Graduate Program

Course Credit: 4

Theory of groups and representations is key to many branches of sciences and mathematics such as studying symmetries in geometry, Conservation laws of physics are related to the symmetry of physical laws under various transformations, group theory predicted the existence of many elementary particles before they were found experimentally. The structure and behavior of molecules and crystals depends on their different symmetries. Group theory shows up in many other areas of geometry and Topology. Examples include different kinds of groups, such as the fundamentalgroup of a space. Classical problems in algebra have been resolved with group theory. Cryptography uses a lot of group theory. Different cryptosystems use different groups. This a course that may be studied for its own sake or from view point of applications.

Course Credit: 4

A ring is an important fundamental concept in algebra and includes integers, polynomials and matrices as some of the basic examples. Ring theory has applications in number theory and geometry. A module over a ring is a generalization of vector space over a field. The study of modules over a ring R provides us with an insight into the structure of R. In this course we shall develop ring and module theory leading to the fundamental theorems of Wedderburn and some of its applications.

Course Credit: 4

Number theory is a broad subject with many strong connections with other branches of mathematics. The idea of the course is to give a solid introduction to quadratic fired and algebraic number theory. It will be bridging the gap between elementary number theory and the systematic study of advanced topics.

Course Credit: 4

This is a continuation of a course on introduction to measure theory in n-dimensional Euclidian space offered in the senior year of the undergraduate program. The general abstract theory of measure, integration, and their applications is in order for a complete knowledge of the subject.

Course Credit: 4

Complex analysis is indeed is one of the classical subjects with most of the main results extending back into the nineteenth century and earlier. Yet, the subject is far from dormant. It is a launching point for many areas of research and it continues to find new areas of applicability, from pure mathematics to applied physics. A graduate course in Complex Analysis has the potential to address many learning outcomes that are important in studying mathematics. The subject has connections to several other mathematical areas and it provides students with opportunities to build a deeper cognitive mathematical framework.

Course Credit: 4

Geometry has grown out of efforts to understand the world around us, and has been a central part of mathematics from the ancient times to the present. Topology has been designed to describe, quantify, and compare shapes of complex objects. Together, geometry and topology provide a very powerful set of mathematical tools that is of great importance in mathematics and its applications. This module will introduce the students to the mathematical foundation of modern geometry based on the notion of distance. We will study metric spaces and their transformations. Through examples, we will demonstrate how a choice of distance determines shapes, and will discuss the main types of geometries. An important part of the course will be the study of continuous maps of spaces. A proper context for the general discussion of continuity is the topological space, and the students will be guided through the foundations of topology. Geometry and topology are actively researched by mathematicians and we shall indicate the most exciting areas for further study

Course Credit: 4

This course will cover the foundations of functional analysis in the context of topological linear spaces and normed linear spaces. This course is a natural follow of the course Topology; while the main focus of Convex sets and hyperplanes, seminorms, locally convex spaces, weak topology, compact convex sets, duality in Banach spaces. Then the linear analysis is on Hilbert spaces with its rich geometrical structures will work with normed linear spaces. The Big Theorems (Uniform Boundedness, Open Mapping and Closed Graph) will be presented and several applications will be analyzed. The important notion of duality will be developed in Banach and Hilbert spaces and an introduction to spectral theory for compact operators will be given. Moreover, Bilinear and quadratic forms, symmetric operators, normal and self-adjoint operators and spectral analysis in Hilbert Spaces for bounded self-adjoint operators and unbounded self-adjoint operators will be analyzed. Despite working in this more general framework many results on Nonlinear Compact Operators and Monotonicity will be re-introduced in this course in more general form. For example, Banach Fixed point theorem with applications, Schauder fixed point theorem, Frechet derivative, Newton’s method for nonlinear operators, positive and monotone operators will be introduced.

Course Credit: 4

Lie groups are continuous groups of symmetries, like the group of rotations of n-dimensional space or the group of invertible n-by-n matrices. In studying such groups we can use tools from calculus to linearize our problems, which leads us to the notion of a Lie algebra: a vector space with an antisymmetric product associated to any Lie group, which remembers everything about its algebraic structure. For example, the Lie algebra associated with the group of rotations of 3-space is just 3-dimensional Euclidean space with (twice) the vector cross product.

Course Credit: 4

A fuzzy set can be defined mathematically by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set.The aim is to introduce the main components of fuzzy set theory and to overview some of its applications. In constructing a model, we always attempt to maximize its usefulness.The aim to construct a model is closely connected with the relationship among three key charactistics, such as, complexity, credibility, and uncertainty.Lotfi A. Zadeh [1965b] has introduced a theory whose object ‘fuzzy set’ are sets with boundaries that are not precise. The membership in a fuzzy set is not a matter of affirmation or denial, but rather a matter of a degree.Fuzzy set theory has become an essential part of the curriculum in different discipline of science, especially in applied mathematics, engineering, and technology. The concept of fuzzy sets, fuzzy topology, and fuzzy algebra have been used in decision making in fuzzy environment, fuzzy cluster analysis, fuzzy control system.The concepts of fuzzy subgroupoids, fuzzy subgroups, fuzzy monoid, fuzzy normal subgroups, fuzzy automata theory and pattern recognition, and coding theory have been dealt with.

Course Credit: 4

This course is intended to develop rigorous practical and analytic skills in differential and integral equations (DIE). It is intended to illustrate various applications of differential and integral equation to technical problems as well. The laws of nature are expressed as differential and integral equations. Scientists and engineers must know how to model the world in terms of differential and integral equations, and how to solve those equations and interpret and analyze the solutions. This course focuses on theoretical aspects of linear and nonlinear differential and integral equations and their applications in science and engineering. More details are given in the course goals below.

Course Credit: 4

Operations Research (OR), also called Management Science, is the study of scientific approaches to decision-making problems. Through mathematical modeling, it seeks to design, improve and operate complex systems in the best possible way. This is a comprehensive course covering several areas of OR. The module covers topics that include: linear programming, bounded variable simplex algorithm, transportation and assignment problem, job sequencing, network model, dynamic programming, integer programming, game theory and nonlinear programming. Analytic techniques and computer packages will be used to solve problems facing different real life application oriented decision making problems.

Course Credit: 4

There are a lot of naturally occurred processes which can be described using ordinary and partial differential equations (ODEs and PDEs). A thorough knowledge of these processes are acquired solving the relevant equations. This course deals with numerical methods of various types of ordinary and partial differential equations. In particular, finite difference methods (FDMs) for linear and nonlinear ordinary differential equations as well as for elliptic, parabolic, hyperbolic partial differential equations will be discussed. Moreover, students will learn finite element methods (FEMs) in details.

Course Credit: 4

Geometry of Differential Manifolds is based on three dimensional basic vector geometry of curves and surfaces with calculus. Understanding of this course students will precede to learn other areas of mathematics such as Differentiable Manifolds, Riemannian Manifolds, Theory of Relativity and cosmology etc. Upon the successful completion of this course students will able to apply the concepts of surfaces to find which surface are minimal surfaces and also to know Weingarten, Gauss and Codazzi equations, Theorema Egreegium, fundamental theorem of surface theory etc. Students will know the concepts of developable surfaces, ruled surfaces, Gaussian curvature, Geodesics, Geodesic curvature, Liouvilles formula, Clairaut’s theorem, Bonnet’s formula and Gauss-Bonnet theorem. Students will learn about Conformal, isometric and geodesic mapping, Tissot’s theorem. Theory of differential functions, charts, atlases, differentiable manifolds, smooth map on Manifolds, Tangent space, Tangent bundles, - vector fields and Lie brackets of vector fields on Manifolds, φ - related vector fields.

Course Credit: 4

Dynamics deals with change which evolve in time. Whether the system in question settles down to equilibrium, keep repeating in cycles, or does something more complicated, it is the dynamics that we use to analyze the behavior in various places of science.

Course Credit: 4

To provide students with the mathematical tools used to study and solve a variety of problems in biology at different scales. Mathematical Biology is one of the most rapidly growing and exciting areas of Applied Mathematics. This is because recently developed experimental techniques in the biological sciences, are generating an unprecedented amount of quantitative data.

Course Credit: 4

Operations activities, such as forecasting, choosing a location for an office or plant, allocating resources, designing products and services, scheduling activities, and assuring and improving quality are core activities and often strategic issues in business organizations. Production Management or Operations Management is the management of systems or processes that create goods and/or services. The material in this course is intended as an introduction to the field of operations management. The field of operations management is dynamic, and very much a part of the good things that are happening in business organizations. Much of what the students learn will have practical application.

Course Credit: 4

It is a follow up course with prerequisites of ‘Stochastic Calculus’ and ‘Introduction to Mathematical Finance’. The idea and concept of risk management in Banks, Insurance and other financial institutions are covered from stochastic modelling perspectives. Stock markets volatility modeling and stock derivative price modelling are good part of the course. Back-testing risk measures for stock return and other risk measures for derivative investments are covered. In addition to celebrated Black and Scholes model several non-normal derivative pricing models are covered

Course Credit: 4

Any mathematical topic not covered in other courses may be offered under this title. The course-teacher will prepare an outline of the course and obtain the approval of the departmental academic committee.

Course Credit: 8

Each thesis student is required to work on a specific topic in different fields of mathematics and its applications, prepare a thesis report as a partial fulfillment of requirements for the degree. The thesis work may include original research, review work, and applications of mathematics and may involve fieldwork and use of technology.

Course Credit: 4

Viva Voce on courses taught in the M. S. program.