Overview

Abstract algebra, among the ancient branches of mathematics, serves as a foundational and unifying framework within the vast landscape of mathematics, establishing crucial connections with diverse mathematical disciplines. It explores abstract structures such as groups, rings, and fields, providing a common language and set of tools applicable across various mathematical contexts. In number theory, abstract algebra plays a pivotal role in understanding the properties and relationships among integers. In geometry, it manifests as symmetries and transformations, while in topology, algebraic structures like groups are instrumental in characterizing spaces. Furthermore, abstract algebra finds applications in cryptography, coding theory, and mathematical physics. Its versatility and broad-reaching impact highlight the interconnected nature of mathematical disciplines, demonstrating that the abstract concepts developed within algebraic frameworks have profound implications and applications throughout the mathematical landscape.

In the realm of linear algebra, the connections with abstract algebra are particularly noteworthy. Linear algebra, a branch of mathematics focused on vector spaces and linear mappings, is deeply intertwined with abstract algebraic concepts. Groups of linear transformations and matrices form algebraic structures, showcasing the interplay between the two fields. Abstract algebra provides a theoretical foundation for understanding the algebraic properties of vector spaces, paving the way for applications in computer science, physics, and engineering. The study of linear algebraic structures, such as vector spaces and linear transformations, underscores the harmonious relationship between abstract algebra and linear algebra, emphasizing their collective impact on diverse mathematical applications.

Abstract and linear algebras not only enrich our theoretical understanding of mathematical structures but also empowers us to address real-world challenges in various scientific and technological domains. Its enduring importance underscores the interconnected and interdisciplinary nature of mathematics, highlighting how such mathematical branch continues to shape and advance the entire mathematical landscape.

Research Approaches

Department of Mathematics has a well-established stronghold in algebra, possessing expertise that covers a wide range of the subject. Currently, we are actively involved in research projects associated with:

• Finite Groups Theory
• Semigroup Theory
• Graded Rings and Modules
• Quantum Groups
• Derivation on Rings
• Coding Theory
• BCK-Algebra
• Lie Algebra
• Number Theory
• Representation Theory
• Fuzzy Set Theory and its applications

Overview

Analysis is a fundamental component of modern mathematics, contributing to numerous other academic disciplines. This field encompasses the study of various spaces and functions, ranging from elementary to advanced stages. The challenges encountered in analysis have increased the growth of diverse areas within science and technology. Its extensive applications and interdisciplinary connections span a wide array of disciplines, such as physics, engineering, economics, signal processing, and medical imaging.

Research Approaches

The Analysis section is distinguished as one of the largest research sections in the Department of Mathematics at KAU, specializing in a broad array of areas including complex and harmonic analysis, real and functional analysis, special functions, sequences spaces, operator theory, fixed point theory, and approximation theory. Our research not only holds significant potential for local industry applications but also offers practical solutions to a variety of complex phenomena. Our section is renowned for enhancing the university's global reputation through innovative and enterprising research. It plays a pivotal role in fostering relationships with local industries in the Kingdom, thereby significantly impacting the national economy. Furthermore, our collaborations extend beyond national borders, engaging with international scholars on similar research topics, thereby establishing a wide-reaching network of academic and practical research. Our section’s influence is shown in our dynamic research culture that attracts PhD students and visiting researchers globally. These collaborations are central to our pursuit of novel and emerging ideas in the field of analysis. Supported by numerous grants and projects, our researchers have made notable advancements across various domains. The success of our programs is reflected in the achievements of our graduates, who have completed their master’s and doctoral degrees in analysis, now contributing significantly to the advancement of higher education in the Kingdom.

Overview

Applied mathematics includes important practical problems.

Research Approaches

Our areas of specialization encompass a wide array of fundamental themes within the realm of applied mathematics, addressing various pertinent aspects such as: mathematical biology, financial mathematics, dynamical systems, elasticity, fluid dynamics, optimal control theory, spectral theory, power systems, optimization, artificial intelligence, crowd dynamics. We collaborate with other research groups, linking our work through application of optimal control theory in epidemiology, virology and ecology, application of control theory in chaotic systems and secure communications, mathematical modeling and control of infectious diseases, study of crowd dynamics in Hajj and Umrah, application of artificial intelligence in electric power systems and renewable energy, application of optimization in energy efficiency.

Overview

Differential equations, fractional differential equations, dynamical systems, stability and instability properties of dynamical systems, Impulsive hybrid state dependent delay systems, nonlinear boundary value problems, existence and uniqueness theory, application of fixed-point theory, analytic and numerical methods, mathematical modeling and numerical simulation of partial differential equations with applications in Biomathematics, blood flow problems, Newtonian and non-Newtonian fluid mechanics, mathematical biology.

Research Approaches

Our areas of research cover a variety of important fields such as analytic approximation of solutions of nonlinear problems associated with the real world problems, stability theory of differential and integro-differential systems, investigation of fractional differential equations with the sphere of study ranging from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions, noise reduction problems, structural stability, plated structures and shells, nonlinear wave phenomena, wavelets, solitons theory, singular perturbation in the diffusion process. These research fields have a great potential in the industry and highly useful to provide the descent solution to the practical problems arising in various phenomena of transport theory, the transfer of neutrons through thin plates and membranes in nuclear reactors, in the propagation of radiation through the atmosphere of planets and stars, and in several other transport problems, nonlinear optics, unsteady aerodynamics and aero elastic phenomena, biophysics, blood flow phenomena, visco-elasticity, visco-elastic panel in supersonic gas flow, fluid dynamics, electro dynamics of complex medium, many models of population growth, polymer rheology, neural network modeling, sandwich system identification, materials with fading memory, mathematical modeling of the diffusion of discrete particles in a turbulent fluid, heat conduction in materials with memory, theory of lossless transmission lines, theory of population dynamics, compartmental systems, nuclear reactors and mathematical modeling of a hereditary phenomenon, etc. In view of the extensive applications of differential equations, our research areas have a great impact in the research activities of the university. Furthermore, the researchers working on different branches of differential equations have extensive international research collaboration with their counterparts in the USA, France, Greece, China, Pakistan, Egypt, India, Italy, Romania, Turkey, etc. In fact, researcher working on the topic of differential equations have the potential to interact with the chemical, mechanical and electronics industries of the Kingdom and thus the research carried out on differential equations and allied fields have a great impact on the economy of the Kingdom. The researchers working on different classes of differential equations developed a vibrant and productive environment at the Mathematics Department, King Abdulaziz University (KAU) by hosting many international researchers, MSc, and PhD students. Visitors across the globe visit King Abdulaziz University on a regular basis to discuss novel and emerging ideas on differential equations and their applications with their counterparts at KAU. The publication rate of the researchers working on differential equations and their applications is quite impressive and contributed significantly to the overall research output of the university. There also emerged a research group: Nonlinear Analysis and Applied Mathematics (NAAM) because of continuous and enormous efforts of the researchers working in the areas of different disciplines of differential equations. One of the significant outcomes of the NAAM-research group was the launching of an international journal: Bulletin of Mathematical Sciences (BMS), which received a world-wide reputation during a short span of time and the world-renowned mathematicians published their excellent works in this journal. It is worth-mentioning that several students received their MSc and PhD degrees in the research area of differential equations under the umbrella of the NAAM-Research Group and they are now serving in different universities of the Kingdom of Saudi Arabia (KSA) and neighboring countries. This clearly reflects the contribution of the NAAM-Research Group toward building infrastructure in the higher education sector of the KSA.

Overview

Numerical Analysis is the mathematics branch responsible for designing effective ways to find numerical solutions to complex mathematical problems. Most mathematical problems from science and engineering are very complex and sometimes cannot be solved directly. Therefore, measuring complex mathematical problems is very important to make it easier to solve. Due to the great advances in computational technology, numeracy has become very popular and is a modern tool for scientists and engineers. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Numerical analysis finds application in all fields of engineering, physical sciences, life and social sciences, medicine, business and even the arts. Some of the real world applications are optimizing engineering designs, predicting economic trends, advancing medical diagnostics, weather prediction, etc. Many great mathematicians, like Newton, Lagrange, Gauss, Euler, developed important algorithms in numerical analysis.

Research Approaches

Our areas of expertise cover many of the common themes of numerical analysis, including: numerical computation, numerical methods, numerical simulation, computational mathematics, scientific computing depending on which aspect of this broad subject is being emphasized. Members of our group have been working on a difficult issue (challenging problems) in science and engineering that is directly related to numerical computation. Our researchers in Mathematical Modelling and Applied Computation Research Group (MMAC) at KAU cover a wide range of activities in numerical computing, with particular emphasis on the development, analysis, and root finding. Although we emphasize the numerical solution of differential equations, fluid flow problems, thermal and petroleum engineering problems. To enhance our research, we collaborate with other researchers at other universities throughout the globe. Our researchers were awarded numerous grants and projects to advance research in many areas.

Overview

Topology and differential geometry are foundational branches of mathematics, each playing a crucial role in understanding the structure and properties of spaces. Topology focuses on the intrinsic properties that remain unchanged under continuous deformations, providing a geometric perspective on space that is independent of specific metric considerations. Differential geometry, on the other hand, studies the interplay between curvature and geometry, particularly in the context of smooth manifolds. Both fields find profound connections with other mathematical disciplines. In physics, topology is instrumental in describing the properties of matter and space, while differential geometry is fundamental to the formulation of general relativity. In algebraic geometry, algebraic topology, and mathematical physics, these disciplines offer essential tools for exploring the underlying structures. The integration of topology and differential geometry with other mathematical areas exemplifies the interdisciplinary nature of mathematics, demonstrating their collective impact on advancing our understanding of mathematical and physical phenomena.

Research Approaches

The primary focus of our research section spans a broad spectrum within the fields of geometry and topology. Specifically, within geometry, our exploration extends across various branches of differential geometry, encompassing classical differential geometry, manifold theory, submanifold theory, Riemannian geometry, singularity theory, and geometric analysis. We also investigate the practical applications of these branches in theoretical physics, relativity theory, computer graphics, computer vision, and other scientific disciplines. In the realm of topology, our coverage includes diverse areas such as general topology, algebraic topology, differential topology, and geometric topology. Furthermore, we explore the applications of these topological areas in both pure and applied mathematics, as well as other scientific domains.