Catastrophe theory provides a powerful framework for describing bifurcation phenomena, offering insights into universal and stable structures that emerge across physics. General Relativity is no exception, yet this elegant mathematical tool remains underutilized in gravitational physics.
In this talk, I will introduce the fundamental concepts of catastrophe theory and explore its applications in two distinct relativistic settings. First, we will apply catastrophe theory to gravitational lensing by a rotating star in the wave-optics regime, uncovering novel insights into the diffraction patterns produced by such systems. Second, we will examine the transition from inspiral to plunge in an extreme mass-ratio inspiral through the lens of catastrophe theory, revealing its potential to characterize this critical transition. By showcasing these applications, we aim to highlight the untapped potential of catastrophe theory in gravitational physics and inspire further exploration of its role in strong-field dynamics.