"Throughout this course, student will explore the algebraic structures of groups, rings, fields, and vector spaces, and develop the skills to apply abstract algebra techniques in solving both mathematical and practical problems. § Group theory: sub groups, cosets, Lagrange's theorem, homomorphisms, normal subgroups and quotient groups, permutation groups, simple groups. § Rings and fields: matrix rings, quaternions, ideals and homomorphisms, quotient rings, polynomial rings, principal ideal rings, Euclidean rings and unique factorization. § Fields and vector spaces: finite-dimensional vector spaces, algebraic field extensions, finite fields."

"The topics covered in this course include the existence and uniqueness of initial value problems, systems of linear differential equations with constant coefficients, matrix exponentials, the Laplace transform method for solving ODEs, stability analysis of linear and nonlinear systems of ODEs, and phase plane analysis."

"The main objectives are § To comprehend the properties and significance of reflexive spaces in functional analysis. § Exploring Weak Convergence and Weak Topology: To investigate weak convergence and its relation to the weak topology in functional analysis. § To grasp important theorems such as Mazur's lemma, the Banach-Alaoglu theorem in different cases, Tychonov's theorem, and the general Banach-Alaoglu theorem. § To delve into compact linear operators, their properties, as well as Fredholm and Hilbert- Schmidt operators. § To understand the concepts of spectrum and resolvent in the context of operators and the spectral mapping theorem. § To study spectral theory for various types of operators such as self-adjoint, compact, bounded, and self-adjoint operators. Including the spectral family of a bounded self-adjoint operator. § To explore the spectral representation of bounded self-adjoint operators. § To delve into Banach algebras, spectral theory within Banach algebras, and commutative algebra in the context of functional analysis."

"This course is a continuation of Numerical Analysis, in which postgraduate students will study advanced algorithms to obtain approximate numerical results for mathematical problems as mentioned below: 1. Numerical methods for solving nonlinear systems: Fixed-point method, Newton's method. 2. Numerical methods for solving initial value problems (single-step methods and multistep methods along with studying error, stability, and convergence of the algorithms). 3. Numerical methods for solving boundary value problems: Shooting method, method of finite differences along with studying error, stability, and convergence of the algorithms. 4. Programming of studied algorithms using MATLAB & Python."

"Within the Master of Science in Mathematics program, this course delves into the ethical dimensions and editing practices fundamental to scientific research within the realm of mathematics. It aims to equip students with a profound understanding of ethical challenges specific to mathematical research, ethical editing norms, and the integration of ethical considerations in mathematical inquiry."

"Introduction to matrices: Groups of matrices, Groups of matrices as metric spaces, Matrixgroups, Some examples of matrix groups, Complex matrix groups as real matrix groups, Continuous homomorphisms of matrix groups, Continuous group actions, The matrix exponential and logarithm functions. Lie algebras for matrix groups: Differential equations in matrices, One parameter subgroups, Curves, tangent spaces and Lie algebras, Some Lie algebras of matrix groups, SO (3) and SU (2) , SL2(C) and the Lorentz group. Quaternions, Clifford algebras and some associated groups: Algebras, Linear algebra over a division algebra, Quaternions, Quaternionic matrix groups, The real Clifford algebras, The spinor groups, Thecenters of spinor groups, Finite subgroups of spinor groups. Matrix groups as Lie groups: Smooth manifolds, Tangent spaces and derivatives, Liegroups, Some examples of Lie groups, Some useful formula in matrix groups, Matrix groups are Lie groups, Not all Lie groups are matrix groups. Homogeneous spaces: Homogeneous spaces as manifolds, orbits, Projectivespaces, Grassmannians. Connectivity of matrix groups: Connectivity of manifolds, Examples of path-connected matrix groups, The path components of a Lie group, Another connectivity result. Compact connected Lie groups and their maximal tori;Tori, Maximal tori in compact Lie groups, The normalizer and Weyl group of a maximal torus."

"An introduction to methods of Harmonic Analysis. Covers convergence of Fourier series, Hilbert transform, Caldéron-Zygmund theory, Littlewood-Paley theory, Fourier restriction, and applications."

Differentiable functions, inverse, and implicit function theorems. Theory of manifolds: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields and differential forms: Frobenius' theorem, integration on manifolds, Stokes' theorem in n dimensions, de Rham cohomology. Lie Groups

Arithmetic functions, partial summation, Dirichlet series and Euler products. The Riemann zeta function, its functional equation and analytic continuation. Evaluation at even integers. Zeros of the zeta function, and the Riemann Hypothesis. The Prime Number Theorem. Dirichlet characters and Gauss sums. Dirichlet L-functions, their analytic continuation and functional equations.

"The course covers the following: Unbounded Linear operators. Closed and Closable Linear Operators. Perturbations of closed operators. Invertible unbounded operators. Spectral Theory for Unbounded Linear Operators. Symmetric and Self-Adjoint Linear Operators. Definition and Elementary Properties of Maximal Monotone Operators. Sobolev spaces: Weak derivatives, Sobolev spaces, Properties of weak derivatives, Completeness of Sobolev spaces, The Hilbert-Sobolev spaces structure. Approximation by smooth functions, Local approximation in Sobolev spaces , Global approximation in Sobolev spaces . Sobolev spaces with zero boundary values.Sobolev inequalities. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension."

"Stability theory of differential equations and Lyapunov techniques, the La Salle""s invariance principle, region of attraction, linearization, feedback control, controllability and obeservability of linear and nonlinear systems."

"Topics include Laplace’s Equation, Harmonic Functions, Poisson Integral, Maximum Principle, NeumannProblem , Classical Dirichlet Problem, ‎Harnack's Inequality, Super Harmonic Functions and their properties, Perron-Wiener Method , Boundary Limit Theorem, Harmonic Measure , Green ‎functions, Negligible Sets, Capacity and Energy.‎"

"Ring, subring, ideal, operations on ideals, zero divisor elements, nilpotent elements, invertible elements, integral domain, field, prime ideal, maximal ideal, homomorphism of rings, quotient ring (with partitions and equivalence relations), isomorphism Theorem, The Chinese remainder Theorem, partially ordered sets, Zorn’slemma and existence of maximal ideals, nilradical, Jacobson ideal, localization of rings, Polynomial ring, Power series ring. Introduction to Noetherian rings: Module, submodule, operations on submodules, homomorphism of modules, finitely generated module, Noetherian modules, Noetherian rings, Hilbert’s Basis Theorem, Cohen’s criterion, primary decomposition. Introduction to UFD (unique factorization domain): Euclidean domain, PID (principal ideal domain), associate elements, irreducible element, prime element, connections between prime and irreducible, unique factorization domain, Hierarchy among UFD, PID, and Euclidean domains. Polynomial ring over UFD, Eisenstein’s Criterion. Introduction to integral dependence and valuations: Integral elements, integrally closed domain, going up, going down, totally ordered group, valuation, valuation domain, Prüfer domain, fractional ideal, invertible ideal, Dedekin domain."

Topics include Laplace’s Equation, Harmonic Functions, Poisson Integral, Maximum Principle, Neumann Problem , Classical Dirichlet Problem, ‎Harnack's Inequality, Super Harmonic Functions and their properties, Perron-Wiener Method , Boundary Limit Theorem, Harmonic Measure , Green ‎functions, Negligible Sets, Capacity and Energy.‎

Introduction to finite difference methods, Elementary finite difference quotients, Basic Aspects of finite-difference equations, Stability, Consistency, Convergence, Explicit and implicit methods, Crank Nicolson implicit scheme, Basics properties ofholomorphic functions in one dimension: Maximum Modulus principle, Liouville Theorem, Zeros, Normal Family. Holomorphic functions in several variables (Domain of holomorphy, Riemann mapping Theorem, Hartogs extension theory) also including the difference between one and several variable theory. Plurisubharmonic functions and potential analysis in several complex variables.

"The Master thesis course in Mathematics is designed to engage students in advanced scientific research within specific areas of mathematics. This course aims to cultivate skilled mathematicians capable of conducting rigorous research and contributing to the field's advancement. Students will focus on a specialized research topic approved by the Department Council of Mathematics. Under the guidance of a chosen supervisor, students will delve into theoretical and practical aspects of their chosen area, honing their research abilities and fostering higher cognitive and self-learning skills. Throughout the thesis process, students will emphasize ethics, professionalism, and adherence to academic standards. By the course's conclusion, students are expected to produce a comprehensive thesis project and deliver an oral presentation showcasing their research findings. The course offers flexibility for students to select a supervisor and project topic from a list of proposals, ensuring a tailored and enriching research experience. Through this thesis course, students will develop critical thinking, communication, and analytical skills essential for success in advanced mathematical research and professional endeavors"