This research investigates the area law conjecture in quantum physics, which proposes that information shared within quantum systems scales with boundaries rather than total particle number. By developing new mathematical tools for tracking and compressing quantum information, the work aims to simplify the analysis of extremely complex systems in physics, chemistry, and materials science.

This research develops rigorous mathematical foundations for consensus-based optimization algorithms, where large groups of interacting particles collaboratively search for optimal solutions. Using mean-field theory and propagation of chaos, the work proves long-term stability and improves optimization methods for applications including robotics, aircraft design, and drug discovery under real-world constraints.

This research presents a new fractional mathematical model for cardiovascular dynamics that maintains the accuracy of traditional methods while greatly reducing complexity. Using only five interpretable parameters instead of twenty, the model analyzes blood pressure in the frequency domain, providing clearer insight into heart function and offering potential improvements for diagnosis and treatment.