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Uygulamalı Matematik Yüksek Lisans programı

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  • Program tanımları
    PROGRAM ÖZELLİKLERİ

    Uygulamalı Matematik Yüksek Lisans programı Türkçe yürütülen bir programdır. Programa katılacak öğrencinin seçimine bağlı olarak tezli veya tezsiz olarak tamamlanabilir. Tezli program toplam 24 kredilik ders, seminer ve tez; tezsiz program ise toplam 30 kredilik ders ile proje çalışmasından oluşur.

    PROGRAMIN AMAÇLARI VE HEDEF KİTLESİ

    Bu programın amacı, matematiğin modern dünyada gelişen sistemlerde "birlikte" kullanıldıkları alt dalları geliştirebilecek ve yönetebilecek uzmanlar yetiştirebilmektir. Kuramsal matematik konularının yanısıra, programda matematiksel finans ve yatırım araçlarının modellenmesi, portföy analizi ve finansal risk yönetimi, istatistik, veri analizi, hesaplamalı bilimler ve bioistatistik konularında alt uzmanlık dalları oluşturulacağı gibi Fen Bilimleri Enstitüsü bünyesinde yürütülmekte olan diğer programlardan ders alınarak bilgisayar dersleri ile tümleşik program da oluşturulabilecektir.

    Bu programa öncelikli olarak Matematik, Matematik-Bilgisayar, İstatistik, Matematik Mühendisliği, Matematik Eğitimi, İşletme/Ekonomi, Fizik ve Fizik Mühendisliği veya herhangi başka bir mühendislik bölümü mezunu olup Matematik alanında altyapısını güçlendirmak isteyenlerin talebine açık olacaktır.

    PROGRAM YAPISI

    Uygulamalı Matematik programına katılacak olan öğrenciler üç farklı uzmanlık dalından birini odak noktası olarak seçebilir. Bunlar Matematik Modülü, Hesaplamalı Bilimler Modülü ve Finans Matematiği Modülüdür.Öğrencilerin uzmanlaşmak istedikleri alana bağlı olarak zorunlu, alan seçmelileri ve serbest seçmeli ders havuzları belirlenmiştir. Programda zorunlu ders sayısı en azda tutularak öğrenciye seçim hakkı tanınmak istenmştir. Seçimlik dersler uygulamalı matematik seçimliklerinden alınabileceği gibi Fen Bilimleri veya Sosyal Bilimler Enstitüsünde yürütülmekte olan programların ilgili derslerinden de seçilebilir.

    Farklı lisans diploması ile programa katılmak isteyenlerin altyapısı yetersiz görüldüğü durumda ön hazırlık dersleri alarak programa katılmak mümkündür.

    Ders Listesi

    Yazılım Mühendisliği Matematiği     
    Ayrık Matematik    
    Soyut Cebir    
    Grup Teorisi    
    Sayılar Teorisi    
    Kriptolojiye Giriş    
    Sonlu Cisim Uygulamaları    
    Lineer Cebir ve Uygulamaları    
    Diferansiyel Geometri    
    Kinematik    
    Topoloji    
    Reel Analiz
    Kompleks Analiz    
    İleri Analiz    
    Fonksiyonel Analiz    
    İleri Düzey Matematiksel Analiz    
    Nümerik Analiz    
    Uygulamalı Matematiğin Temelleri
    Kısmi Diferansiyel Denklemler    
    Diferansiyel Denklemlerin Sayısal Çözümleri    
    Kısmi Diferansiyel Denklemlerin Sayısal Çözümleri    
    Sınır Değer Problemleri    
    Matematiksel Programlama ve Modelleme    
    Nümerik Optimizasyon    
    Matematiksel İstatistik    
    Olasılık Teorisi    
    İstatistik Teorisi    
    Uygulamalı İstatistiksel Analiz    
    İleri Düzey Olasılık Teorisi    
    Bioistatistik Yöntemler    
    Zaman Seri Analizi    
    Stokastik Süreçler    
    Parametrik Olmayan İstatistik    
    İleri İstatistiksel Modeller    
    Matematiksel Finansa Giriş    
    Risk Yönetimi    
    Finansal Matematik
    Finansta Stokastik Hesaplamalar    
    Finansta Hesaplamalı Modeller    
    Sabit Gelirli Menkul Kıymet. & Kredi Riskine Giriş    
    Finansal Risk Analizi    
    Seminer    
    Yüksek Lisans Tezi    
    Proje

    Ders Tanımları

    Yazılım Mühendisliği Matematiği
    Logic: propositions, logical connectives, rules of inference. Sets, elements of a set, set of sets, Cartesian product. Sequences; sequence operators; sequence of functions; structural induction. Principle of induction, recursions.

    Ayrık Matematik
    Foundations of discrete mathematics, logic and proof, sets, functions, sequences, algorithms, complexity of algorithms; integers and divisibility, prime numbers, matrices, relations, equivalence relations, order relations; graph theory, trees, Boolean algebra.

    Soyut Cebir
    Divisibility, fundamental theorem of algebra, greatest common divisor, division algorithm, some arithmetical functions, congruence systems, Fermat?s theorem, Chinese remainder theorem, factorization in polynomial rings, field extensions, normal extensions, Galois theory.

    Grup Teorisi
    Fundamental concepts of group theory, finite groups, C groups, symmetric groups, unitary groups, applications to physics and other sciences.

    Sayılar Teorisi
    General introduction to numerical sets: Definitions and properties of natural numbers, integers, rational numbers, real numbers, complex numbers. Divisibility in integers: Concept of divisibility and properties of divisibility of integers, fundamental theorem of arithmetic, canonical representation of integers. Arithmetic functions: integer value function, tau, sigma, Möbiüs and Euler functions. Congruence: definition and properties of congruence, prime and reduced remainder systems, Euler and Fermat theorems, linear congruence equations, Chinese remainder theorem, high order congruence equations. Quadratic congruence systems, quadratic remainders, Legendre symbols, Gauss lemma, quadratic reciprocal theorem, Jacobi symbols, primitive roots and indices. Diophantine equations, continuous fractions, quadratic forms, Pell equations, distribution of prime numbers, algebraic numbers, algebraic integers, units and primes in a numerical set, ideals, algebra in ideals, prime ideals.

    Kriptolojiye Giriş
    Historical introduction to cryptography: general principles, services, mechanisms and attacks. Classical coding methods, symmetric coding methods. Block Ciphers: diffusion, confusion, Feistel structure. Introduction to finite groups and number theory. Cryptographic criteria. Public key cryptography, and hash functions. Discrete logarithm. RSA, key management, Diffie-Hellman key switch, elliptic curve cryptography. Digital signatures, verification protocols, digital signature criteria.

    Sonlu Cisim Uygulamaları
    Groups, rings and fields. Polynomials. Field extensions. Characterization of finite fields. Roots of irreducible polynomials. Norms and basis. Representations of elements of finite fields. Wedderburn theorem. Irreducible polynomials. Construction of Irreducible Polynomials.

    Lineer Cebir ve Uygulamaları

    Vector spaces, inner product spaces, orthonormal vector systems, concepts of basis and dimension, linear transformations, vector spaces of linear transformations. Matrices, matrices and linear transformations, rank of a linear transformation, solutions of linear system of equations, permutations and determinants. Matrix polynomials, eigenvalues and eigenvectors, diagonalization of matrices, quadratic forms, special transformation in inner product spaces.

    Diferansiyel Geometri
    Affine space, Euclidean space, topological space, topological manifolds, differentiable manifolds, tangent vectors and tangent spaces, vector fields, directional derivative and covariant derivative, cotangent space, 1-forms, gradient, divergence and rotational functions, curvature theory, Serret-Frenet vectors, curvatures of curve and geometric interpretations, curvature axis, curvature sphere, spherical curves, involute and evolute curves, pairs of Bertrand curves, hypersurfaces, shape operator and fundamental forms.

    Kinematik
    Affine space, Euclidean space, coordinate systems and change of coordinates, Isometries of Euclidean spaces, motion and motion groups, 1-Parametered motions, derivative equations, velocity and acceleration, pole point and pole curves, Euler-Savary Theorem, envelopes, high order velocity, acceleration and acceleration poles, closed motions , areas of orbits, Holditch Theorem, spherical and spatial kinematics.

    Topoloji
    Topological spaces, neighborhoods, basis, subspace topology, Product and quotient topologies, Compactness, Tychonoff theorem. Heine-Borel theorem, Urysohn?s lemma, Tietze extension theorem, Stone-Cech compactification, Alexandroff single point compactification, Convergence of sequences and nets, connectedness, metrizable spaces.

    Reel Analiz
    Lebesgue measure. Measure theory and integration. Point set topology, Radon-Nykodym theorem, outer measure. Fubini?s theorem.

    Kompleks Analiz
    Complex numbers, metric spaces, Topology of complex numbers, Main properties and examples of analytical functions, complex integration, maximum modulus theorem, Cauchy integral formula, linear integrals, conformal mappings.

    İleri Analiz
    General measure and integration theory, general convergence theorems, Radon-Nikodym theorem, outer measure, Caratheodory extensions theorem, product measures; Fubini?s theorem, Riesz representation theorem.

    Fonksiyonel Analiz
    Linear spaces, basis, norms, completeness, linear transformations, continuity, Hahn-Banach theorem, separation of convex spaces, uniform boundedness, compactness, unbounded and closed operators, kernels and image spaces, weak, strong and uniform convergence, Hilbert spaces, projections, Riesz representation theorem, Fourier series.

    İleri Düzey Matematiksel Analiz
    Power series, directional derivative and gradient, extreme values and Lagrange multipliers, multivariable integration; uniform convergence of functions, open mapping theorem, closed graph theorem, functions defined via integrals.

    Nümerik Analiz
    Eigenvalue problems, Generalized eigenvalue problems, prediction of eigenvalues, Hyman method, Reducing to Frobenius form. Ordinary differential equations, initial, boundary value problems, Finite difference methods. Iterative methods for linear system of equations.

    Uygulamalı Matematiğin Temelleri
    Applications of linear algebra: network structures, least squares method, matrix factorization in eigenvalue problems, optimization problems, Lagrange multipliers, numerical solutions of linear and nonlinear systems, solutions of ordinary and partial differential equations.

    Kısmi Diferansiyel Denklemler
    Cauchy-Kowalevski theorem. First order linear and nonlinear equations, second order elliptic, parabolic and hyperbolic equations, existence uniqueness theorems. Well-posed problems, Green's function.

    Diferansiyel Denklemlerin Sayısal Çözümleri
    Definition of differential equations, definition of initial value problems in ordinary differential equations. Elementary theory, fundamental concepts of ordinary differential equations. Series and numerical methods. Review of single and multi-step methods for ODE, Runge Kutta methods, error measurement and Runge Kutta Fehlerg method, multi-step methods, higher order differential equations and their numerical solutions.

    Kısmi Dif. Denklemlerin Sayısal Çözümleri
    Finite difference methods: stability, convergence and qualitative properties; initial and boundary conditions, nonsmooth boundaries, parabolic equations. Explicit and implicit methods, stability, accuracy, variable coefficients, derived boundary conditions, solutions of tridiagonal systems, elliptic equations, iterative methods, speed of convergence; hyperbolic equations; Lax-Wendroff method, variable coefficients, conservations laws, stability and finite elements method.

    Sınır Değer Problemleri
    Numerical methods for solutions of ordinary differential equations, boundary value problems for ordinary differential equations, boundary value problems for partial differential equations, Fourier integrals and transforms, numerical methods.

    Matematiksel Programlama ve Modelleme
    Modeling techniques, modeling in linear programming, solution techniques and modeling in linear programming, sensitivity analysis in linear programming, dynamical programming.

    Nümerik Optimizasyon
    Linear programming. Modeling, solution methods. Duality theory in linear programming; nonlinear programming: first and second order conditions for unconstrained optimization problems, Lagrange multipliers, convexity in mathematical programming, Kuhn-Tucker theorem; discrete optimization.

    Matematiksel İstatistik
    One-sample and two-sample problems. Multivariate normal distribution. Mean and covariance estimates. Maximum likelihood estimation of mean vector and variance-covariance matrix, determining the outliers and normality check. Confidence intervals, Behrens-Fisher problem, test for a subvector, tests for linear restrictions, principle component analysis. Factor analysis, classification analysis, discriminant analysis, clustering analysis, correlation analysis, multivariable regression analysis, robust multivariable methods.

    Olasılık Teorisi
    Random variables, axioms of probability, expected value, characteristic functions, moments, distributions and distribution functions, moment generating functions, sums of random variables, sequences of random variables, independence, convergence and statistical applications. Introduction to abstract probability spaces and measure theory.

    İstatistik Teorisi
    General introduction to statistics, statistical modeling, features of random sampling, data reduction, properties of point estimators, hypothesis testing, interval estimation and decision theory.

    Uygulamalı İstatistiksel Analiz
    This course is designed for statistical analysis and applications. It includes the concepts of population and sampling, sampling techniques, classification of variables, definition of the data, sampling distributions, estimation of the population mean and variance, confidence intervals, testing of the population mean and variance, applications of correlation and regression analysis.

    İleri Düzey Olasılık Teorisi
    Sigma algebras, measure theory and probability; Lebesgue and Lebesgue-Stieltjes measures and distribution functions; measurable functions, random variables, integration theorems; comparison of Lebesgue and Riemann integrals. Jensen?s, Holder?s (Schwartz) and Minkowski inequalities, Lp spaces; Jordan-Hahn and Lebesgue measures and Radon-Nikodym theorem, signed measures; convergence of sequence of random variables, uniform integrability; product spaces and Fubini theorem. Independence, conditional expectation, conditional probability. Sums of random variables: law of large numbers and three series theorem. Martingales and Martingale convergence theorem, Poisson approximation, stochastic orderings.

    Bioistatistik Yöntemler
    Usage of SPSS program in medical problems, clinical experiments, cases analysis and forecasts. Logistic and Poisson regression, applications of generalized linear methods to medical data.

    Zaman Seri Analizi
    Autocovariance and autocorrelation functions, trends, seasonal and uniform effects, stationary processes, forecast and spectral analysis.

    Stokastik Süreçler

    Parametrik Olmayan İstatistik
    Single sample case. Chi-square test, Kolmogorov-Smirnov test, Run test. Correlated two sample case, Mc-Nemar test, Sign test, Walsh test. Uncorrelated two sample tests, Chi-square test, Median test, Mann-Whitney test, Wolfowitz test, Correlated k-sample case, Cochran q-test, Friedman test. Uncorrelated k-sample case, Kruskal Wallis test. Nonparametric correlation coefficients, Spearman correlation coefficient, Kendal correlation coefficient, concordance coefficient and related hypothesis tests.

    İleri İstatistiksel Modeller

    Matematiksel Finansa Giriş
    The aim of this course is to provide an introduction to the mathematical modeling of financial markets with particular emphasis on the pricing of derivative securities and the management of risk. Topics covered will include an introduction financial instruments and markets, fixed-income securities and rates of return, utility functions and optimal investment, simple models of random variation in prices, the fundamental concepts of arbitrage, replication, and completeness, and the use of arbitrage-free models for the valuation of securities and for the management of risk.

    Risk Yönetimi
    This course presents the most important mathematical concepts, methods and models used to value assets; select, maintain and optimize portfolios; and to manage risks. Topics covered include the following: returns, risks and utilities; quantification of risk-variance, shortfall risk, value at risk; portfolio analysis, diversification, correlations, principal components, sensitivity measures (greeks); asset valuation and pricing methods as capital markets theory, capital asset pricing model, efficient frontiers, arbitrage pricing theory, consumption/ accumulation and equilibrium models; risk management techniques ?diversification, immunization, insurance/ reinsurance, hedging; optimal asset allocation, portfolio optimization and dynamic delta hedging.

    Finansal Matematik
    Basic micro and micro economic principles, time value of money, simple and compound interest, accumulated value and present value, solution of interest problems, basic and general annuities, profit ratios, discounted cash flow, investment planning, amortization tables and loan funds, evaluation of investment returns, basic rules for portfolio diversification.

    Finansta Stokastik Hesaplamalar
    This course introduces martingales, Brownian motion, Ito integrals and Ito?s formula. This is done within the context of the Black-Scholes option pricing model and includes a detailed examination of this model. The course also develops Girsanov?s Theorem, which is used for change-of-measure arguments in finance. Applications presented are risk-neutral pricing and its connection with partial differential equations, currency options and forward measures in fixed-income models. Jump processes and their application to option pricing will be introduced.

    Finansta Hesaplamalı Modeller
    This course covers numerical methods relevant to solving the partial differential equations, which arise in option pricing. Exact solutions including Black Scholes and its relatives, finite difference methods, the connection with binomial models, interest rate models, early exercise, and techniques for calibration will be explained. This course initially presents standard topics in simulation including random variable generation, variance reduction methods and statistical analysis of simulation output. The course then addresses the use of Monte Carlo simulation in solving applied problems on derivative pricing discussed in the current finance literature. Application areas include the pricing of American options, pricing interest rate dependent claims, and credit risk.

    Sabit Gelirli Menkul Kıymet. & Kredi Riskine Giriş
    First half of this course introduces the most important securities traded in fixed income markets and the valuation models used to price them. Payoff characteristics and quotation conventions will be explained for treasury bills and bonds, STRIPS, defaultable bonds, mortgage-backed securities like Collateralized Mortgage Obligations and derivative securities like swaps, caps, floors, and swaps. Basic concepts will be explained such as the relation between yields and forward rates, duration, convexity, and factor models of yield curve dynamics. Second half of the course provides techniques for modeling credit risk. Pricing techniques for credit derivatives like Credit default swaps, basket default swaps and collateralized debt obligations (CDO?s) will be examined.

    Finansal Risk Analizi
    The content of this course consist of two parts. First part is pricing of financial derivative securities: option pricing with binomial and trinomial trees, Black-Scholes formula, exotic options, swaps, volatility, dynamical hedging strategies. Second part is financial risk measurement: market risk, credit risk, liquidity risk, model risk. This course also covers utility theory and insurance, individual risk theory, prioritization of risks.

    Seminer
    This course is designed to provide students with a chance to prepare and present a professional seminar on subjects of their own choice.
    Yüksek Lisans Tezi
    Program of research leading to M.S. degree arranged between the student and a faculty member. Students register to this course in all semesters while the research program or write up of thesis is in progress.

    Proje
    M.S. students working on a common area choose a research topic to study and present to a group under the guidance of a faculty member.

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