Graduate Program Course List

Compulsory Departmental Courses

Elective Courses



Compulsory Departmental Courses

ENG 101 - English Communication Skills

MATH541 - Algebra
Groups: quotient groups, isomorphism theorems, alternating and dihedral groups, direct products, finitely generated abelian groups, actions, Sylow theorems, nilpotent and solvable groups. Rings: ring homomorphisms, ideals, factorization in commutative rings, rings of quotients, PIDs, Euclidean domains, UFDs, factorization in polynomial rings. Modules: homomorphisms, exact sequences, vector spaces, tensor products. Fields: field extensions, the fundamental theorem of Galois theory, splitting fields, algebraic closure, the Galois group of a polynomial.

MATH552 - Complex Analysis
Analytic Functions as Mappings, Conformal Mappings, Complex Integration, Harmonic Functions, Series and Product Developments, Entire Functions, Analytic Continuation, Algebraic Functions.

MATH557 - Functional Analysis
Sets and mappings, countable sets, Metric Spaces, Complete Metric Spaces, Baire Category Theorem Compactness, Connectednes, Normed Spaces, Linear Topological Invariants, Hilbert Spaces, Cauchy Schwartz Inequality, Linear Operators, Bounded Operators, Unbounded Operators, Inverse Operators, Hahn-Banach Extension Theorems, Open mapping and Closed Graph Theorems, The Banach-Steinhaus theorem.

MATH587 - Applied Mathematics
Calculus of Variations: Euler-Lagrange equation, the first and second variations, necessary and suffcient conditions for extrema, Hamilton's principle, and applications to Sturm-Liouville problems and mechanics. Integral Equations: Fredholm and Volterra integral equations, the Green’s function, Hilbert-Schmidt theory, the Neumann series and Fredholm theory and Applications.

MATH589 - Graduation Seminar
Presentation involving current research given by graduate students and invited speakers.

MATH598 - Special Studies on Thesis Subject
Determination of the deficiencies about the Thesis topic. Deciding the activities (lecture listening and/or performing, studying with supervisor, participation in the seminars, conference, etc.) to be performed to compensate the deficiencies. Deciding whether or not the deficiencies have been compensated in a way deemed appropriate by the supervisor.

MATH599-1 - M.Sc. Thesis
Students with thesis option register to this course in all semesters starting from the beginning of their second semester after deciding to their thesis subject. The thesis subject is arranged between a student and a department member. The work consists of writing an overview of available results, solving original problems within the selected area and writing and defending a thesis.The student must pass an exam consisting of an oral defense of the thesis.

MATH599-2 - M.Sc. Thesis
Students with thesis option register to this course in all semesters starting from the beginning of their second semester after deciding to their thesis subject. The thesis subject is arranged between a student and a department member. The work consists of writing an overview of available results, solving original problems within the selected area and writing and defending a thesis. The student must pass an exam consisting of an oral defense of the thesis.

MATH599-3 - M.Sc. Thesis
Students with thesis option register to this course in all semesters starting from the beginning of their second semester after deciding to their thesis subject. The thesis subject is arranged between a student and a department member. The work consists of writing an overview of available results, solving original problems within the selected area and writing and defending a thesis.The student must pass an exam consisting of an oral defense of the thesis.

Elective Courses

MATH521 - Numerical Analysis I
Matrix and vector norms, Error analysis, Solution of Linear systems: Gaussian elimination and LU decomposition, Condition number, Stability analysis and computational complexity, Least square problems: Singular value decomposition, QR algorithm, Stability analysis, Matrix eigenvalue problems, Iterative methods for solving linear systems: Jacobi, Gauss-Seidel and Relaxation methods, Conjugate gradient type methods, Convergence analysis.

MATH522 - Numerical Analysis II
Iterative methods for nonlinear equations and nonlinear systems, Interpolation and approximation: polynomial trigonometric, spline interpolation, Least squres and minimax approximations, Numerical differentiation and integration: Newton-Cotes, Gauss, Romberg methods, Extrapolation, Error analysis

MATH524 - Finite Difference Methods for Partial Differential Equations
Finite difference method. Parabolic equations: explicit and implicit methods, Richardson, Dufort-Frankel and Crank-Nicolson schemes. Hyperbolic equations: Lax-Wendroff, Crank-Nicolson, box and leap-frog schemes. Elliptic equations. Consistency, stability and convergence of finite different methods for numerical solutions of partial differential equations.

MATH542 - Algebraic Number Theory
Integers, Norm, Trace, Discriminant, Algebraic integers, Quadratic integers, Dedekind domains, Valuations, Ramification in an extension of Dedekind domains, Different, Ramification in Galois extensions, Ramification and arithmetic in quadratic fields, The Quadratic Reciprocity Law. Ramification and integers in cyclotomic fields, Kronecker-Weber Theorem on Abelian extensions, Dirichlet’s Theorem on the finiteness of the class group, Dirichlet’s Theorem on units, Hermite-Minkowski Theorem, Fermat’s Last Theorem.

MATH543 - Group Theory I
Review of Elementary Group Theory, Groups of Matrices, Normal Closure and Core of a group, Group Actions on Sets, The Wreath Product of Permutation Groups, Decompositions of a Group, Series and Composition Series, Chain Conditions, Some Simple Groups, Sylow's Theorem, The Simplicity of the Projective Special Linear Groups, Solvable groups and Nilpotent groups, Locally Finite Groups, Locally Nilpotent Groups

MATH545 - Introduction to Authenticated Encryption
Fundamentals of cryptography, block ciphers, DES, AES competition, authentication, mode of operations, cryptographic hash functions, collision resistance, birthday paradox, Merkle Damgard construction, MD5, SHA-1, SHA-3 competition, Keccak, authenticated encryption, CAESAR competition, success probability of cryptanalytic attacks, LLR method, hypothesis testing, randomness testing.

MATH547 - Algebraic Geometry
Affine spaces, the Hilbert's basissatz, the Hilbert's nullstellensatz, the Zariski's topology, irreducible sets, algebraic varieties, curves, surfaces, sheafs, ringed spaces, preschemes, affine schemes, the equivalence between affine schemes and commutative rings, projective varieties, dimension, singular points, divisors, differentials.

MATH555 - Bernstein polynomials
Uniform continuity, uniform convergence, Bernstein polynomials, Weierstrass approximation theorem, positive linear operators, Popoviciu theorem, Voronovskaya theorem, simultaneous approximation, shape-preserving properties, De Casteljau Uniform continuity, uniform convergence, Bernstein polynomials, Weierstrass approximation theorem, positive linear operators, Popoviciu theorem, Voronovskaya theorem, simultaneous approximation, shape-preserving properties, De Casteljau algorithm, complex Bernstein polynomials, Kantorovich polynomials.

MATH556 - Applied Functional Analysis
Review on basics in functional analysis, spectral theory of self-adjoint operators in Hilbert spaces, semigroups of operators and their applications to evolution equations, optimal control in Banach spaces.

MATH562 - Theory of Differential Equations
Initial Value Problem: Existence and Uniqueness of Solutions; Continuation of Solutions; Continuous and Differential Dependence of Solutions. Linear Systems: Linear Homogeneous And Nonhomogeneous Systems with Constant and Variable Coefficients; Structure of Solutions of Systems with Constant and Periodic Coefficients; Higher Order Linear Differential Equations; Sturmian Theory, Stability: Lyapunov Stability and Instability. Lyapunov Functions; Lyapunov's Second Method; Quasilinear Systems; Linearization; Stability of an Equilibrium and Stable Manifold Theorem for Nonautonomous Differential Equations.

MATH563 - Difference Equations
The difference calculus, first order linear difference equations, second order linear difference equations, the discrete Sturmian theory, Green’s functions, disconjugacy, the discrete Riccati equation, oscillation, the discrete Sturm-Liouville eigenvalue problem, linear difference equations of higher order, systems of difference equations.

MATH564 - Impulsive Differential Equations
General Description of IDE: Description of mathematical model. Systems with impulses at fixed times. Systems with impulses at variable times. Discontinuous dynamical systems. Impulsive oscillator. Linear Systems of IDE: General properties of solutions. Stability of solutions. Adjoint systems, Perron theorem. Linear Hamiltonian systems of IDE. Stability of Solutions of IDE: Stability criterion based on first order approximation. Stability in systems of IDE with variable times of impulsive effect. Direct Lyapunov method. Periodic and Almost Periodic Systems of IDE: Nonhomogeneous linear periodic systems. Nonlinear periodic systems. Almost periodic functions and sequences. Almost periodic IDE. Integral Sets of Systems of IDE: Bounded solutions of nonhomogeneous linear systems. Integral sets of quasilinear systems with hyperbolic linear part and with non-fixed moments of impulse actions.

MATH565 - Dynamic Systems on Time Scales
Differentiation on time scales. Integration on time scales. The first-order linear differential equations on time scales, Initial value problem. The exponential function on time scales., The second-order linear differential equations on time scales, Boundary value problem. Green’s function. The Sturm-Liouville eigenvalue problem.

MATH571 - Topology
Topological spaces, homeomorphisms, and homotopy, Product and quotient topologies, Separation axioms, Compactness, Connectedness, Metric spaces and metrizability, Covering spaces, Fundamental groups, The Euler characteristic, Classification of surfaces, Homology of surfaces, Simple applications to Geometry and Analysis.

MATH572 - Differentiable Manifolds
Topological manifolds, Differentiable manifolds, Tangent and cotangent bundles, Differential of a map, Vector fields, Submanifolds, Tensors, Differential forms, Orientations on manifolds, Integration on manifolds, Stoke's theorem

MATH573 - Algebraic Topology
The fundamental group, covering spaces, the singular homology, the cellular homology, the simplicial homology, cohomology, universal coefficient theorems for homology, cup product and cross productand cohomology, the Künneth formula.

MATH574 - Riemannian Geometry
Review of Differentiable Manifolds and Tensor Fields, Riemannian Metrics, The Levi-Civita Connections, Geodesics and Exponential Map, Curvature Tensor, Sectional Curvature, Ricci Tensor, Scalar Curvature, Riemannian Submanifolds, The Gauss and Codazzi Equations.

MATH575 - Calculus on Manifolds
Euclidean Spaces, Manifolds, The Tangent Spaces, Vector Fields, Differential Forms, Integration on Manifolds, Stokes’ Theorem

MATH576 - Differential Topology
Manifolds and differentiable structures, tangent space, vector bundles, immersions, submersions, embeddings, transversality, the Sard’s theorem, the Whitney’s embedding theorem, the exponential map and tubular neighborhoods, manifolds with boundary, the Thom’s transversality theorem.

MATH577 - An Introduction to Low Dimensional Topology
Knots, Links and Their Invariants, Seifert Surfaces, Braids, Mapping Class Groups, Heegaard Decompositions, Lens Spaces and Surface Homeomorphisms, Surgery of 3-Manifolds, Branched Coverings

MATH580 - Special Studies I
Conducting literature survey on the research subject. Understanding the latest developments in literature on the chosen subject. Writing and presentation of a report on the subject.

MATH581 - Special Studies II
Conducting literature survey on the research subject. Understanding the latest developments in literature on the chosen subject. Writing and presentation of a report on the subject.

MATH582 - Approximation Theory
Uniform convergence, uniform approximation, Weierstrass approximation theorems, best approximation, Chebyshev polynomials, modulus of continuity, rate of approximation, Jackson’s theorems, positive linear operators, Korovkin’s theorem, Müntz theorems

MATH584 - Nonlinear Problems in Applied Mathematics
Boundary value problems for nonlinear second order ordinary differential equations on finite intervals, reducing nonlinear boundary value problems to a fixed point problem, application of the Banach and Schauder fixed point theorems, boundary value problems for nonlinear difference equations, application of the Brouwer fixed point theorem, positive solutions of nonlinear boundary value problems.

MATH585 - Mathematical Modelling
Modeling with first order differential equations: radioactivity, rate of growth and decay; single-species population models, a heat flow model, modeling RL and RC electric circuits; modeling with second order DEs: the motion of a mass on an elastic spring, modeling RLC electric circuits, diffusion models; modeling with systems of DEs: multiple-species population models, model of an electrical network; modeling with partial DEs: vibrating string, vibrating membrane, modeling of heat flow.

MATH591 - Analytical Probability Theory
Definition and properties of probability, conditional probability and independence, random variables, probability distributions, their types, classical distributions, moments, random vectors, independent random variables, moment-generating and characteristic function, sums of independent random variables, limit theorems.