Project 1: Topological Photonic Crystal Design and Optimization
Status: Ongoing Ph.D. Research
This project focuses on the mathematical design, analysis, and optimization of topological photonic crystals. By combining semidefinite programming, finite element methods, spectral theory, and topological analysis, the research seeks to develop photonic structures that exhibit large spectral band gaps, robust wave-guiding capabilities, and desirable topological properties. Applications include optical communications, integrated photonics, sensing technologies, and emerging quantum systems.
Research Components:
Shared spectral band-gap optimization
Valley-Hall photonic systems
Topological invariants (Berry Curvature, Berry Phase, and Valley Chern number)
Bloch wave propagation
Finite element modeling of periodic structures
Computational optimization of photonic materials
Project 2: Numerical Methods for Nonlinear Systems of Equations
Status: Previous Research / Continuing Interest
This project investigates the development and analysis of iterative algorithms for solving nonlinear systems of equations. Particular emphasis is placed on Broyden-like methods, quasi-Newton techniques, convergence analysis, and efficient computational implementations for large-scale scientific applications.
Research topics include:
Broyden-like methods
Quasi-Newton methods
Nonlinear systems
Numerical analysis
Scientific computing
Convergence theory
Project 3: Scientific Computing for Large Scale Mathematical Models
Status: Ongoing Research Interest
This research interest focuses on the development of computational techniques for large-scale mathematical and engineering problems. Areas of interest include high-performance computing, numerical linear algebra, finite element methods, computational electromagnetics, and scientific simulation.
My long-term research goal is to develop rigorous mathematical and computational frameworks for the analysis, optimization, and design of complex physical systems. By integrating optimization, numerical analysis, scientific computing, and topological wave physics, I aim to contribute both to fundamental mathematical understanding and to technological advances in photonics, communications, sensing, and computational engineering.