Overview
My research lies at the intersection of applied mathematics, optimization, numerical analysis, and scientific computing. I am interested in developing mathematical models, computational algorithms, and optimization frameworks that address challenging problems arising in science and engineering. My work combines rigorous mathematical analysis with modern computational techniques to design efficient methods for solving large-scale problems involving wave propagation, nonlinear systems, and complex physical structures.
A central theme of my research is the use of optimization and numerical methods to understand and control the behavior of physical systems. Through the integration of mathematical theory, finite element analysis, and high-performance computing, I seek to develop computational tools that not only advance theoretical understanding but also contribute to practical technological applications.
Current Research: Topological Photonics and Photonic Crystal Optimization
My doctoral research focuses on the mathematical design and optimization of topological photonic crystals. These engineered periodic structures possess unique wave-guiding properties that can enable robust light transport for applications in optical communications, sensing technologies, integrated photonic devices, and emerging quantum systems.
The primary objective of my current work is the development of optimization frameworks for maximizing shared spectral band gaps in topological photonic crystals with distinct topological characteristics. This research combines semidefinite programming, finite element methods, spectral theory, and topological analysis to design structures that simultaneously achieve desirable spectral and topological properties.
A major component of this work involves the study of valley-Hall photonic systems, Berry curvature, topological invariants, and wave propagation in periodic media. By leveraging mathematical optimization techniques, I investigate how geometric and material parameters influence the formation of band gaps and topological edge states. The ultimate goal is to establish computational methodologies capable of designing photonic structures with enhanced robustness and performance.
Previous Research: Nonlinear Systems of Equations
Prior to my doctoral studies, my research focused on nonlinear systems of equations and iterative numerical methods. In particular, I investigated the development and analysis of Broyden-like methods for solving high-dimensional nonlinear systems arising in scientific and engineering applications.
This work involved the construction of robust quasi-Newton algorithms, convergence analysis, and the development of efficient computational techniques for large-scale problems. The resulting research contributed to several peer-reviewed publications and provided practical algorithms for solving nonlinear systems more efficiently and reliably.
My continued interest in nonlinear numerical methods reflects a broader commitment to developing computational approaches that balance mathematical rigor with practical effectiveness.
Research Methods
My research integrates tools and techniques from several areas of applied mathematics, including:
Semidefinite Programming (SDP)
Numerical Optimization
Finite Element Methods (FEM)
Spectral Theory and Eigenvalue Problems
Scientific Computing
High-Performance Computing (HPC)
Mathematical Modeling
Computational Electromagnetics
These methodologies provide a foundation for addressing both theoretical and computational challenges across a variety of scientific disciplines.
Future Research Directions
My long-term research vision is to develop advanced optimization and computational frameworks for the design and analysis of complex physical systems. I am particularly interested in expanding mathematical approaches for topological wave phenomena, photonic materials, and computational engineering applications.
Future research directions include the optimization of topological materials, inverse design of photonic structures, computational methods for wave propagation, and the development of scalable numerical algorithms for large-scale scientific computing. Through these efforts, I aim to contribute to the advancement of mathematical sciences while supporting technological innovation in photonics, communications, energy systems, and related engineering fields.
Research Mission
My research mission is to develop rigorous mathematical and computational methods that address fundamental challenges in science and engineering. By combining optimization, numerical analysis, scientific computing, and mathematical modeling, I seek to create efficient computational frameworks that advance both theoretical understanding and technological innovation. Through interdisciplinary collaboration, I aim to contribute to emerging areas such as photonics, wave engineering, and large-scale computational science.