Research Themes in Numerical Analysis, Scientific Computing

The research interests of the Numerical Analysis in the Mathematics Department showcase a broad and complementary range of topics. Below is a summary emphasizing their interconnected themes:

1. Numerical Methods and Computational Mathematics

• Numerical methods (e.g., finite element methods, spectral approximation, and computational methods) are central to solving mathematical models in fundamental sciences and engineering.
• The focus includes initial and boundary value problems for Patial differential Equations and Functional Differential Equations, with applications in physics and engineering.
• Parallel computing and high-performance computation enhance the efficiency of these methods for large-scale problems.

2. Fractional Calculus

• Fractional calculus extends the analytical and numerical study of PDEs and ODEs, providing innovative solutions for modeling complex systems

3. Mathematical Optimization and AI

• Optimization methods form a critical intersection between mathematical theory and applied sciences, addressing problems in various domains such as parameters estimations and model validations.
• Artificial intelligence (AI) and machine learning (ML) leverage advanced optimization and numerical methods to develop models that mimic human reasoning and learning using neural network.
• Genetic algorithms and artificial neural networks (ANNs) demonstrate the fusion of numerical computation, optimization, and AI for innovative solutions.

Current research directions in geometry and topology

1. Geometry of Riemannian (resp. Spacelike) Hypersurfaces in Riemannian (resp. Lorentzian) manifolds. This research is conducted as part of project RSPD2024R1053 under the Researchers Support Program, funded by the Deanship of Scientific Research at the university.

• Riemannian and Spacelike hypersurfaces: Characterization, in terms of the Ricci and scalar curvatures as well as the mean curvature, of Riemannian and spacelike hypersurfaces (especially the compact ones) in Riemannian and Lorentzian manifolds (especially Generalized Robertson-Walker (GRW) spacetimes), respectively. 

2. Geometry of Lie Groups and Lie Algebras

• Geometry of Lie groups: Examination of Lie groups and their homogeneous structures, with a focus on left-invariant pseudo-Riemannian metrics, curvature, and subgroup structures. Specific investigations of semi-simple Lie groups and their applications in physics.
• Geodesic flows: Research into geodesic completeness of left-invariant pseudo-Riemannian (especially Lorentzian) metrics on Lie groups and their relevance to dynamical systems.

3. Ricci Solitons on hypersurfaces of Riemannian and Lorentzian manifolds. This research is conducted as part of project RSPD2024R824 under the Researchers Support Program, funded by the Deanship of Scientific Research at the university.

• Riemannian Ricci solitons: Study of Ricci solitons as solutions to the Ricci flow, focusing on their properties, classification, and rigidity. Ricci solitons on Riemannian hypersurfaces in both Riemannian and Lorentzian manifolds.
• Almost Ricci solitons and Yamabe solitons: Investigation of almost Ricci solitons and Yamabe solitons, focusing on their properties and classification for Riemannian hypersurfaces in both Riemannian and Lorentzian manifolds.

4. Groups with difference sets; Symmetric designs; The friendship theorem; Strongly regular graphs.

5. Nonlinear partial differential equation on a manifold. This research is conducted as part of project RSP-2024/57 under the Researchers Support Program, funded by the Deanship of Scientific Research at the university.

• In the general field of partial differential equation on a noncompact complete Riemannian manifold, considerable efforts have been taken to understanding the blow-up structural mechanism for solutions of evolution equations. For applications, we refer to reaction–diffusion processes of fluid mechanics and turbulence flows, which are often represented by nonlinear equations. This research topic continues to attract the interest of both pure mathematicians and scientists in applied theories. The first challenge is to depict the behavior of solutions and to carry out an asymptotic analysis of solutions near any kinds of singularities. The second one is to establish sufficient conditions for the existence and nonexistence of global or local solutions

   

List of Research Interests of the Faculty Members in Applied Mathematics of the Mathematics Department

1. Partial Differential Equations (PDEs) and Functional Differential Equations (FDEs)

This area of research remains foundational for modeling real-world phenomena, serving as a cornerstone in the mathematical analysis of various disciplines, including physics, engineering, finance, and biology. Spectral methods and fractional calculus further extend its analytical and numerical study.

2. Computational Mathematics
   This branch involves the development and application of numerical algorithms to solve mathematical problems, particularly initial and boundary value problems for different classes of Partial Differential Equations.
3. Mathematical Biology
   This area uses mathematical models to understand and solve problems in biology. It covers a wide range of applications, from modeling population dynamics and ecosystems to understanding the spread of diseases and the genetic basis of evolution.
4. Nanotechnology
   This area of applied mathematics focuses on problems related to designing, producing, and using structures, devices, and systems by manipulating atoms and molecules at the nanoscale.
5. Mathematical Optimization
   Also known as mathematical programming, this area of research focuses on finding the best solution from a set of feasible solutions.
6. Multifractal Analysis
   This field is concerned with local signal regularity and provides a powerful classification tool in various domains. It has been successfully applied in biomedical signal processing, geophysics, finance, and image processing.
7. Mathematical Modeling
   Mathematical modeling involves describing real-world problems using ordinary and partial differential equations, as well as other types of functional differential equations, to understand and uncover new insights into the posed problem.
8. Actuarial and Financial Mathematics
   This area focuses on the development, analysis, and application of mathematical models and techniques to solve problems in finance, insurance, pensions, investment, and risk management.
9. Stochastic Analysis
   Stochastic processes are widely used as mathematical models of systems and phenomena that exhibit random behavior. This includes, in particular, stochastic calculus, Stochastic Differential Equations (SDEs), and Stochastic Partial Differential Equations (SPDEs).