Program DescriptionThe 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 AreasAlgebra 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 2factorizations 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 nonconvex 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 FourierStieltjes 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
 PseudoHermitian 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 multispecies 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
 padic methods in arithmetical algebraic geometry, Ramification theory of arithmetic varieties
Geometry and Topology
 Topology of lowdimensional manifolds, in particular Lefschetz fibrations, symplectic and contact structures, Stein fillings
 Symplectic topology and geometry, SeibergWitten 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 531532, Math 521522, 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 503Applied Mathematics
Review of Linear Algebra and Vector Fields: Vector Spaces, Eigenvalue Problems, Quadratic Forms, Divergence Theorem and Stokes' Theorem. SturmLiouville 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 504Numerical 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 506Numerical 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 521Algebra I
Free groups, group actions, group with operators, Sylow theorems, JordanHölder 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 522Algebra 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, WedderburnArtin theorem, finitely generated modules over a principal ideal domain, basis theorem for finite abelian groups.
MATH 525Algebraic 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 527Number Theory
Method of descent, unique factorization, basic algebraic number theory, diophantine equations, elliptic equations, padic numbers, Riemann zeta function, elliptic curves, modular forms, zeta and Lfunctions, ABC conjecture, heights, class numbers for quadratic fields, a sketch of Wiles' proof.
MATH 528Analytic Number Theory
Primes in arithmetic progressions, Gauss' sum, primitive characters, class number formula, distribution of primes, properties of the Riemann zeta function and Dirichlet Lfunctions, 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 531Real Analysis I
Lebesgue measure and Lebesgue integration on Rn, general measure and integration, decomposition of measures, RadonNikodym theorem, extension of measures, Fubini's theorem.
MATH 532Real Analysis II
Normed and Banach spaces, Lpspaces and duals, HahnBanach 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 533Complex 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 534Complex 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 535Functional Analysis
Topological vector spaces, locally convex spaces, weak and weak* topologies, duality, Alaoglu's theorem, KreinMilman theorem and applications, Schauder fixed point theorem, KreinShmulian theorem, EberleinShmulian theorem, linear operators on Banach spaces.
Prerequisite: MATH 532 or consent of the instructor.
MATH 536Applied 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 boundaryvalue problems for parabolic and hyperbolic equations.
MATH 537Applied Functional Analysis II
Existence and uniqueness of solutions of abstract evolutionary equations. Global nonexistence and blow up theorems. Applications to the study of the solvability and asymptotic behavior of solutions of initial boundaryvalue problems for reaction diffusion equations, NavierStokes equations, nonlinear KleinGordon equations and nonlinear Schrödinger equations.
MATH 538
Differential Geometry
Differentiable manifolds; differentiable forms; integration on manifolds; de Rhamm cohomology; connections and curvature
MATH 541Probability 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, DoobMeyer decomposition, Doob's martingale convergence theorem, characterization of square integrable martingales, RadonNikodym 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 BlackScholes equation, stochastic differential equations, FeynmanKac theorem, CoxIngersollRoss and Vasicek term structure models, Girsanov's theorem and risk neutral measures, HeathJarrowMorton term structure model, exchangerate 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; boundaryvalue problems and SturmLiouville theory.
MATH 551
Partial Differential Equations I
First order equations, method of characteristics; the CauchyKovalevskaya 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 boundaryvalue problem, the maximum principle; wave equation: the Cauchy problem, the domain of dependence, initial boundaryvalue problem.
MATH 552
Partial Differential Equations II
Review of functional spaces and embedding theorems; existence and regularity of solutions of boundaryvalue problems for secondorder elliptic equations; maximum principles for elliptic and parabolic equations; comparison theorems; existence, uniqueness and regularity theorems for solutions of initial boundaryvalue problems for secondorder parabolic and hyperbolic equations.
Prerequisite: MATH 551 or consent of the instructor.
MATH 563
Algebraic Coding Theory
Error correcting coding theory. Hamming, Golay, cyclic, 2error correcting BCH codes, ReedSolomon, Convolutional, ReedMuller 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 HopkinsLevitzki Theorem. The WedderburnArtin 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 KrullRemakSchmidtAzumaya 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 handson 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
Noncredit presentation of topics of interest in mathematics through seminars offered by faculty, guest speakers and graduate students.