulamalı Matematikten Seçme Konular)

(3+0+0) 3

 

Math 589

Selected Topics in Combinatorics (Kombinatorikten Seçme Konular)

(3+0+0) 3

 

Math 590

Readings in Mathematics (Matematikte Okumalar)

(0+0+2) 1

 

Literature survey and presentation on a subject to be determined by the instructor.

 

Önkoşul:

Yok

 

Math 601

Measure Theory (Ölçü Teorisi)

(4+0+0) 4

 

Fundamentals of measure and integration theory, Radon-Nikodym Theorem, Lp spaces, modes of convergence, product measures and integration over locally compact topological spaces.

 

Önkoşul:

Yok

 

Math 611

Differential Geometry I (Diferansiyel Geometri 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.

 

Önkoşul:

Yok

 

Math 612

Differential Geometry II (Diferansiyel Geometri 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.

 

Önkoşul:

Math 611

 

Math 623

Integral Transforms (İntegral Dönüşümleri)

(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, Fourier-Bessel and Hankel transforms for circular regions, Abel transform for dual integral equations.

 

Önkoşul:

Yok

 

Math 624

Numerical Solutions of Partial Differential and Integral Equations (Kısmi Türevli Diferansiyel Denklemlerle İntegral Denklemlerin Sayısal Çözülmesi)


(4+0+0) 4

 

Parabolic differential equations, explicit and implicit formulas, elliptic equations, hyperbolic systems, finite elements characteristics, Volterra and Fredholm integral equations.

 

Önkoşul:

Yok

 

Math 627

Optimization Theory I (Eniyileme Kuramı I)

(4+0+0) 4

 

Fundamentals of linear and nonlinear optimization theory. Unconstrained optimization, constrained optimization, saddlepoint conditions, Kuhn-Tucker conditions, post-optimality, duality, convexity, quadratic programming, multistage optimization.

 

Önkoşul:

Yok

 

Math 628

Optimization Theory II (Eniyileme Kuramı II)

(4+0+0) 4

 

Design and analysis of algorithms for linear and non-linear optimization. The revised simplex method, algorithms for network problems, dynamic programming, quadratic programming techniques, methods for constrained nonlinear problems.

 

Önkoşul:

Math 627

 

Math 631

Algebraic Topology I (Cebirsel Topoloji 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.

 

Önkoşul:

Yok

 

Math 632

Algebraic Topology II (Cebirsel Topoloji II)

(4+0+0) 4

 

Singular homology, exact sequences, the Mayer-Vietoris exact sequence, the Lefschetz fixed-point theorem, cohomology, cup and cap products, duality theorems, the Hurewicz theorem, higher homotopy groups.

 

Önkoşul:

Yok

 

Math 635

An Introduction to Nonlinear Analysis (Doğrusal Olmayan Analize Giriş)

(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. Palais-Smale condition. Mountain Pass Theorem.

 

Önkoşul:

Math 535 ya da eşdeğeri

 

Math 643

Stochastic Processes I (Rassal Süreçler 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 second-order processes, linear prediction and filtering, Ito and Saratonovich integrals, Ito calculus, stochastic differential equations, diffusion processes, Gaussian measures, recursive estimation.

 

Önkoşul:

Math 552 ya da öğretim üyesinin onayı

 

Math 644

Stochastic Processes II (Rassal Süreçler II)

(4+0+0) 4

 

Tightness, Prohorov's theorem, existence of Brownian motion, Martingale characterization of Brownian motion, Girsanov's theorem, Feynmann-Kac formulas, Martingale problem of Stroock and Varadhan, applications to Mathematics of finance.

 

Önkoşul:

Math 643

 

Math 645

Mathematical Statistics (Matematiksel İstatistik)

(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 Neyman-Pearson lemma, sequential decision problem.

 

Önkoşul:

Math 552 ya da öğretim üyesinin onayı

 

Math 660

Number Theory (Sayılar Kuramı)

(4+0+0) 4

 

Basic algebraic number theory; number fields, ramification theory, class groups, Dirichlet unit theorem; zeta and L-functions; Riemann, Dedekind zeta functions, Dirichlet, Hecke L-functions, primes in arithmetic progressions, prime number theorem; cyclotomic fields, reciprocity laws, class field theory, ideles and adeles, modular functions and modular forms.

 

Önkoşul:

Yok

 

Math 680

Seminar in Pure Mathematics I (Sırfi Matematik Semineri I)

(4+0+0) 4

 

Recent developments in pure Mathematics.

 

Önkoşul:

Yok

 

Math 681

Seminar in Pure Mathematics II (Sırfi Matematik Semineri II)

(4+0+0) 4

 

Recent developments in pure Mathematics.

 

Önkoşul:

Yok

 

Math 682

Seminar in Applied Mathematics I (Uygulamalı Matematik Semineri I)

(4+0+0) 4

 

Recent developments in applied Mathematics.

 

Önkoşul:

Yok

 

Math 683

Seminar in Applied Mathematics II (Uygulamalı Matematik Semineri II)

(4+0+0) 4

 

Recent developments in applied Mathematics.

 

Önkoşul:

Yok

 

Math 690

M.S. Thesis (Yüksek Lisans Tezi)

 

 

Math 699

Guided Research (Yönlendirilmiş Araştırmalar)

(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.

 

Önkoşul:

Yok

 

Math 790

Ph.D. Thesis (Doktora Tezi)

 

 

Ana Sayfa Lisansüstü Lisansüstü Ders Kataloğu