Master of Science (M.Sc.) in Mathematics - Sarıyer - İstanbul - Koç Üniversitesi - I477

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Master of Science (M.Sc.) in Mathematics
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Koç Üniversitesi

Master of Science (M.Sc.) in Mathematics - Sarıyer - İstanbul

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Master of Science (M.Sc.) in Mathematics - Sarıyer - İstanbul Master of Science (M.Sc.) in Mathematics - Sarıyer - İstanbul
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Program Description

The Department of Mathematics offers graduate courses leading to M.Sc., and eventually to Ph.D., degree in Mathematics. The Master of Science program aims to provide a sound foundation for the students who wish to pursue a research career in mathematics as well as other related areas. The department emphasizes both pure and applied mathematics. Research in the department covers algebra, number theory, combinatorics, differential equations, functional analysis, abstract harmonic analysis, mathematical physics, stochastic analysis, biomathematics and topology.

Research Areas

Algebra and Number Theory
  • Ring Theory and Module Theory, especially Krull dimension, torsion theories, and localization
  • Algebraic Theory of Lattices, especially their dimensions (Krull, Goldie, Gabriel, etc.) with applications to Grothendieck categories and module categories equipped with torsion theories
  • Field Theory, especially Galois Theory, Cogalois Theory, and Galois cohomology
  • Algebraic Number Theory, especially rings of algebraic integers
Combinatorics
  • Combinatorial design theory, in particular metamorphosis of designs, perfect hexagon triple systems
  • Graph theory, in particular number of cycles in 2-factorizations of complete graphs
  • Coding theory, especially relation of designs to codes
  • Random graphs, in particular, random proximity catch graphs and digraphs
Differential Equations
  • Nonlinear ordinary differential equations of molecular dynamics
  • PDE’s of quantum mechanics: time dependent Schrodinger equation
  • Weak, in particular viscosity solutions, of second order equations
  • Asymptotic analysis of reaction diffusion equations
  • Gamma limits of non-convex functionals
  • Geometric flows and level set equations
  • Global behavior of solutions to nonlinear PDE’s
  • Dissipative dynamical systems generated by evolutionary PDE’s
  • PDE’s modeling nonlinear problems of continuum mechanics
Analysis
  • Banach algebras, especially the structure of the second Arens duals of Banach algebras
  • Abstract Harmonic Analysis, especially the Fourier and Fourier-Stieltjes algebras associated to a locally compact group
  • Geometry of Banach spaces, especially vector measures, spaces of vector valued continuous functions, fixed point theory, isomorphic properties of Banach spaces
Mathematical Physics
  • Differential geometric, topologic, and algebraic methods used in quantum mechanics
  • Geometric phases and dynamical invariants
  • Supersymmetry and its generalizations
  • Pseudo-Hermitian quantum mechanics
  • Quantum cosmology
Probability and Stochastic Processes
  • Mathematical finance
  • Stochastic optimal control and dynamic programming
  • Stochastic flows and random velocity fields
  • Lyapunov exponents of flows
  • Unicast and multicast data traffic in telecommunications
  • Probabilistic Inference
Statistics
  • Spatial Statistics, mostly on nearest neighbor methods and multi-species spatial patterns of segregation and association
  • Statistical Pattern Recognition, Classification
  • Statistical Depth
  • Statistics of Medicine concerning morphometric changes in organs and tissues, say, due to a disease
  • Scale, size, and shape comparisons of organs or tissues based on MRI data
  • Linear Models
  • Computationally Intensive Methods: Bootstrap and Randomization
Algebraic Geometry
  • Arithmetical Algebraic Geometry, Arakelov geometry, Mixed Tate motives
  • p-adic methods in arithmetical algebraic geometry, Ramification theory of arithmetic varieties
Geometry and Topology
  • Topology of low-dimensional manifolds, in particular Lefschetz fibrations, symplectic and contact structures, Stein fillings
  • Symplectic topology and geometry, Seiberg-Witten theory, Floer homology
  • Foliation and Lamination Theory, Minimal Surfaces, and Hyperbolic Geometry
Faculty
  • Attila Askar, Differential Equations
  • Mine Caglar, Probability and Stochastic Processes
  • Emre Alkan, Number Theory 
  • Elvan Ceyhan, Probability and Statistics
  • Tolga Etgu, Topology
  • Varga Kalantarov, Differential Equations
  • Sinan Unver, Algebraic Geometry
  • Selda Kucukcifci, Combinatorics
  • Ali Mostafazadeh, Mathematical Physics
  • Burak Ozbagci, Topology
  • Baris Coskunuzer, Geometric Topology 
  • Ali Ulger, Functional Analysis
  • Emine Sule Yazici, Combinatorics
Curriculum

The requirements for the degree of Master of Science in Mathematics are as follows.

The master program in Mathematics consists of a) at least 21 credit hours of course work, b) a master thesis and c) graduate seminar course. All students have to take
  • MATH 521 Algebra I (4 credits) (*)
  • MATH 531 Real Analysis I (4 credits) (*)
  • MATH 590 Graduate Seminar (0 credit)
  • MATH 595 Thesis

The total credit of these required courses is 8. In addition to these courses students have to take two sequence courses among the courses in areas listed below as a) to g), such as Math 531-532, Math 521-522, etc... These courses are not allowed to be 400 level courses. The areas are;
  • Algebra and Number Theory
  • Analysis
  • Topology and Geometry
  • Probability and Statistics
  • Discrete Mathematics
  • Applied Mathematics
  • Logic and Foundations of Science

To complete the credit requirements the courses may be chosen among courses offered by the department of Mathematics and the other departments of the Graduate School of Sciences and Engineering. At most 2 of the elective courses can be 400 level undergraduate courses. The choice of courses must be approved by the student's graduate advisor.

Students who have TA assignments  must take TEAC 500: Teaching Experience during the semesters of their assignments.  Students must also take ENGL 500: Graduate Writing course.
  • Math 503 Applied Mathematics
  • Math 504 Numerical Methods I
  • Math 506 Numerical Methods II
  • MATH 521 Algebra I
  • MATH 522 Algebra II
  • MATH 525 Algebraic Number Theory
  • MATH 527 Number Theory
  • MATH 528 Analytic Number Theory
  • MATH 531 Real Analysis I
  • MATH 532 Real Analysis II
  • MATH 533 Complex Analysis I
  • MATH 534 Complex Analysis II
  • MATH 535 Functional Analysis
  • MATH 536 Applied Functional Analysis I
  • MATH 537 Applied Functional Analysis II
  • MATH 538 Differential Geometry
  • MATH 541 Probability Theory
  • MATH 544 Stochastic Processes and Martingales
  • MATH 545 Mathematics of Finance
  • MATH 550 Advanced Ordinary Differential
  • MATH 551 Partial Differential Equations I
  • MATH 552 Partial Differential Equations II
  • MATH 563 Algebraic Coding Theory
  • MATH 564 Combinatorial Design Theory
  • MATH 565 Graph Theory
  • MATH 566 Topics in Module and Ring Theory
  • MATH 571 Topology
  • MATH 572 Algebraic Topology
  • MATH 579 Readings in Mathematics
  • MATH 580 Selected Topics in Topology I
  • MATH 581Selected Topics in Analysis I
  • MATH 582 Selected Topics in Analysis II
  • MATH 583 Selected Topics in Foundations of Mathematics
  • MATH 584 Selected Topics in Algebra and Topology
  • MATH 585 Selected Topics in Probability and Statistics
  • MATH 586 Selected Topics in Differential Geometry
  • MATH 587 Selected Topics in Differential Equations
  • MATH 588 Selected Topics in Applied Mathematics
  • MATH 589 Selected Topics in Combinatorics

Course Descriptions

Math 503
Applied Mathematics
Review of Linear Algebra and Vector Fields: Vector Spaces, Eigenvalue Problems, Quadratic Forms, Divergence Theorem and Stokes' Theorem. Sturm-Liouville Theory and Orthogonal Polynomials, Methods of Solution of Boundary Value Problems for the Laplace Equation, Diffusion Equation and the Wave Equation. Elements of Variational Calculus.

Math 504
Numerical Methods I
Review of Linear Algebra: linear spaces, orthogonal matrices, norms of vectors and matrices, singular value decomposition. Projectors, QR Factorization Algorithms, Least Squares, Conditioning and Condition Numbers, Floating Point Representation, Stability, Conditioning and Stability of Least Squares, Conditioning and Stability Analysis of Linear Systems of Equations.

Math 506
Numerical Methods II
Numerical Solution of Functional Equations, the Cauchy Problem and Boundary Value Problems for Ordinary Differential Equations. Introduction to the Approximation Theory of One Variable Functions. Finite - difference Methods for Elementary Partial Differential Equations. Monte Carlo Method and Applications.

MATH 521
Algebra I
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.

MATH 522
Algebra II
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.

MATH 525
Algebraic Number Theory
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.

MATH 527
Number Theory
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.

MATH 528
Analytic Number Theory
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.
Prerequisite: MATH 533 or consent of the instructor.

MATH 531
Real Analysis I
Lebesgue measure and Lebesgue integration on Rn, general measure and integration, decomposition of measures, Radon-Nikodym theorem, extension of measures, Fubini's theorem.

MATH 532
Real Analysis II
Normed and Banach spaces, Lp-spaces and duals, Hahn-Banach theorem, Baire category and uniform boundedness theorems, strong, weak and weak*-convergence, open mapping theorem, closed graph theorem. Prerequisite: MATH 531 or consent of the instructor.

MATH 533
Complex Analysis I
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.

MATH 534
Complex Analysis II
Runge's theorem, analytic continuation, Riemann surfaces, harmonic functions, entire functions, the range of an analytic function.
Prerequisite: MATH 533 or consent of the instructor.

MATH 535
Functional Analysis
Topological vector spaces, locally convex spaces, weak and weak* topologies, duality, Alaoglu's theorem, Krein-Milman theorem and applications, Schauder fixed point theorem, Krein-Shmulian theorem, Eberlein-Shmulian theorem, linear operators on Banach spaces.
Prerequisite: MATH 532 or consent of the instructor.

MATH 536
Applied Functional Analysis I
Review of linear operators in Banach spaces and Hilbert spaces; Riesz -Schauder theory; fixed point theprems of Banach and Schauder; semigroups of linear operators; Sobolev spaces and basic embedding theorems; boundary - value problems for elliptic equations; eigenvalues and eigenvectors of second order elliptic operators; initial boundary-value problems for parabolic and hyperbolic equations.

MATH 537
Applied Functional Analysis II
Existence and uniqueness of solutions of abstract evolutionary equations. Global non-existence and blow up theorems. Applications to the study of the solvability and asymptotic behavior of solutions of initial boundary-value problems for reaction diffusion equations, Navier-Stokes equations, nonlinear Klein-Gordon equations and nonlinear Schrödinger equations.

MATH 538
Differential Geometry

Differentiable manifolds; differentiable forms; integration on manifolds; de Rhamm cohomology; connections and curvature

MATH 541
Probability Theory
An introduction to measure theory, Kolmogorov axioms, independence, random variables, product measures and joint probability, distribution laws, expectation, modes of convergence for sequences of random variables, moments of a random variable, generating functions, characteristic functions, distribution laws, conditional expectations, strong and weak law of large numbers, convergence theorems for probability measures, central limit theorems.

MATH 544 Stochastic Processes and Martingales
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.
Prerequisite: MATH 541 or consent of the instructor.

MATH 545
Mathematics of Finance
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.

MATH 550
Advanced Ordinary Differential
Existence and uniqueness theorems; continuation of solutions; continuous dependence and stability, Lyapunovs direct method; differential inequalities and their applications; boundary-value problems and Sturm-Liouville theory.

MATH 551
Partial Differential Equations I
First order equations, method of characteristics; the Cauchy-Kovalevskaya theorem; Laplace's equation: potential theory and Greens's function, properties of harmonic functions, the Dirichlet problem on a ball; heat equation: the Cauchy problem, initial boundary-value problem, the maximum principle; wave equation: the Cauchy problem, the domain of dependence, initial boundary-value problem.

MATH 552
Partial Differential Equations II
Review of functional spaces and embedding theorems; existence and regularity of solutions of boundary-value problems for second-order elliptic equations; maximum principles for elliptic and parabolic equations; comparison theorems; existence, uniqueness and regularity theorems for solutions of initial boundary-value problems for second-order parabolic and hyperbolic equations.
Prerequisite: MATH 551 or consent of the instructor.

MATH 563
Algebraic Coding Theory
Error correcting coding theory. Hamming, Golay, cyclic, 2-error correcting BCH codes, Reed-Solomon, Convolutional, Reed-Muller and Preparata codes. Interaction of codes and combinatorial designs.

MATH 564
Combinatorial Design Theory
Balanced incomplete block designs, group divisible designs and pairwise balanced designs. Resolvable designs, symmetric designs and designs having cyclic automorphisms. Pairwise orthogonal latin squares. Affine and projective geometries. Embeddings and nestings of designs.

MATH 565
Graph Theory
Matchings, edge colorings and vertex colorings of graphs. Connectivity, spanning trees, and disjoint paths in graphs. Cycles in graphs, embeddings. Planar graphs, directed graphs. Ramsey Theory, matroids, random graphs.

MATH 566
Topics in Module and Ring Theory
Generalities on modules, categories, and functors. The socle and the Jacobson radical of a module. Semisimple modules. Chain conditions on modules. The Hopkins-Levitzki Theorem. The Wedderburn-Artin Theorem and its applications to linear representations of finite groups. The "Hom" functors and exactness. Injective modules. Essential monomorphisms, injective hulls. Projective modules. Superfluous epimorphisms, projective covers. Indecomposable direct sum decompositions of modules. The Krull-Remak-Schmidt-Azumaya Theorem. Krull dimension and Goldie dimension of modules and lattices.

MATH 571
Topology
Fundamental concepts, basis, subbasis of a topology, 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.

MATH 572
Algebraic Topology
Basic notions on categories and functors, the fundamental group, homotopy, covering spaces, the universal covering space, covering transformations, simplicial complexes and their homology.

MATH 579
Readings in Mathematics
Literature survey and presentation on a subject determined by the instructor.

MATH 580
Selected Topics in Topology I

MATH 581
Selected Topics in Analysis I

MATH 582
Selected Topics in Analysis II

MATH 583
Selected Topics in Foundations of Mathematics

MATH 584
Selected Topics in Algebra and Topology

MATH 585
Selected Topics in Probability and Statistics

MATH 586
Selected Topics in Differential Geometry

MATH 587
Selected Topics in Differential Equations

MATH 588
Selected Topics in Applied Mathematics

MATH 589
Selected Topics in Combinatorics

MATH 590
Graduate Seminar

TEAC 500Teaching Experience
Provides hands-on teaching experience to graduate students in undergraduate courses. Reinforces students' understanding of basic concepts and allows them to communicate and apply their knowledge of the subject matter.  

ENGL 500 Graduate Writing
This is a writing course specifically designed to improve academic writing skills as well as critical reading and thinking. The course objectives will be met through extensive reading, writing and discussion both in and out of class. Student performance will be assessed and graded by Satisfactory/Unsatisfactory

Non-credit presentation of topics of interest in mathematics through seminars offered by faculty, guest speakers and graduate students.
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